Problem 46031. Construct dimensionless parameters

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Summary</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to construct dimensionless parameters from a list of variables, a matrix indicating their dimensions, and a vector identifying the variables to use to make the other variables dimensionless. More details are given in \"Problem statement\" below.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Introduction</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Dimensional analysis and the Buckingham </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:rPr><w:i/></w:rPr><w:t> theorem</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Buckingham_π_theorem\"><w:r><w:t>Buckingham π theorem</w:t></w:r></w:hyperlink><w:r><w:t> is a key result exploited in</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Dimensional_analysis\"><w:r><w:t>dimensional analysis</w:t></w:r></w:hyperlink><w:r><w:t> in engineering, physics, and applied mathematics. It states that if a problem involves</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> variables and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> physical dimensions (e.g., length, time, mass), then the problem can be described with</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m-n\"/></w:customXmlPr><w:r><w:t>m-n</w:t></w:r></w:customXml><w:r><w:t> dimensionless groups, called </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:t> groups. In</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Buckingham_π_theorem\"><w:r><w:t>other words</w:t></w:r></w:hyperlink><w:r><w:t>, the \"physics does not depend on a specific unit system\".</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>To find the dimensionless groups, we follow these steps:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Choose</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> variables that cannot form a dimensionless group by themselves</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Form the product of one of the remaining</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m-n\"/></w:customXmlPr><w:r><w:t>m-n</w:t></w:r></w:customXml><w:r><w:t> variables with the variables in step 1, each raised to different exponents</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Determine the exponents that would make the product (or group) dimensionless</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Repeat steps 2 and 3 for the remaining variables</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Example</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Suppose we are interested in the drag</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"F\"/></w:customXmlPr><w:r><w:t>F</w:t></w:r></w:customXml><w:r><w:t> (i.e., a force) on a cylinder of length</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> and diameter</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t> immersed in a fluid of density</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"rho\"/></w:customXmlPr><w:r><w:t>\\rho</w:t></w:r></w:customXml><w:r><w:t> and dynamic viscosity</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"mu\"/></w:customXmlPr><w:r><w:t>\\mu</w:t></w:r></w:customXml><w:r><w:t> flowing at velocity</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>. Because the problem involves 6 variables and 3 dimensions (length, time, and mass), it can be described with 3 dimensionless groups. For step 1 above, choose</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"rho\"/></w:customXmlPr><w:r><w:t>\\rho</w:t></w:r></w:customXml><w:r><w:t>. We could choose other lists of 3, but notice that we could not choose</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> because</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> form a dimensionless group themselves (and we have no way of removing mass dimensions.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For step 2, start with the drag</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>F</w:t></w:r><w:r><w:t>. Following the step, we get</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi_1 = F V^{a1} d^{a2} rho^{a3}\"/></w:customXmlPr><w:r><w:t>\\pi_1 = F V^{a_1} d^{a_2} \\rho^{a3}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a1</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a2</w:t></w:r><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a3</w:t></w:r><w:r><w:t> are exponents. Because </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi1\"/></w:customXmlPr><w:r><w:t>\\pi_1</w:t></w:r></w:customXml><w:r><w:t> must be dimensionless and because the dimensionless of</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"F\"/></w:customXmlPr><w:r><w:t>F</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"rho\"/></w:customXmlPr><w:r><w:t>\\rho</w:t></w:r></w:customXml><w:r><w:t> are </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"ML/T^2\"/></w:customXmlPr><w:r><w:t>\\rm ML/T^2</w:t></w:r></w:customXml><w:r><w:t>, L/T, L, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"M/L^3\"/></w:customXmlPr><w:r><w:t>\\rm M/L^3</w:t></w:r></w:customXml><w:r><w:t>, respectively, we can solve a system of three equations to find the exponents and write</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi1 = F V^{-2} d^{-2} rho^{-1}\"/></w:customXmlPr><w:r><w:t>\\pi_1 = F V^{-2}d^{-2}\\rho^{-1}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>which is related to a quantity in fluid mechanics called the drag coefficient. We then repeat steps 2 and 3 with the two remaining variables and find</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi2 = L/d\"/></w:customXmlPr><w:r><w:t>\\pi_2 = L d^{-1}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>and</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi3 = mu/(V d rho)\"/></w:customXmlPr><w:r><w:t>\\pi_3 = \\mu V^{-1} d^{-1} \\rho^{-1}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>or a geometric ratio and the inverse of the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Reynolds_number\"><w:r><w:t>Reynolds number</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Problem statement</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that constructs dimensionless parameters from a cell array</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>vars</w:t></w:r><w:r><w:t> with the variable names, a matrix</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dims</w:t></w:r><w:r><w:t> with the exponents for the dimensions in rows (length, time, mass) for each variable in the columns, and a vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>indx</w:t></w:r><w:r><w:t> indicating the variables chosen in step 1 above. The function should return a cell array</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Pi</w:t></w:r><w:r><w:t> with the dimensionless groups in strings and a vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> of size</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>m x m-n</w:t></w:r><w:r><w:t> with the exponents of the normalizing variables. For the example above, the input could be specified as</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ vars = {'F', 'V', 'd', 'L', 'rho', 'mu'};\n indx = [2 3 5];\n dims = [ 1 1 1 1 -3 -1;\n -2 -1 0 0 0 -1;\n 1 0 0 0 1 1];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>and the output would be</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ Pi = {'F V^{-2} d^{-2} rho^{-1}', 'L d^{-1}', 'mu V^{-1} d^{-1} rho^{-1}'} \n a = [-2 0 -1;\n -2 -1 -1;\n -1 0 -1]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Further specifications:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the variables chosen in</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>indx</w:t></w:r><w:r><w:t> will not work (i.e., the number is incorrect or some form a dimensionless group themselves), return</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a = NaN</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Pi = {'Error in choosing normalizing variables'}</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent of a variable is zero, omit the variable from the string.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent of a variable is one, keep the variable but do not show the exponent.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent of a variable is a positive integer, indicate it with a caret (^) and no braces.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent is anything else, indicate it with a caret and braces.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent is not an integer, express it as a fraction. Hint: MATLAB has a function that will help.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 1429.07px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 714.533px; transform-origin: 407px 714.533px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.9px 7.8px; transform-origin: 31.9px 7.8px; unicode-bidi: normal; ">Summary</div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.783px 7.8px; transform-origin: 376.783px 7.8px; unicode-bidi: normal; ">Write a function to construct dimensionless parameters from a list of variables, a matrix indicating their dimensions, and a vector identifying the variables to use to make the other variables dimensionless. More details are given in "Problem statement" below.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8167px 7.8px; transform-origin: 40.8167px 7.8px; unicode-bidi: normal; ">Introduction</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 133.45px 7.8px; transform-origin: 133.45px 7.8px; unicode-bidi: normal; ">Dimensional analysis and the Buckingham π<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.6167px 7.8px; transform-origin: 27.6167px 7.8px; unicode-bidi: normal; "> theorem</div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; ">The<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem">Buckingham π theorem</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.0833px 7.8px; transform-origin: 82.0833px 7.8px; unicode-bidi: normal; "> is a key result exploited in<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Dimensional_analysis">dimensional analysis</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115.167px 7.8px; transform-origin: 115.167px 7.8px; unicode-bidi: normal; "> in engineering, physics, and applied mathematics. It states that if a problem involves<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> m<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5833px 7.8px; transform-origin: 43.5833px 7.8px; unicode-bidi: normal; "> variables and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.633px 7.8px; transform-origin: 174.633px 7.8px; unicode-bidi: normal; "> physical dimensions (e.g., length, time, mass), then the problem can be described with<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="m-n" style="width: 38.5px; height: 18px;" width="38.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.7667px 7.8px; transform-origin: 93.7667px 7.8px; unicode-bidi: normal; "> dimensionless groups, called π<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 33.0667px 7.8px; transform-origin: 33.0667px 7.8px; unicode-bidi: normal; "> groups. In<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem">other words</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.65px 7.8px; transform-origin: 96.65px 7.8px; unicode-bidi: normal; ">, the "physics does not depend on a specific unit system".</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.267px 7.8px; transform-origin: 174.267px 7.8px; unicode-bidi: normal; ">To find the dimensionless groups, we follow these steps:</div><ol style="block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 51.0833px; transform-origin: 391px 51.0833px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">Choose n variables that cannot form a dimensionless group by themselves</li><li style="block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4333px; text-align: left; transform-origin: 363px 20.4333px; white-space: pre-wrap; margin-left: 56px; ">Form the product of one of the remaining <img src="data:image/png;base64,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" alt="m-n" style="width: 38.5px; height: 18px;" width="38.5" height="18"> variables with the variables in step 1, each raised to different exponents</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">Determine the exponents that would make the product (or group) dimensionless</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">Repeat steps 2 and 3 for the remaining variables</li></ol><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; ">Example</div><div style="block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 118.267px 7.8px; transform-origin: 118.267px 7.8px; unicode-bidi: normal; ">Suppose we are interested in the drag<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> F<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.633px 7.8px; transform-origin: 111.633px 7.8px; unicode-bidi: normal; "> (i.e., a force) on a cylinder of length<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42.8px 7.8px; transform-origin: 42.8px 7.8px; unicode-bidi: normal; "> and diameter<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.4px 7.8px; transform-origin: 70.4px 7.8px; unicode-bidi: normal; "> immersed in a fluid of density<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> ρ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.0167px 7.8px; transform-origin: 70.0167px 7.8px; unicode-bidi: normal; "> and dynamic viscosity<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> μ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 56.8px 7.8px; transform-origin: 56.8px 7.8px; unicode-bidi: normal; "> flowing at velocity<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.783px 7.8px; transform-origin: 191.783px 7.8px; unicode-bidi: normal; ">. Because the problem involves 6 variables and 3 dimensions (length, time, and mass), it can be described with 3 dimensionless groups. For step 1 above, choose<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> ρ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.2px 7.8px; transform-origin: 16.2px 7.8px; unicode-bidi: normal; ">. We could choose other lists of 3, but notice that we could not choose<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.4px 7.8px; transform-origin: 28.4px 7.8px; unicode-bidi: normal; "> because<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; "> and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.0167px 7.8px; transform-origin: 70.0167px 7.8px; unicode-bidi: normal; "> form a dimensionless group themselves (and we have no way of removing mass dimensions.)</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 7.8px; transform-origin: 91px 7.8px; unicode-bidi: normal; ">For step 2, start with the drag<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.28333px 7.8px; transform-origin: 4.28333px 7.8px; unicode-bidi: normal; ">F<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.8px 7.8px; transform-origin: 84.8px 7.8px; unicode-bidi: normal; ">. Following the step, we get</div><div style="block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi_1 = F V^{a1} d^{a2} rho^{a3}" style="width: 102px; height: 26px;" width="102" height="26"></div><div style="block-size: 42.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.4167px; text-align: left; transform-origin: 384px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.0667px 7.8px; transform-origin: 19.0667px 7.8px; unicode-bidi: normal; ">where<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 7.8px; transform-origin: 7.78333px 7.8px; unicode-bidi: normal; ">a1<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 7.8px; transform-origin: 7.78333px 7.8px; unicode-bidi: normal; ">a2<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 7.8px; transform-origin: 7.78333px 7.8px; unicode-bidi: normal; ">a3<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.35px 7.8px; transform-origin: 79.35px 7.8px; unicode-bidi: normal; "> are exponents. Because <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABCUlEQVRYhe2WUQ2EMAyGPw84wAAGUHAKcDAHOMACGpBwHrCABixwD7ShtwMCt+Verl/SBDq2f/SnC+A4juM4P6YEGqC9EFUu0QLogeVGlLmER2ACnmbxp8Qo95PJ9TmEkcWCXFdGWAmSa3MJKgXv3mnpg8lNksvm8dFGYj+1EtOF+TXw+FZcy2tL3kmuO5lXAsOF507R8lpvZ8nVB3Nq1k0niddsJVdvH3I/s1pyRpsiri02mtywk+vY7++vxK1fC+vppqgN+rGFaCNJ4rbU8RsSjWkceX9bvDGLznz2sa1I3PvJ4rB+VIF9HwsZaw/Gk8Vz4eL/J66H1PBL0ZKtIzQaMv3lOI5zmxdP4XNErSZ5SAAAAABJRU5ErkJggg==" alt="pi1" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.9px 7.8px; transform-origin: 180.9px 7.8px; unicode-bidi: normal; "> must be dimensionless and because the dimensionless of<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> F<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> ρ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 7.8px; transform-origin: 14px 7.8px; unicode-bidi: normal; "> are <img src="data:image/png;base64,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" alt="ML/T^2" style="width: 47px; height: 19.5px;" width="47" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.5px 7.8px; transform-origin: 38.5px 7.8px; unicode-bidi: normal; ">, L/T, L, and <img src="data:image/png;base64,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" alt="M/L^3" style="width: 38px; height: 19.5px;" width="38" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 269.8px 7.8px; transform-origin: 269.8px 7.8px; unicode-bidi: normal; ">, respectively, we can solve a system of three equations to find the exponents and write</div><div style="block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi1 = F V^{-2} d^{-2} rho^{-1}" style="width: 107.5px; height: 21px;" width="107.5" height="21"></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 363.433px 7.8px; transform-origin: 363.433px 7.8px; unicode-bidi: normal; ">which is related to a quantity in fluid mechanics called the drag coefficient. We then repeat steps 2 and 3 with the two remaining variables and find</div><div style="block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi2 = L/d" style="width: 64px; height: 21px;" width="64" height="21"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.6833px 7.8px; transform-origin: 11.6833px 7.8px; unicode-bidi: normal; ">and</div><div style="block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi3 = mu/(V d rho)" style="width: 105.5px; height: 21px;" width="105.5" height="21"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.517px 7.8px; transform-origin: 129.517px 7.8px; unicode-bidi: normal; ">or a geometric ratio and the inverse of the<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Reynolds_number">Reynolds number</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0167px 7.8px; transform-origin: 63.0167px 7.8px; unicode-bidi: normal; ">Problem statement</div><div style="block-size: 106.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 53.4px; text-align: left; transform-origin: 384px 53.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 231.7px 7.8px; transform-origin: 231.7px 7.8px; unicode-bidi: normal; ">Write a function that constructs dimensionless parameters from a cell array<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.8px; transform-origin: 15.4px 7.8px; unicode-bidi: normal; ">vars<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104.25px 7.8px; transform-origin: 104.25px 7.8px; unicode-bidi: normal; "> with the variable names, a matrix<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.8px; transform-origin: 15.4px 7.8px; unicode-bidi: normal; ">dims<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> with the exponents for the dimensions in rows (length, time, mass) for each variable in the columns, and a vector<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.8px; transform-origin: 15.4px 7.8px; unicode-bidi: normal; ">indx<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> indicating the variables chosen in step 1 above. The function should return a cell array<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.8px; transform-origin: 7.7px 7.8px; unicode-bidi: normal; ">Pi<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105.033px 7.8px; transform-origin: 105.033px 7.8px; unicode-bidi: normal; "> with the dimensionless groups in strings and a vector<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.8px; transform-origin: 3.85px 7.8px; unicode-bidi: normal; ">a<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.1667px 7.8px; transform-origin: 22.1667px 7.8px; unicode-bidi: normal; "> of size<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.95px 7.8px; transform-origin: 26.95px 7.8px; unicode-bidi: normal; ">m x m-n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 255.567px 7.8px; transform-origin: 255.567px 7.8px; unicode-bidi: normal; "> with the exponents of the normalizing variables. For the example above, the input could be specified as</div><div style="background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 51.0833px; transform-origin: 404px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 161.7px 8.25px; transform-origin: 161.7px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> vars = {<span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'F', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'V', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'d', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'L', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 19.25px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 19.25px 8.25px; ">'rho', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 15.4px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 15.4px 8.25px; ">'mu'};</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 61.6px 8.25px; transform-origin: 61.6px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> indx = [2 3 5];</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 111.65px 8.25px; transform-origin: 111.65px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> dims = [ 1 1 1 1 -3 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 111.65px 8.25px; transform-origin: 111.65px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> -2 -1 0 0 0 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 115.5px 8.25px; transform-origin: 115.5px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> 1 0 0 0 1 1];</div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.7px 7.8px; transform-origin: 74.7px 7.8px; unicode-bidi: normal; ">and the output would be</div><div style="background-color: rgb(247, 247, 247); block-size: 81.7333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40.8667px; transform-origin: 404px 40.8667px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 292.6px 8.25px; transform-origin: 292.6px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> Pi = {<span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 100.1px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 100.1px 8.25px; ">'F V^{-2} d^{-2} rho^{-1}', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 38.5px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 38.5px 8.25px; ">'L d^{-1}', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 103.95px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 103.95px 8.25px; ">'mu V^{-1} d^{-1} rho^{-1}'} </div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 8.25px; transform-origin: 69.3px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> a = [-2 0 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 8.25px; transform-origin: 69.3px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> -2 -1 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 8.25px; transform-origin: 69.3px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> -1 0 -1]</div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.4667px 7.8px; transform-origin: 68.4667px 7.8px; unicode-bidi: normal; ">Further specifications:</div><ul style="block-size: 143.933px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.9667px; transform-origin: 391px 71.9667px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 41.7667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.8833px; text-align: left; transform-origin: 363px 20.8833px; white-space: pre-wrap; margin-left: 56px; ">If the variables chosen in indx will not work (i.e., the number is incorrect or some form a dimensionless group themselves), return a = NaN and Pi = {'Error in choosing normalizing variables'}.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent of a variable is zero, omit the variable from the string.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent of a variable is one, keep the variable but do not show the exponent.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent of a variable is a positive integer, indicate it with a caret (^) and no braces.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent is anything else, indicate it with a caret and braces.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent is not an integer, express it as a fraction. Hint: MATLAB has a function that will help.</li></ul></div></div>

*Summary*

Write a function to construct dimensionless parameters from a list of variables, a matrix indicating their dimensions, and a vector identifying the variables to use to make the other variables dimensionless. More details are given in "Problem statement" below.

*Introduction*

_Dimensional analysis and the Buckingham π theorem_

The <https://en.wikipedia.org/wiki/Buckingham_π_theorem Buckingham π theorem> is a key result exploited in <https://en.wikipedia.org/wiki/Dimensional_analysis dimensional analysis> in engineering, physics, and applied mathematics. It states that if a problem involves _m_ variables and _n_ physical dimensions (e.g., length, time, mass), then the problem can be described with _m-n_ dimensionless groups, called π groups. In <https://en.wikipedia.org/wiki/Buckingham_π_theorem other words>, the "physics does not depend on a specific unit system".

To find the dimensionless groups, we follow these steps:

# Choose _n_ variables that cannot form a dimensionless group by themselves
# Form the product of one of the remaining _m-n_ variables with the variables in step 1, each raised to different exponents
# Determine the exponents that would make the product (or group) dimensionless
# Repeat steps 2 and 3 for the remaining variables

_Example_

Suppose we are interested in the drag _F_ (i.e., a force) on a cylinder of length _L_ and diameter _d_ immersed in a fluid of density _ρ_ and dynamic viscosity _μ_ flowing at velocity _V_. Because the problem involves 6 variables and 3 dimensions (length, time, and mass), it can be described with 3 dimensionless groups. For step 1 above, choose _V_, _d_, and _ρ_. We could choose other lists of 3, but notice that we could not choose _V_, _d_, and _L_ because _d_ and _L_ form a dimensionless group themselves (and we have no way of removing mass dimensions.)

For step 2, start with the drag _F_. Following the step, we get

π_1 = F V^a1 d^a2 ρ^a3

where _a1_, _a2_, and _a3_ are exponents. Because π_1 must be dimensionless and because the dimensionless of _F_, _V_, _d_, and _ρ_ are ML/T^2, L/T, L, and M/L^3, respectively, we can solve a system of three equations to find the exponents and write

π_1 = F V^{-2} d^{-2} ρ^{-1}

which is related to a quantity in fluid mechanics called the drag coefficient. We then repeat steps 2 and 3 with the two remaining variables and find

π_2 = L d^{-1}   and   π_3 = μ V^{-1} d^{-1} ρ^{-1}

or a geometric ratio and the inverse of the <https://en.wikipedia.org/wiki/Reynolds_number Reynolds number>.

*Problem statement*

vars = {'F', 'V', 'd', 'L', 'rho', 'mu'};
 indx = [2 3 5];
 dims = [ 1  1  1  1  -3  -1;
         -2 -1  0  0   0  -1;
          1  0  0  0   1   1];
  
and the output would be
  
 Pi = {'F V^{-2} d^{-2} rho^{-1}', 'L d^{-1}', 'mu V^{-1} d^{-1} rho^{-1}'} 
 a  = [-2   0  -1;
       -2  -1  -1;
       -1   0  -1]

Further specifications:

* If the variables chosen in |indx| will not work (i.e., the number is incorrect or some form a dimensionless group themselves), return |a = NaN| and |Pi = {'Error in choosing normalizing variables'}|.
* If the exponent of a variable is zero, omit the variable from the string.
* If the exponent of a variable is one, keep the variable but do not show the exponent.
* If the exponent of a variable is a positive integer, indicate it with a caret (^) and no braces.
* If the exponent is anything else, indicate it with a caret and braces. 
* If the exponent is not an integer, express it as a fraction. Hint: MATLAB has a function that will help.

Solve

Solution Stats

5 Solutions
4 Solvers

Last Solution submitted on Dec 23, 2020

Last 200 Solutions

Problem Comments

Problem Recent Solvers4

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 46031. Construct dimensionless parameters

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers4

Suggested Problems

More from this Author190

Problem Tags

Community Treasure Hunt

Problem 46031. Construct dimensionless parameters

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers4

Suggested Problems

More from this Author190

Problem Tags

Community Treasure Hunt

Players