{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-16T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":60749,"title":"Compute the dispersion coefficient","description":"A contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.\r\nG.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with \r\n\r\nwhere  is the width of the stream,  is the transverse mixing coefficient, and  is the deviation of the velocity profile from the cross-sectional average velocity\r\n\r\nWrite a function that takes a (normalized) velocity profile  specified at several points and computes the quantity \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; 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: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 375px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 187.5px; transform-origin: 407px 187.5px; vertical-align: baseline; \"\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eG.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"240\" height=\"44\" alt=\"K = -(1/hD) integral(u' integral(integral(u'(y2),0\u003c=y2\u003c=y1),0\u003c=y1\u003c=y),0\u003c=y\u003c=h)\" style=\"width: 240px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the width of the stream, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eD\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the transverse mixing coefficient, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"64\" height=\"19\" alt=\"u' = u - bar(u)\" style=\"width: 64px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the deviation of the velocity profile from the cross-sectional average velocity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"103\" height=\"44\" alt=\"ubar = (1/h) integral(u(y),0\u003c=y\u003c=h)\" style=\"width: 103px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a (normalized) velocity profile \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADwAAAAkCAYAAADVeVmEAAAAAXNSR0IArs4c6QAABRhJREFUaEPtmXnIplMYxn9D9j0kIRTZsguRpUSWULLvk33ftxCy7/vOH8hOzVgKYw8hZA2RNSRKkizZzm+6Tz0ezzrvMmrmqbf55v3Oc59znfu6r/s655vADPZMmMHwMhPwCDM+F7AR8Dzw6xDmWQpYAngF+LtrvHFleFHgNOAG4P2ui2sZNwuwOzAbcEeK/0eXuOMAvFhayJnAVUMEm7G5/l2BObuCHjVgaXwh8DJwdx/qdclWjHGOc4FHgKfb3hs1YHd/U+A44Oe2xQzw+xViY48GPmuKM0rAiydRuRm4KIRqADytr4rjGGBB4Czgz7o3Rgl4P2AT4LCU4Z9alzz4gNWA64EDgffGDXgB4BbgSeDGwbF0ijAfcC3wJnB5nV6MKsNrAw+GgipY43pOAdYC9k8U/7Fq0lEBPgjYO/rk5+NCC2wHnAPsnDb7g2EDnh2QRj7W6O/x86zA2cDyTTs9ok2QWQ8D+wBTugIWyMrAsullA6wP3Bp9tBhDUbJOnwX2SGLxdfxybuDKUEqV85cWcCsBawJrpF66DrAisFeyoU8V3rPXnhesUQTvralR25O/uwy4vQ/ghdJCTwqp/wbYNn1eLwTIorQjcF302eyPF07ZvTPR640E+vSmFhHx5gXmSBnZMNzY0vHvicBvMWZ+4Kbk1HYBHo0N+aEC0HJpw+5JGzg5WPafIU01rAC4q4/FBN8X3l43ZWRSUmB77cElJc6AX6ybtCbjruXI1EuvAN4KcB8WxurHdVTayLpW1zp3HeAs8VLr/FKmrNEz4ruq7LdO2kDxVYD7Ey2l+Z7BlOJwhdD4tp2qp3XuOsC5FlYHtk9S/1AhurWtL16vJvt9KV1c+DzANcke7hugTi6IobQ3w/cBr9YAzuu2jk1UZ0pvDjxRQa1s4S6NSO50cVF+3Ve0youypXmMLIuh2T8ktKXOl2fAF1ewY+o8VRn2u1Oj/jxnFutFqpldM+9TRbtB21JuLYrZVoBaYMzjgddK6l3eLIXvhbC0XjR0ynBRgVVZaeSNgnSzv/rYbj4CVOm3K+Lapg6dRuOxSJxttywIoj5Zj2znaDp1qeInhMP7uCtggz8QxmGLaOBmXbtme3BB0tgxdRbOGLaHAyJDNSVX+bU+4ILYVOvQjycgu0Jl1iJKZpad4/C6jamitPbMPvZSIUOeehQvW4Z9d5tQabO/Q6qtb4NKGUEWn3eajHzDLqjGt8Wm6smluVdEuS9XvZqZYSI0SpVPGbD/t+X40UUdFfXqZNaQ1FYlV40N+ATYLfp1mWpuhDWuI6syCU1Z190pmj6WzMQooaZ3NoskqPC1lwBlwFo4lVeltPmbZempuXg33JCi4KN99MBtXZnh8qNb83yqyMmYPo9u664EeoMwPTq3pptJW5YXDa7RRNWOraL0TtEWXKDtQRp/F6vV7Vyd+uDWQTeZ8GUDEkvBDMuUPlnOvfyLqOW266GNQySPKKy1E6X7ZKHLWIXk2Biooa+9eikF09zINO1t27Wut6Ky7ZJoW43rGtV5uDipZeLlmhfmz3S4uZSenmk93uU6rgPRN/bY/tTiwqSbAvR4A2gToAguGUe8psv1HFOb+VyHjZy6aePIcM6OfylYBrAuMxBNjqKo2nue1mx4FtdPN7UgY9qGZMNXXWorjxkn4PK6iqcuj4FayE/TLYo+ve3SoA/Gf42dnoBdiNZUsTHrGn4P939NM5oOL05vwB2WONwhMwEPdz//f9H+AX/HFDSie/GkAAAAAElFTkSuQmCC\" width=\"30\" height=\"18\" alt=\"u(y)\" style=\"width: 30px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e specified at several points and computes the quantity \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"215\" height=\"44\" alt=\"I = -integral(u' integral(integral(u'(y2),0\u003c=y2\u003c=y1),0\u003c=y1\u003c=y),0\u003c=y\u003c=h)\" style=\"width: 215px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function I = computeK(y,u)\r\n  I = -integral(u'*integral(integral(u',0,y1),0,y),0,h);\r\nend","test_suite":"%%\r\nny = 1000;\r\ny = linspace(0,1,ny);\r\nu = y.*(1-y);\r\nI = computeK(y,u);\r\nI_correct = 1/7560;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6);\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = 2*y;\r\nu(y\u003e1/2) = 2*(1-y(y\u003e1/2));\r\nI = computeK(y,u);\r\nI_correct = 1/480;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6);\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = sin(pi*y);\r\nI = computeK(y,u);\r\nI_correct = 5/(6*pi^2)-8/pi^4;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = sin(pi*y);\r\nI = computeK(y,u);\r\nI_correct = 5/(6*pi^2)-8/pi^4;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = sin(pi*y);\r\nI = computeK(y,u);\r\nI_correct = 5/(6*pi^2)-8/pi^4;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\na = 2.5; \r\nb = 2.5;\r\ny = linspace(0,1,ny);\r\nu = gamma(a+b)*y.^(a-1).*(1-y).^(b-1)/(gamma(a)*gamma(b));\r\nI = computeK(y,u);\r\nI_correct = 0.00788915;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\na = 2.5; \r\nb = 3;\r\ny = linspace(0,1,ny);\r\nu = gamma(a+b)*y.^(a-1).*(1-y).^(b-1)/(gamma(a)*gamma(b));\r\nI = computeK(y,u);\r\nI_correct = 0.01168232;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\na = 3.2; \r\nb = 3.2;\r\ny = linspace(0,1,ny);\r\nu = gamma(a+b)*y.^(a-1).*(1-y).^(b-1)/(gamma(a)*gamma(b));\r\nI = computeK(y,u);\r\nI_correct = 0.01192484;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-10-16T01:19:16.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-10-16T01:18:30.000Z","updated_at":"2024-10-27T15:58:26.000Z","published_at":"2024-10-16T01:19:16.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eG.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K = -(1/hD) integral(u' integral(integral(u'(y2),0\u0026lt;=y2\u0026lt;=y1),0\u0026lt;=y1\u0026lt;=y),0\u0026lt;=y\u0026lt;=h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK = -\\\\frac{1}{hD}\\\\int_0^h u^\\\\prime \\\\int_0^y \\\\int_0^{y_1} u^\\\\prime(y_2) dy_2 dy_1 dy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the width of the stream, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the transverse mixing coefficient, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u' = u - bar(u)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu^\\\\prime = u-{\\\\bar u}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the deviation of the velocity profile from the cross-sectional average velocity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ubar = (1/h) integral(u(y),0\u0026lt;=y\u0026lt;=h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\bar u} = \\\\frac{1}{h} \\\\int_0^h u(y) dy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a (normalized) velocity profile \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u(y)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu(y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e specified at several points and computes the quantity \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = -integral(u' integral(integral(u'(y2),0\u0026lt;=y2\u0026lt;=y1),0\u0026lt;=y1\u0026lt;=y),0\u0026lt;=y\u0026lt;=h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = -\\\\int_0^h u^\\\\prime \\\\int_0^y \\\\int_0^{y_1} u^\\\\prime(y_2) dy_2 dy_1 dy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":60749,"title":"Compute the dispersion coefficient","description":"A contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.\r\nG.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with \r\n\r\nwhere  is the width of the stream,  is the transverse mixing coefficient, and  is the deviation of the velocity profile from the cross-sectional average velocity\r\n\r\nWrite a function that takes a (normalized) velocity profile  specified at several points and computes the quantity \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; 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: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 375px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 187.5px; transform-origin: 407px 187.5px; vertical-align: baseline; \"\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eG.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"240\" height=\"44\" alt=\"K = -(1/hD) integral(u' integral(integral(u'(y2),0\u003c=y2\u003c=y1),0\u003c=y1\u003c=y),0\u003c=y\u003c=h)\" style=\"width: 240px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the width of the stream, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eD\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the transverse mixing coefficient, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"64\" height=\"19\" alt=\"u' = u - bar(u)\" style=\"width: 64px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the deviation of the velocity profile from the cross-sectional average velocity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAM4AAABYCAYAAAC9IMiaAAAAAXNSR0IArs4c6QAAEBdJREFUeF7tnQXsNkcRxp/iUtytQHEIFIqmECju7k6Q4hDcPcVdC4Wgxd0Kxd2LBxqcBrfgxblfs/Nlerm792Tv7r33ZpN/SPnuvZ2b3WfHZ/dSjOBAcKAzB/bq/Iv4QXAgOKAATmyC4EAPDgRwejAtfhIcCODEHggO9OBAAKcH02b8ySkk7SfpApIOkPRySZ+bkZ7VTh3AWd7Sn0rSiySdV9KtJf1oeZ+wfIoDOMtbw7NKOqwAzVGSHiDp78v7hOVTHMBZ3hperlDTjigA8zBJh0j63/I+YfkUB3CWt4Z3L8DyOEm3kvTJ5ZG/GxQHcJa1jieV9BxJ55d0+8LWOYOkx0q6mqRHJtsnJNAEaxrAmYDJGac4ZwGa10v6kqTDJV1G0tskPV/SzyXdW9KfM84Xr6rhQABnWVvj8gkwh0o6WtJLJe2f/r/nSnqCpP8s65OWSW0AZ3vW7XzFpn+MpF9KQiV7haSvlcjDi/ZsSe9L0uUnkrB5cBLcpIjxvGN7Pme3KQngbMf6nkvSq5L9AihuIOm+kh4i6cuJRIKfxG+wZ+6SpMxJJD1LEpLolslFvR1ftONUBHDmX+ATJBXrQgkQ+0p6T6GGnUXSAxOYoNLsGwKe9yv+fi/JYjrfSzGdv87/OeugIIAz/zpfTNIbJWG34DFD+rwuSZHbpWAnVF49xW8OSqk2eM8umUCGe/qjkk5USJ7vzP9Ju09BAGfeNT5+isncVdL1i7+vJHL2lnTCQor8sXA3/7eQSKzToyTdJsVvvpGeu0P6/f0L79pFk7MASRRjZA4EcEZm8IbXo469RtK/U1zmtzXPm31DntrdJP06PXefQso8WdJbE4DwtMWYgAMBnAmY3DCFqV9PSR61cCXPux6tZw/gtGZV9gdR054k6RGSvC2TfaJ4YX4OBHDy87TtG08v6bWpvsbbN21/H8/NyIEAznzMN48Y5QG3TSkz81ETM3fiwJqBs08KMh5ZVFS+oRPX8jxsEf8XF/llD5J0TJ7Xxlum4MAagQNg7pW8U6edyb6wLGfA44OcU6x5zJGBA2sCzqklES85XhFAPHuSNrBwDsPcIv4HSrpGEez8UIa1jFdMyIE1AYfUFoKJ/F1Q0ptT0HAO4FiWMwmdNysSNC2gOeHSx1RDOLAm4Hg+0SXmTcmjNQdwLMv5wykb4DdDFjF+Oz0HAjjTq2revqGeJhpuTL/vB88YwJkeON6+odyZrIEYC+OAB86JJV1CEjUeXQe5Vl9fUNnunKqadakh/4yGG6iMMRbGAQ+c06UU9mv2+AZAs6RCqjmBY/EbegPgUft8D35v00/IyubgzFHOgMfzypK+muqNtuk7j0NLqGrTqmrUyzw12TWfTY4Byp+XONg7N04Fdi9I4MnxHWdLAeHnSdpa3gRwpgUO7ZzoUkP5M6UAxJWouVnaYN+gYZwjFd8hcXKOM6WGi1sLngDOtMCh2hPA0JiDas+HF/3Q/plzx030LuJQ9ESggO5XI83JHNiA8ChnSTh7/txFYeC/UqegXuQHcKYFDqc0ZdKMpabaIDVRzV6WyrV7bbwWP6Ls4qHF399S37ghjRavK+lKki6VYnffHaomB3CmAw68pjcAf4wbFj3R3t1iA23bI3TYufRE8Sckw0uSzfPtgYzAi0nvuTunEvNB8bMAznTAsfJnWtfSlWaJqTZW6v30ifLrrNgPG2pos8WTScJmwq68RwJPbywGcKYDDvfZoKZRh/Pxhdbg0O+Nk3rK+iHKy6mUHXoXkLXXwn1+7aJr0Gd6oyZ1Txny+6X+do44zhWLGMUnEsOo/Fxan2dzpRMon7J+yA6cJw5Uba2/Q5ZDKyTOdBLHAp9gB7WDvyEG79SHlqUKYZfhEZxq0NmHKxuxcfryzNprIbmyeDOHAIcTCL2dQRR8SW5VRPX7E+2cnjBzzE3sA59Me8fUFmqqzdd2HvbDKSVRgoFd8SfHF3OlP3jgyd+WFnvOeEePhr5S2sCHXZklG74KOBB64eTrRh8nt4oG4OXyYrwrnARZRF9XbvZ4HhXjKsmbdfPCxUn1J4MGfvRt/rSkT0mq623WY8o9PzlNasyBW5RxhaE6dkdiWGeCitQhsbZ4xU6eKmH995rxT4C2bECbqtmGdg5U5iD3ERcwqjFpWfSB8zEZYjXsLeqRiAvVxYRoRs/8NGTcVIKBu5xGjdCJE+Y8Ceg0dGQAHtoID6KvDjgsNFflYQj+otRlksk9grvUzHvbouPaH+dxLlNC7C5l+O+mWyeBve9PSLxJEjYy60Wf6iqVBeP/XUmDKOfR4RDAld7GG4jEAjxILzYsTRTZRzeS9EX33RwkHMibEl5poUUMrCkfkjk5EJkPJwzN6LkflW8mkZags2VrAN5B9DWpahBLl8gPVHSZ5EKjd6bG4F1ce2sFjncMzJlq4z17ZZUFiYybmYbuVVoEwOH2hK7JvL4XNpLjYKf+kdSJ2swtDICrroR809zQTkCZ/UqmAaCxNCBfxlGeHwz3oq8OOD7mUO4yaf2OIaJKGk14kC5mKu8YyGKc9vxy8yxVxZEuIuktDdJo0+atI8nvl6pDGNWKO38e3ZDU2TQ3kgY7leRZ6IfXf3DE+MO6qr9DL/rqgOMnK0e4ieYiXi9bI416runO/qzsGOgioXMyxXuWylKPf0Mt56RmVBnQXVS1Mt1XTZde/aVC7Ud7uUVSsf5R88Ec0uzDKhUXMPA9vLvqQmHr7/DNhjSbzvTVAcdOpnKdTZnBc56eOTfVmO/y9iDzkDM1x23RXosoqyz+MKzTIro4B8r89OqSPzg47bFX4cdHGhYBegEA4P2de+6M6XoUbDPuQSW3rQw+k/ZNZeqd6asCjj+ZyoE6jEqkzX4NJ9OYm3CJ7/Z2xZwFf3UqC6oOdoE5W6rUKfg+xB3tpa7fwNBEjztKyOsyoK1HAyAjG5ukTxtc38jlwYwqNQwnF1476oaaJH1n+qqA409IfzLhvjTmItb75Fut0TnASYmrm8E1heSqeR18qsPAPGZl8CIBcd/Sa47NV3dzgvW6PqJnABRpwYVZ5njgqpLHFxL4YxukjbnySZHx/Rl805O6okCvxm3qz92Jvirg+JoRQzHPkRyHexEGckL18Q5NDRwkJlKyz2CD5ygHZlO+OhEwl2rrb0bwWgTVlhjVby8u7H1actnWZW3bO4h/9ckstl7Z2CK4tFEd8dAR9sBtXDfsxjriPF6d80WBVSlMAA7phou6TRuuTvRVAcdOJo9iTiUYSlo2cQD87yaNEJcEruxU7bNJx/rN3MAplxLM5RjwWoTVAZkGQdzmzCn2YdKIoDAHBxvPq1AY0eSM4TwgiNhl+I1+p2SzUNNjlwPXvQtJwPMcQNhfNvwhXLZfvKeN5+3A4tAnCE4VLo0p/ehEXxk4fqHJCkCnxJ7BVUiqBWkp1gETIP0wZa3iP89ZpddlQbb5WZ/KDp1zOQZ851JoIAiLXUFKDa5ggoYEN9EiSGsBNGxoS0o1HpvNgMqFlOoy7IZsbBoAwGZm7qbLtAD3C9O19Rj/Pi3KG/RcU08mCyowoCGl6eJFxsD+xb2pB6T//mBBLKoh76nSJDrRVwaO1xs5fZA6iEpOym+lU8IkC7UN9GNG1I5VPttlYbbxWb+4c2QMGE+8ncXViaTfAAo2LkFINjHeJ+wNbFf+DSdQVf4eByYp/vfsYatZB1NsPVR/WgA3DVzVJHby7M9KD/r4C/9EwRsBVFQz7Ce0DRwDHPzMhzpI5WqTR7M1fVWqGhMfkojkf1HPLD/IymavE/dOtsKptxfnLCWwdcOmIB8PdYtbqk1dsTVnc9qdonUNOJAC2ENsUtS8LgO16xlJ1WP+psEhDrABsZWbl5/HTgLAVHVySOFAQFN6bzoQyIRA+iBJ+eYfb5izNX1DsqO7MGytz1o8jO9fYilB3bqR9wUASMFBQrUZFgPkNm2fElP1W55FqpFnhplQFxhtM2/bZ7rQd+w14Gsd6MKcnpRElA3FXDzxqTa71rUTLxSOIbxy8HDTIEv6oGQrY1s1ja7v3jR3m3/vQt+qgINHBf0cbwylEhiPiPWxWjR5F3CfmFebxZ77GYzvmyYJgieubmBTkdyJpNnk4gc0eHaf2RKQOXjQhb5j51ubxMEQJoZg/ZqzFDXVrJxPcWkTR8ixAeZ4B62T4avZwewpEkbxUpEfhlTHgYSNh4OpaXDY7FvYLD8dUT0bQt8e2tcGHD68KUM458az5hC4Q+cKfOb8nrbv8lnWuIkJU+Bgwqu4DSMLfWsEjtUZ9cl86LLwFommqnKuwGcXenM9a2UCeF5fmVSuTW7nXHO3eU8W+tYGnKYM4TZM7/KMr6Yc3I6oy8Tx7PgcWBtwNhU15eS4BdOWfitBTp7szLvWBpyqPLwxFtOnb8R1hWNweOZ3rgk43j3sNzM6L3lOJBKy4Qm4kW7SlEO1adl8V5ulNlff9I2r/vc1AcfqSa7ljHXiOcQWSGgkPebAlJ9HG6Ihlxp5lXCuxM5Vb+yxP35NwLG8sb1T3TvxBbK/yQzm2geL8uewSazMOMe7xt4D8f4eHFgTcKzCj9JgKglRoXBNE8n2WeE5kjE31bnDd+66pDjrqFRARjLjoRuKunoscfxkDA6sBTi+ppw0G/LU6N5p9SY+WFnVe6sL7/1cVfEbS2CkNoReYtBAygd219Gp4URTRWQXWuLZkTiwFuD4uhiKqOiGcpirN/FZzFVNH7qw3xwD5HFV1blbR5kvlLqyIBGpKSEliLr+GFvMgbUAx3prEQClkAsVyTJ0fVefHH2wzZb6QUUHVLaCqYzlxuv2u8Nr2hxt8TZaH2lrAY4FI5E25aZ1vh6/rsNLl51hsaKqd/n4TtnbZjXvlFsP9ep1oTee7cGBNQDHG/7YNeWO+TmzCXysqErl8yAtd/0nyxj1EcmzqZVRj6WOn+TkwBqA4w1/atepQ/ejnE3AtRd4vCgFOKYjsy1WhGSpuu7PwEHqfLnrfxOoOpIRj4/NgTUAp6mMwHf1IfWfrAFcyUdWdHhpsxZmS5FGX3XZaxNwfAJqmzto2tATz4zEgV0Hjjf8q8oI/ClP/AZ3MHGdqr5bfgm4gJXLl2gKYTX31nUFSYIdxWVJ5RGq2kgbeerX7jpw6tr5Gp+xf+h+gpThwiPq5333l6r1KIPNrtczNzPdJulRVtUlxsd4wjkw9W7PON+uAycjq/a8yqtUtB8iYEqzD3oX0PABEDb1matzR1vTQNJ0+rSYHeNb4501HAjg9NsaSAt6lCFZiAfRu4C4DJkApNA0DbsBjFJify2Fdd6n+UXXLpn9viJ+1ZsDAZzerNM+RVb19VLzcFoBc4t1mzbAlnKDhAJw2EI0Miflhg6UgKnNe/pTHr8czIEAzmAW9noBXWFoXE8HSkvyJAWHlJs2Pcp6TRo/yseBAE4+XsabVsSBAM6KFjs+NR8HAjj5eBlvWhEHAjgrWuz41HwcCODk42W8aUUcCOCsaLHjU/NxIICTj5fxphVx4P+PHOaGiP/oBAAAAABJRU5ErkJggg==\" width=\"103\" height=\"44\" alt=\"ubar = (1/h) integral(u(y),0\u003c=y\u003c=h)\" style=\"width: 103px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv 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-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a (normalized) velocity profile \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"30\" height=\"18\" alt=\"u(y)\" style=\"width: 30px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan 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: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e specified at several points and computes the quantity \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"215\" height=\"44\" alt=\"I = -integral(u' integral(integral(u'(y2),0\u003c=y2\u003c=y1),0\u003c=y1\u003c=y),0\u003c=y\u003c=h)\" style=\"width: 215px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function I = computeK(y,u)\r\n  I = -integral(u'*integral(integral(u',0,y1),0,y),0,h);\r\nend","test_suite":"%%\r\nny = 1000;\r\ny = linspace(0,1,ny);\r\nu = y.*(1-y);\r\nI = computeK(y,u);\r\nI_correct = 1/7560;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6);\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = 2*y;\r\nu(y\u003e1/2) = 2*(1-y(y\u003e1/2));\r\nI = computeK(y,u);\r\nI_correct = 1/480;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6);\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = sin(pi*y);\r\nI = computeK(y,u);\r\nI_correct = 5/(6*pi^2)-8/pi^4;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = sin(pi*y);\r\nI = computeK(y,u);\r\nI_correct = 5/(6*pi^2)-8/pi^4;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\ny = linspace(0,1,ny);\r\nu = sin(pi*y);\r\nI = computeK(y,u);\r\nI_correct = 5/(6*pi^2)-8/pi^4;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\na = 2.5; \r\nb = 2.5;\r\ny = linspace(0,1,ny);\r\nu = gamma(a+b)*y.^(a-1).*(1-y).^(b-1)/(gamma(a)*gamma(b));\r\nI = computeK(y,u);\r\nI_correct = 0.00788915;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\na = 2.5; \r\nb = 3;\r\ny = linspace(0,1,ny);\r\nu = gamma(a+b)*y.^(a-1).*(1-y).^(b-1)/(gamma(a)*gamma(b));\r\nI = computeK(y,u);\r\nI_correct = 0.01168232;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)\r\n\r\n%%\r\nny = 10000;\r\na = 3.2; \r\nb = 3.2;\r\ny = linspace(0,1,ny);\r\nu = gamma(a+b)*y.^(a-1).*(1-y).^(b-1)/(gamma(a)*gamma(b));\r\nI = computeK(y,u);\r\nI_correct = 0.01192484;\r\nassert(abs(I-I_correct)/I_correct \u003c 1e-6)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-10-16T01:19:16.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-10-16T01:18:30.000Z","updated_at":"2024-10-27T15:58:26.000Z","published_at":"2024-10-16T01:19:16.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eG.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K = -(1/hD) integral(u' integral(integral(u'(y2),0\u0026lt;=y2\u0026lt;=y1),0\u0026lt;=y1\u0026lt;=y),0\u0026lt;=y\u0026lt;=h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK = -\\\\frac{1}{hD}\\\\int_0^h u^\\\\prime \\\\int_0^y \\\\int_0^{y_1} u^\\\\prime(y_2) dy_2 dy_1 dy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the width of the stream, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the transverse mixing coefficient, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u' = u - bar(u)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu^\\\\prime = u-{\\\\bar u}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the deviation of the velocity profile from the cross-sectional average velocity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ubar = (1/h) integral(u(y),0\u0026lt;=y\u0026lt;=h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\bar u} = \\\\frac{1}{h} \\\\int_0^h u(y) dy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a (normalized) velocity profile \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u(y)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu(y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e specified at several points and computes the quantity \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = -integral(u' integral(integral(u'(y2),0\u0026lt;=y2\u0026lt;=y1),0\u0026lt;=y1\u0026lt;=y),0\u0026lt;=y\u0026lt;=h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = -\\\\int_0^h u^\\\\prime \\\\int_0^y \\\\int_0^{y_1} u^\\\\prime(y_2) dy_2 dy_1 dy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"shear 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