Problem 51087. Solve an ODE: equation for a 2D laminar jet

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>Problem</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>In the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f’’’ + 2(f’2+ff”) = 0\"/></w:customXmlPr><w:r><w:t>f\\prime\\prime\\prime + 2(f\\prime^2+ff\\prime\\prime) = 0</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 primes denote differentiation with respect to the independent variable </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=\"eta\"/></w:customXmlPr><w:r><w:t>\\eta</w:t></w:r></w:customXml><w:r><w:t>. The problem has the conditions </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(0) = f''(0) = 0\"/></w:customXmlPr><w:r><w:t>f(0) = f\\prime\\prime(0) = 0</w:t></w:r></w:customXml><w:r><w:t> 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=\"f'(oo) = 0\"/></w:customXmlPr><w:r><w:t>f\\prime(\\infty ) = 0</w:t></w:r></w:customXml><w:r><w:t> and the constraint </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"integral of f'^2 from -infinity to infinity = a\"/></w:customXmlPr><w:r><w:t>\\int_{-\\infty}^\\infty\\left[f\\prime(\\eta)\\right]^2d\\eta=a</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:t> </w:t></w:r><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:t>where </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=\"a\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> is a constant.</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 solve this problem—that is, return values of </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> at specified values of </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=\"eta\"/></w:customXmlPr><w:r><w:t>\\eta</w:t></w:r></w:customXml><w:r><w:t>. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:</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>char('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)</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>Background</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 physical problem involves a jet in the </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=\"x, y\"/></w:customXmlPr><w:r><w:t>x, y</w:t></w:r></w:customXml><w:r><w:t> plane emanating from a source at </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=\"(x,y) = (0,0)\"/></w:customXmlPr><w:r><w:t>(x,y) = (0,0)</w:t></w:r></w:customXml><w:r><w:t>. It can be solved by expressing the velocities </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=\"u\"/></w:customXmlPr><w:r><w:t>u</w:t></w:r></w:customXml><w:r><w:t> 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=\"v\"/></w:customXmlPr><w:r><w:t>v</w:t></w:r></w:customXml><w:r><w:t> in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable </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=\"eta\"/></w:customXmlPr><w:r><w:t>\\eta</w:t></w:r></w:customXml><w:r><w:t>. In the problem above, </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(eta)\"/></w:customXmlPr><w:r><w:t>f(\\eta)</w:t></w:r></w:customXml><w:r><w:t> is proportional to the streamfunction. The conditions at </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=\"eta = 0\"/></w:customXmlPr><w:r><w:t>\\eta = 0</w:t></w:r></w:customXml><w:r><w:t> define the line </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=\"y = 0\"/></w:customXmlPr><w:r><w:t>y = 0</w:t></w:r></w:customXml><w:r><w:t> as a line of symmetry; the transverse velocity </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> and the shear stress are zero there. The condition </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'(oo) = 0\"/></w:customXmlPr><w:r><w:t>f\\prime(\\infty ) = 0</w:t></w:r></w:customXml><w:r><w:t> states that the streamwise velocity </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=\"u\"/></w:customXmlPr><w:r><w:t>u</w:t></w:r></w:customXml><w:r><w:t> is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value </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=\"a = 4/3\"/></w:customXmlPr><w:r><w:t>a = 4/3</w:t></w:r></w:customXml><w:r><w:t> corresponds to the physical problem of the jet. Other values are included for variety.</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: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; 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: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; ">Problem</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: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; ">In the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises</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; "><img src="data:image/png;base64,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" alt="f’’’ + 2(f’2+ff”) = 0" style="width: 147.5px; height: 20px;" width="147.5" height="20"></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: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; ">where primes denote differentiation with respect to the independent variable η<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: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; ">. The problem has the conditions <img src="data:image/png;base64,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" alt="f(0) = f''(0) = 0" style="width: 115px; height: 19px;" width="115" height="19"><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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" alt="f'(oo) = 0" style="width: 69px; height: 19px;" width="69" height="19"><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: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; "> and the constraint </div><div 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: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="integral of f'^2 from -infinity to infinity = a" style="width: 119.5px; height: 44px;" width="119.5" height="44"><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: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; 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: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; ">where 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: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; "> is a constant.</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: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; ">Write a function to solve this problem—that is, return values of 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: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; "> at specified values of η<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: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; ">. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:</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: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; ">char('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)</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: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; ">Background</div><div style="block-size: 147px; 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 73.5px; text-align: left; transform-origin: 384px 73.5px; 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: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; ">The physical problem involves a jet in the <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAkCAYAAAAgh9I0AAABi0lEQVRYhe2WUZGEMAyGPw84wAAGUHAK6gAHOMBCNSABD1hAQy1wD22GwtGDtrD70m9mZ3ebIW3S5CdQKBQKhUKhUCgcqAEF9O730dY7ey4N0AJdwF/l9upinP4AIzABq/uMnl156yt/A4yhBwbAOF/mxF/v7RUViCCBGGxGWmDBBqrZB5dDy3bQ/mCrgDlgu0XnOZcA2tSTXiAHXQJ2OUM0DVsQhmd6IISfsLMeNDnOpV5DGXqKmnDtS+8kMzrHc46Tmyxur8lbq9x6soD4TZWrRHfQbLdeubXsW9Dsg3izJ2Av3w02aVm3oNgyIn2hs4/5P35fyPsqSVbBZmFx37D1hTS3lFnt/dfYWh7YSiEF6YuZSDGp3EFHrDIYbCYEX/6U20AF7Ct5NSwJi34v+G/MM4mrL+zHUSRH02XMSCpdGfZC0bcXdnl+Ik+Oe/bq9BUG0stJeu2t0eYWIgipWZxInFSfQgJI1XTN+xJ+ieL+DchAZ7A98ORY/zH86ThXkr+KqN3b81ihcIdfNuyHpTGjL0oAAAAASUVORK5CYII=" alt="x, y" style="width: 24.5px; height: 18px;" width="24.5" 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: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; "> plane emanating from a source at <img src="data:image/png;base64,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" alt="(x,y) = (0,0)" style="width: 87.5px; height: 19px;" width="87.5" height="19"><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: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; ">. It can be solved by expressing the velocities u<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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "> and 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: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; "> in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable η<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: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; ">. In the problem above, <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAAmCAYAAAB0xJ2ZAAACmElEQVRoge1ZbY3rMBAcDmFQAiEQBEVQBmFQBqEQDIEQDqUQDKHQ+2HPy16U2rvu+qQneaTTqVXW2Y/ZLxdoaGhoaHDBDcBglOkAzPG/B+aox5+iB7ABWKMCO4BRIddFmd5RlxuAl/OZSQwA3gCW+HmMn3eko9ohKGpljAYMSHUmdAiG7uJldwQHrBnZBcBUTzU8EBxcFU/8jr4WD+QZ4oEVQcdq2BEc8DDKbaisWATZWCUVHvFw6wsoVzv6xIZKqbYgGLIZ5V7I1wdPTAhMdcGIQN0nDvpv4rsR6cjeooyW/v2HMzsEJj0RaJ4C08Cl29DQGQf9F/F9rhaQ/hpl5DskY3oczudfyglWp6vA6m/1LOVyQ8ocn33F59nO2N/vOCL7Rj7HSzpVEsx/aytbo1wKA4IDgGAsDaTxg3iODsgxb4dz3SEFrYdqHECQujTwPDVKFua60ArHQigVs7YXiwM4UtPR592CLNRMe5b3ZiFzL1eBv1FEptkV08hCTRBcHTDhcIB1mLEoIqv8uWjK/NcUYW6qLqARJYsGnZdznDTwqnoz/7VFuKReJQ9746jUFjCvc1GTLLtqmQyCxqgOjnNALxSzLkBSPndZwv5/NWbTIM05wMEma726hKzMpRvWhjR7pIFXUZNFWHPro007FTieWhegs0Kpniy3zKtUoUFaHV4oS9ePh5X0fwnOEZ9SaMCxW1yB06HGKKacyyIkqfntheOEsqp8ng5zmOG4A5CaHvdsHUIkrYXUkv8Dft9VmsC7elllOZl53eKW3NzK+T9348ytsQjnFzGX3IpJxB0hFbQVWjuELShr0//ASwfSNNe6vn2X1gkaB0z40njihpACI+r/wKBlADtEagD685/FGhoaGv5r/AAD9vXFC0db4wAAAABJRU5ErkJggg==" alt="f(eta)" style="width: 32px; height: 19px;" width="32" height="19"><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: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; "> is proportional to the streamfunction. The conditions at <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAkCAYAAAAq23xmAAAB10lEQVRoge2ZXbHCMBBGj4c6wEANVAEK6gAHdYAFNCABD1hAQy2Uh2SnO7khSWdKSOfumclTtyH9sn8JYBiGYRjGcRiAyY8R6H67nHY4Ay/giRNpAO7ADFx+uK4mGIEFJ07oMQ//7Fp7Ua1wwnnJgvOi2PPFj77iuprhhvv4mc/55snqYf+KjtU7Hgm7q7I7VVhXM0juyeUYbVecsHtvHLpl5yeciMd0S0yUffig7O4lE0vchq7ZsyY8GVtEmnYY44bfkwq1RaBXbtKbX0iYuHr/8tmPEtcNWXYYRTvs0QKlhO2V3ZyacMAJBE4MEUDEGZRdyQ+H7OFBWzxWC5Qr4UUCCbo3GFm7T0HHdstZXwuUElZ/b5FAF/XCg7/xe+cYfYOsc9ccpCeeifcPkqhbb8+/VsV0UgxjV082sI3aVUwXk2kHOyCvpuxKqnX/RO0qpjvp1Hva07KbrtvuWOaXxJdq3VMLqVnF4AtnMTGOJSu9I0e5Q9lyms96jxYgFos6Vo90NZC6D5KCVFRw9KEtpqaEX1EpbIyRtSqfWW8UixKzoO9rY0h3ffvwvHU6nDg6n+12Jx1210bAUfNPNXSvYH+VRJD+p/Xz188wgTJIhTtKg2gYRvu8AaTzBtP2k26rAAAAAElFTkSuQmCC" alt="eta = 0" style="width: 36px; height: 18px;" width="36" 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: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; "> define the line <img src="data:image/png;base64,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" alt="y = 0" style="width: 37px; height: 18px;" width="37" 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: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; "> as a line of symmetry; the transverse velocity 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: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; "> and the shear stress are zero there. The condition <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIoAAAAmCAYAAAAbWQPxAAAD3klEQVR4nO2aa7GrMBSFlwccYKAGquAowAEOcFAL1VAJeMACGmqB+wPWsMvNYwcSCDP5Zpg5PU0hj521HwQoFAqFQqFQKBQKqAE8le3qgPs+ALx29SgeHXRjKzh4ABgB9ADeAL4AWkfbCcAQcO8eQHWwj0epAHwANBf3IxYVgD/MG6Bb/k7KE/PCf5bP7fL5C/Pivpbv34p70wCvNhJSYTbw5JOamA7z+rwxr98f5nElG1u1PPCL1ZX8YTaE3vKbcfn+obj3iPwWpcbveO/GG+aNyrWckGDOO/yqiQ+6HU37N+zGdjUd8u2bC27iCWaVbpbvoqs4LVDrt+mWfGpSL+1yDR4rhI07F6jmNiPnuCZETB5ofRP0MjxArybaYPcq7tBHiVSTztFuwKoqUfgE3pDW6pM0tnMNJgc48T51zAXGJr4YRLbbPbYWazpFtzOK/7WwG0IDnZxRqULdTrX8ln3x/f5oengXgyY9dAbQiXa2EocXTqq0uo/4v89na1wU7x3ar8lw9TBPSoPV0OU1WNrbYO1oD12EKyRGkuPUGoqmhOFE3ix2wNljXkQtNKzX0pcWa9BGxZMTI2Mr07VtH7OvElcftJc229w+z7VhW+y7vxHGJ7ai2hFcUfkWxglbiWQVdbv4TM9ZOeaEPTbttRPE3+whhqKEuEuOzWfYT0Q0FMp2ilpCyH17uINpRvA0FiqAbUdRnbQFNSrrHQJaaSiusUnFPWQorHFEzbUFIYbiG4ysNmr8s2yvcak0lFzrPRI5D6fEKDIfT1Fe1xrK9h2TDdnfEX6loDvRRPx3MpRTsx5gfamnqYnsIdRQNHUcGX/4dgknSpNR7MnQ5HPOzHrkurkMW9vOCy0zVVWSbzF9SBfoG5DcTT4lDIk7eN89nJ31SGV1KQXHdDhRiZZjWwjJJOh3B9gHxV1vyoJsz9duggHX1lFCXf9p73qYXmqleQ/aF4fAb+GvxzpxrNIy6+HO8xkL3Zl2bKkC+lRQVWyZT4NIaiKLManOY9ClaBaLZ0Nc8ixPx/HgkUwVWe6Xh3k0cNJzOy/jw3UehYpzeEx8SLQ3ixa0b5mB9RScyUhe+H9nVPgN2OTVGdrbeCH9PKRCnjKUJ9y+iGT43I2p5ZYSGKJaVAX6bt9va6wvOTXtt1CN7kqNsJeoamSgc0YlckC+C9Egr7O8WcFdftZhHQbOuZ1NZfX2brFJEirM/kvm28wYzqxCMnPJiQ/ulekkRZZxK6y7O1XtxNeXK55rokU+fcmCB2Z5HbH64ysnqLn4+exDURIDzArkuY0ruTpwzGEOCoVCoVAoFAqFwh7+ATX923/I0qEaAAAAAElFTkSuQmCC" alt="f'(oo) = 0" style="width: 69px; height: 19px;" width="69" height="19"><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: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; "> states that the streamwise velocity u<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: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; "> is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value <img src="data:image/png;base64,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" alt="a = 4/3" style="width: 51.5px; height: 19px;" width="51.5" height="19"><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: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; "> corresponds to the physical problem of the jet. Other values are included for variety.</div></div></div>

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Last Solution submitted on Apr 28, 2022

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William on 21 Mar 2021

The constraint that involves the constant 'a' is missing in the problem description.

ChrisR on 22 Mar 2021

Thanks for pointing out my omission.

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Solution 5142247

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ChrisR on 22 Mar 2021

Well done.

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Problem 51087. Solve an ODE: equation for a 2D laminar jet

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Problem 51087. Solve an ODE: equation for a 2D laminar jet

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