<?xml version="1.0" encoding="UTF-8"?>
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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 4.13</Font></Text-field>
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<Text-field style="Text" layout="Normal">The velocity distribution in the boundary layer is given by:<Font superscript="true">[6]</Font></Text-field>
<Text-field style="Text" layout="Normal">	</Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>	</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">                                                                     u(0,y) =1                                                   </Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>(x,0) = 1 and <Font encoding="UTF-8">u(x,\342\210\236)=1</Font></Text-field>
<Text-field style="Text" layout="Normal">                                                                     v(x,0) =0  </Text-field>
<Text-field style="Text" layout="Normal">	 	</Text-field>
<Text-field style="Text" layout="Normal">where u and v are the x and y components of the velocity.  Next, the stream function is introduced<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"> and v= -<Equation executable="false" style="2D Math" input-equation="" display="LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2Ji1JI21vR0YkNi5RIn5GJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLzYuUSsmUGFydGlhbEQ7RidGMkY1RjhGO0Y9Rj9GQUZDRkVGR0ZKLUkjbWlHRiQ2JlEmJnBzaTtGJy8lJ2l0YWxpY0dGOkYyRjVGNS1GLDYlRkwtRlA2JlEieEYnRlNGMkY1RjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmluLyUpYmV2ZWxsZWRHRjo=">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</Equation>                                                  (4.1.27)</Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">By definition (4.1.27), the stream function \317\210 satisfies the first equation in equation (4.1.26).  The boundary conditions for \317\210 are</Font></Text-field>
<Text-field style="Text" layout="Normal">	<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> (0,y)=1</Text-field>
<Text-field style="Text" layout="Normal">                                                             <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">                                                                 </Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>= 0</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">The following transformation is used to combine the independent variables:</Text-field>
<Text-field style="Text" layout="Normal">	 	<Font encoding="UTF-8">                                                 \316\267=</Font>  <Equation executable="false" style="2D Math" input-equation="" display="LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JC1JI21pR0YkNiVRInlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2Ji1JI21uR0YkNiRRIjJGJ0Y1LUkmbXNxcnRHRiQ2Iy1GLDYkLUYvNiVRInhGJ0YyRjVGNS1GLzYjUSFGJ0Y1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZOLyUpYmV2ZWxsZWRHRjQ=">LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JC1JI21pR0YkNiVRInlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2Ji1JI21uR0YkNiRRIjJGJ0Y1LUkmbXNxcnRHRiQ2Iy1GLDYkLUYvNiVRInhGJ0YyRjVGNS1GLzYjUSFGJ0Y1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZOLyUpYmV2ZWxsZWRHRjQ=</Equation>                                                       (4.1.29)</Text-field>
<Text-field style="Text" layout="Normal">Next, the following transformation is introduced:</Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">	 	                                                  \317\210=</Font><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNiQtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRidGPQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNiQtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRidGPQ==</Equation><Font encoding="UTF-8"> f(\316\267)                                                     (4.1.30)</Font></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The boundary conditions for f(\316\267) are:</Font></Text-field>
<Text-field style="Text" layout="Normal">	 	                                      </Text-field>
<Text-field style="Text" layout="Normal">                                                                      f(0)=0 and <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">                                                                                                                                                  (4.1.31)</Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font encoding="UTF-8">(\342\210\236) = 1</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Using Maple the transformation involved in the governing equation and boundary conditions in example 4.13 is solved below:</Text-field>
</Input>
</Group>
<Group labelreference="L3" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(student):</Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Enter the governing equation:</Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=u(x,y)*diff(u(x,y),x)+v(x,y)*diff(u(x,y),y)-diff(u(x,y),y$2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsKComLUkidUdGJDYkSSJ4R0YkSSJ5R0YkIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JEYnRipGLEYsKiYtSSJ2R0YkRilGLC1GLjYkRidGK0YsRiwtRi42JEYnLUkiJEdGLzYkRisiIiMhIiI=">LCgqJi1JInVHNiI2JEkieEdGJkkieUdGJiIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJEYoRipGKiomLUkidkdGJkYnRiotRiw2JEYkRilGKkYqLUYsNiRGMkYpISIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The steam function is introduced:</Text-field>
</Input>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:={u(x,y)=diff(psi(x,y),y),v(x,y)=-diff(psi(x,y),x)};</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjwkLy1JInVHRiQ2JEkieEdGJEkieUdGJC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSRwc2lHRiRGKUYrLy1JInZHRiRGKSwkLUYtNiRGMEYqISIi">PCQvLUkidUc2IjYkSSJ4R0YmSSJ5R0YmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJHBzaUdGJkYnRikvLUkidkdGJkYnLCQtRis2JEYuRighIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation for the stream function is:</Text-field>
</Input>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=subs(vars,eq);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsKComLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJHBzaUdGJDYkSSJ4R0YkSSJ5R0YkRi8iIiItRig2JUYrRi9GLkYwRjAqJi1GKDYkRitGLkYwLUYoNiRGKy1JIiRHRik2JEYvIiIjRjAhIiItRig2JEYrLUY5NiRGLyIiJEY8">LCgqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSRwc2lHNiI2JEkieEdGKkkieUdGKkYtIiIiLUYlNiRGJEYsRi5GLiomLUYlNiRGKEYsRi4tRiU2JEYkRi1GLiEiIi1GJTYkRjRGLUY2</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the transformation defined in equation (4.1.30) is introduced:</Text-field>
</Input>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),eq);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCwqKC0tSSJER0koX3N5c2xpYkc2IjYjSSJmR0YoNiMtSSRldGFHRig2JEkieEdGKEkieUdGKCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGLEYwRjEsKComRiRGMUYyRjFGMSoqRi9GMS0tLUkjQEBHNiRGNEYnNiRGJiIiI0YpRitGMUYyRjEtRjM2JEYsRi9GMUY/KihGL0YxRiRGMS1GMzYkRkBGMEYxRj9GMSNGMUY/KiYsJi1GKkYrRjEqKEYvRjFGJEYxRkBGMUY/RjEsJiomRjlGMUYyRj9GMSomRiRGMS1GMzYkRjJGMEYxRjFGMSMhIiJGPyooRi9GRS0tLUY8NiRGJiIiJEYpRitGMUYyRlZGUCoqRi9GRUY5RjFGMkYxRk1GMSEiJCooRi9GRUYkRjEtRjM2JEZNRjBGMUZQ</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The independent variables are combined using the transformation defined in equation (4.1.29):</Text-field>
</Input>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1:=(simplify(subs(eta(x,y)=y/x^(1/2),eq))):</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1:=subs(y=eta*x^(1/2),eq1):</Text-field>
</Input>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1:=simplify(eq1*x):</Text-field>
</Input>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation for f in the combined variable is:</Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=convert(-eq1,diff);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqKCMiIiIiIiNGKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJmR0YkNiNJJGV0YUdGJC1JIiRHRiw2JEYxRilGKEYuRihGKC1GKzYkRi4tRjM2JEYxIiIkRig=">LCYqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiU2JC1JImZHNiI2I0kkZXRhR0YsRi5GLiIiIkYqRi8jRi8iIiMtRiU2JEYkRi5GLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Nest, the velocity variables u and v (i.e., derivatives of the stream function) are expressed in terms of the combined variable and f:</Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v(eta):=-diff(psi(x,y),x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInZHNiI2I0kkZXRhR0YlLCQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkkcHNpR0YlNiRJInhHRiVJInlHRiVGMCEiIg==">LCQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkkcHNpRzYiNiRJInhHRilJInlHRilGKyEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v(eta):=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),v(eta)):</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v(eta):=expand(subs(eta(x,y)=y/x^(1/2),v(eta))):</Text-field>
</Input>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v(eta):=subs(y=eta*x^(1/2),v(eta)):</Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v(eta):=factor(v(eta));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInZHNiI2I0kkZXRhR0YlLCQqKCMiIiIiIiNGKywmLUkiZkdGJUYmISIiKiYtLUkiREdJKF9zeXNsaWJHRiU2I0YvRiZGK0YnRitGK0YrKUkieEdGJUYqRjBGKw==">LCQqJiwmLUkiZkc2IjYjSSRldGFHRichIiIqJi0tSSJER0koX3N5c2xpYkdGJzYjRiZGKCIiIkYpRjFGMUYxSSJ4R0YnI0YqIiIjI0YxRjQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(eta):=diff(psi(x,y),y);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2I0kkZXRhR0YlLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJHBzaUdGJTYkSSJ4R0YlSSJ5R0YlRjA=">LUklZGlmZkclKnByb3RlY3RlZEc2JC1JJHBzaUc2IjYkSSJ4R0YoSSJ5R0YoRis=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(eta):=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),u(eta)):</Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(eta):=expand(subs(eta(x,y)=y/x^(1/2),u(eta))):</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(eta):=subs(y=eta*x^(1/2),u(eta));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2I0kkZXRhR0YlLS1JIkRHSShfc3lzbGliR0YlNiNJImZHRiVGJg==">LS1JIkRHSShfc3lzbGliRzYiNiNJImZHRiY2I0kkZXRhR0Ym</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L54" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the boundary conditions are expressed in terms of f:</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=subs(eta=0,v(eta))=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLywkKigjIiIiIiIjRiktSSJmR0YkNiMiIiFGKSlJInhHRiRGKCEiIkYxRi4=">LywkKiYtSSJmRzYiNiMiIiEiIiJJInhHRicjISIiIiIjRixGKQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=-bc1*2*x^(1/2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JImZHRiQ2IyIiIUYp">Ly1JImZHNiI2IyIiIUYn</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=subs(eta=0,u(eta))=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLy0tSSJER0koX3N5c2xpYkdGJDYjSSJmR0YkNiMiIiFGLQ==">Ly0tSSJER0koX3N5c2xpYkc2IjYjSSJmR0YnNiMiIiFGKw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc3:=subs(eta=infinity,u(eta))=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMzRzYiLy0tSSJER0koX3N5c2xpYkdGJDYjSSJmR0YkNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiIg==">Ly0tSSJER0koX3N5c2xpYkc2IjYjSSJmR0YnNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The length of the domain is taken to be 5:</Text-field>
</Input>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc3:=subs(infinity=5,bc3);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMzRzYiLy0tSSJER0koX3N5c2xpYkdGJDYjSSJmR0YkNiMiIiYiIiI=">Ly0tSSJER0koX3N5c2xpYkc2IjYjSSJmR0YnNiMiIiYiIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The numerical solution for the Blassius equation is obtained as:</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eq2,bc1,bc2,bc3},f(eta),type=numeric);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">odeplot(sol,[eta,f(eta)],0..5,thickness=3,title=&quot;Figure Exp. 4.1.26.&quot;,axes=boxed);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">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</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the velocity profiles are obtained by converting the corresponding expression to 'diff' form:</Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(eta):=convert(u(eta),diff);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2I0kkZXRhR0YlLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImZHRiVGJkYn">LUklZGlmZkclKnByb3RlY3RlZEc2JC1JImZHNiI2I0kkZXRhR0YoRio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v(eta):=convert(v(eta),diff);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInZHNiI2I0kkZXRhR0YlLCQqKCMiIiIiIiNGKywmLUkiZkdGJUYmISIiKiYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkRi5GJ0YrRidGK0YrRispSSJ4R0YlRipGMEYr">LCQqJiwmLUkiZkc2IjYjSSRldGFHRichIiIqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJUYpIiIiRilGMEYwRjBJInhHRicjRioiIiMjRjBGMw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L58" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Since v is a function of x, v<Font superscript="true">*</Font>x<Font superscript="true" size="10">1/2</Font><Font size="10"> </Font>is plotted:</Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">odeplot(sol,[eta,u(eta)],0..5,thickness=4,color=blue,title=&quot;Figure Exp. 4.1.27.&quot;,axes=boxed,labels=[eta,u]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">NiotJSdDVVJWRVNHNiM3VDckJCIjNSEjNSQiMiVcJFIpKnlCOk8kISNFNyQkIio9My8tIiEiKiQiMjpIcEY6JzRJTSEjPTckJCIqTTszLyMhIiokIjFGL1kpZVApZm8hIzw3JCQiKVhBaEkhIikkIjJRQW5RekkpRzUhIzw3JCQiKm1LOzMlISIqJCIxW2RKTnVScjghIzs3JCQiKiQzLy1eISIqJCIyUWNcMiIqcE1yIiEjPDckJCIqKipbQzcnISIqJCIyaj4menFaeGE/ISM8NyQkIio6ZEc5KCEiKiQiMkFGJkc1KylcUiMhIzw3JCQiKktsSzspISIqJCIxaSQ9RnYncExGISM7NyQkIipbdE89KiEiKiQiMjpOQ1NvIltxSSEjPDckJCIrayIzLy0iISIqJCIyOURxJFIuJVtTJCEjPDckJCIrIikqW0M3IiEiKiQiMSJcZzBBSml0JCEjOzckJCIrKHoqW0M3ISIqJCIyOStXY3pxUzElISM8NyQkIis4MWBFOCEiKiQiMjpEeSM+JVJ4USUhIzw3JCQiKlZyJkc5ISIpJCIxRStsNCIpZTFaISM7NyQkIitZQWhJOiEiKiQiMSUzL0g9WSo+XSEjOzckJCIraUlsSzshIiokIjEmb1A3ckhySyYhIzs3JCQiK3lRcE08ISIqJCIxJFFPbGtddWkmISM7NyQkIismcE1uJD0hIiokIjFucjJDbEE/ZiEjOzckJCIrNmJ4UT4hIiokIjFKND01MHovaSEjOzckJCIrRmoiMy8jISIqJCIwaj9kbSwwWychIzo3JCQiK1dyJkc5IyEiKiQiMXNGMzpedlluISM7NyQkIioneipbQyMhIikkIjFqd1AlKT0qSCsoISM7NyQkIit3KFFwTSMhIiokIjFPcWIiPjMoW3MhIzs3JCQiKyNmeipbQyEiKiQiMCcqPS0pUVkkWyghIzo3JCQiKzQvLV5EISIqJCIxcEdiVigqKW9xKCEjOzckJCIrRDcxYEUhIiokIjFaYXRKUHA9eiEjOzckJCIrVD81YkYhIiokIjF4YFQpb2wnPSIpISM7NyQkIitlRzlkRyEiKiQiLyYpUU8vbzEkKSEjOTckJCIrdU89ZkghIiokIjE+USNbOypwI1spISM7NyQkIipcQzcxJCEiKSQiMWhVcT8hcG5rKSEjOzckJCIrMmBFakohIiokIjEvej5iNS0qeikhIzs3JCQiK0JoSWxLISIqJCIxKVEnKTNnbSdSKikhIzs3JCQiK1JwTW5MISIqJCIxaHVcYkAqKm8hKiEjOzckJCIrY3hRcE0hIiokIjFILlZCSE4oPSohIzs3JCQiK3MmRzlkJCEiKiQiMVk5dlBfOyZIKiEjOzckJCIrKVFwTW4kISIqJCIxInBIWzYpKkdSKiEjOzckJCIrLy1edlAhIiokIjE6PnAqPWs1WyohIzs3JCQiK0A1YnhRISIqJCIxd2JxSS5AZyYqISM7NyQkIitQPWZ6UiEiKiQiMWwmR2w+MzRqKiEjOzckJCIrYEVqIjMlISIqJCIxUS5tLF11JHAqISM7NyQkIipadE89JSEiKSQiMSZSJT0ncDkkXCgqISM7NyQkIisnRzlkRyUhIiokIjF3TktbKDQjKXoqISM7NyQkIistXnYoUSUhIiokIjEkKXkiM244NSUpKiEjOzckJCIrPWZ6KlslISIqJCIxWSU0QFElSHkpKiEjOzckJCIrTm4kPWYlISIqJCIxakZtJ1spZjUqKiEjOzckJCIrXnYoUXAlISIqJCIxMG5xTmxXUSoqISM7NyQkIjIwKytxTz1meiUhIzskIjEiR2glWyRIQicqKiEjOzckJCIrJT1meipbISIqJCIxdWs+JlswRikqKiEjOzckJCIjXSEiIiQiMSgqKioqKioqKioqKioqKiohIzstJStBWEVTTEFCRUxTRzYnSSRldGFHNiItSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJ1NiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSVGT05URzYlJSFHJSFHIiM1JStIT1JJWk9OVEFMRyUrSE9SSVpPTlRBTEctJSZUSVRMRUc2JC0lKV9UWVBFU0VURzYjLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1LUkjbXNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2I1E0RmlndXJlfkV4cC5+NC4xLjI3LjYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EmZmFsc2U2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdub3JtYWw2Ii0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIjNSEiIi0lKlRISUNLTkVTU0c2IyIiJS0lKkFYRVNTVFlMRUc2IyUkQk9YRy0lKFNDQUxJTkdHNiMlLlVOQ09OU1RSQUlORURHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkKyYhIiItJSlCT1VORFNfWUc2IyQiJHEjISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlNU0hIiItJS5CT1VORFNfSEVJR0hURzYjJCIlcUEhIiItJSlDSElMRFJFTkc2Ig==</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">odeplot(sol,[eta,v(eta)*x^(1/2)],0..5,thickness=4,color=brown,title=&quot;Figure Exp. 4.1.28.&quot;,axes=boxed,labels=[eta,&quot;v*x^(1/2)&quot;]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">NiotJSdDVVJWRVNHNiM3VDckJCIjNSEjNSQiK2I5QC8lKSEjSDckJCIqPTMvLSIhIiokIiswdU1dKCkhIzg3JCQiKk07My8jISIqJCIrbGwhKSpcJCEjNzckJCIpWEFoSSEiKSQiKl9mRih5ISM2NyQkIiptSzszJSEiKiQiKzddKSopUiIhIzY3JCQiKiQzLy1eISIqJCIrKXpHVj0jISM2NyQkIioqKltDNychIiokIittRi5VSiEjNjckJCIqOmRHOSghIiokIitgJzQtRiUhIzY3JCQiKktsSzspISIqJCIqJm9DbWIhIzU3JCQiKlt0Tz0qISIqJCIqKDNxRXEhIzU3JCQiK2siMy8tIiEiKiQiKi4tc2spISM1NyQkIisiKSpbQzciISIqJCIrKT1QQS8iISM1NyQkIisoeipbQzchIiokIithY2RNNyEjNTckJCIrODFgRTghIiokIislKXkoNFciISM1NyQkIipWciZHOSEiKSQiK1k2ZGc7ISM1NyQkIitZQWhJOiEiKiQiK2R0TyMqPSEjNTckJCIraUlsSzshIiokIitsYEVOQCEjNTckJCIreVFwTTwhIiokIis7ZjApUSMhIzU3JCQiKyZwTW4kPSEiKiQiK3klSCVcRSEjNTckJCIrNmJ4UT4hIiokIis4byl6IkghIzU3JCQiK0ZqIjMvIyEiKiQiKy1DRCM+JCEjNTckJCIrV3ImRzkjISIqJCIqUSFwcU0hIio3JCQiKid6KltDIyEiKSQiK0hGc15QISM1NyQkIit3KFFwTSMhIiokIis3JFxQLiUhIzU3JCQiKyNmeipbQyEiKiQiKyc+cF5KJSEjNTckJCIrNC8tXkQhIiokIitiRlMlZiUhIzU3JCQiK0Q3MWBFISIqJCIqQTkqcFshIio3JCQiK1Q/NWJGISIqJCIrJilRQlNeISM1NyQkIitlRzlkRyEiKiQiKzoielJTJiEjNTckJCIrdU89ZkghIiokIip0dClmYyEiKjckJCIqXEM3MSQhIikkIjEqKioqKlwpXHcxZiEjOzckJCIrMmBFakohIiokIjEqKioqKlwkempWaCEjOzckJCIrQmhJbEshIiokIipqRSdwaiEiKjckJCIrUnBNbkwhIiokIilALSVlJyEiKTckJCIrY3hRcE0hIiokIjEqKioqKio0bUYneSchIzs3JCQiK3MmRzlkJCEiKiQiMSwrK2JWK3dwISM7NyQkIispUXBNbiQhIiokIipteUg6KCEiKjckJCIrLy1edlAhIiokIiooZTc8dCEiKjckJCIrQDVieFEhIiokIipCOyZvdSEiKjckJCIrUD1melIhIiokIipPX3RnKCEiKjckJCIrYEVqIjMlISIqJCIqaWJSdCghIio3JCQiKlp0Tz0lISIpJCIqN14oW3khIio3JCQiKydHOWRHJSEiKiQiKnNeQSZ6ISIqNyQkIistXnYoUSUhIiokIipkVF0vKSEiKjckJCIrPWZ6KlslISIqJCIpJ2Z4NykhIik3JCQiK05uJD1mJSEiKiQiKkIkMywjKSEiKjckJCIrXnYoUXAlISIqJCIoOGRFKSEiKDckJCIyMCsrcU89ZnolISM7JCIqRGVCSykhIio3JCQiKyU9ZnoqWyEiKiQiKipcc3IkKSEiKjckJCIjXSEiIiQiKkUyWFQpISIqLSUrQVhFU0xBQkVMU0c2J0kkZXRhRzYiUSp2KnheKDEvMik2Ii0lJUZPTlRHNiUlIUclIUciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lJlRJVExFRzYkLSUpX1RZUEVTRVRHNiMtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjUtSSNtc0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYjUTRGaWd1cmV+RXhwLn40LjEuMjguNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSZmYWxzZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lJkNPTE9SRzYmJSRSR0JHJCIxX01tWCUpZXFrISM7JCIyd21vVigpZXFrIiEjPCQiMndtb1YoKWVxayIhIzwtJSpUSElDS05FU1NHNiMiIiUtJSpBWEVTU1RZTEVHNiMlJEJPWEctJShTQ0FMSU5HRzYjJS5VTkNPTlNUUkFJTkVERy0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJDUqISIiLSUpQk9VTkRTX1lHNiMkIiRJIyEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStJISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJTVCISIiLSUpQ0hJTERSRU5HNiI=</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The solution at \316\267 = 0 is obtained as:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol(0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="NyYvSSRldGFHNiIkIiIhRicvLUkiZkdGJTYjRiRGJi8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRilGJEYmLy1GLjYkRiktSSIkR0YvNiRGJCIiIyQiM0FjJVIpKnlCOk8kISM9">NyYvSSRldGFHNiIkIiIhRicvLUkiZkdGJTYjRiRGJi8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRilGJEYmLy1GLjYkRi1GJCQiM0FjJVIpKnlCOk8kISM9</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Stress is related to the Reynolds number (re) and the flux at y = 0:</Text-field>
</Input>
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">S:=re*diff(u(x,y),y);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiU0c2IiomSSNyZUdGJCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1R0YkNiRJInhHRiRJInlHRiRGMEYn">KiZJI3JlRzYiIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkieUdGJEYuRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The velocity gradient is terms of the stream function is:</Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">subs(u(x,y)=diff(psi(x,y),y),S);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="KiZJI3JlRzYiIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJHBzaUdGJDYkSSJ4R0YkSSJ5R0YkLUkiJEdGKDYkRi4iIiNGJQ==">KiZJI3JlRzYiIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJzYkLUkkcHNpR0YkNiRJInhHRiRJInlHRiRGMEYwRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The second derivative of the stream function is expressed in terms of f and \316\267.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d:=diff(psi(x,y),y$2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiZEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSRwc2lHRiQ2JEkieEdGJEkieUdGJC1JIiRHRic2JEYtIiIj">LUklZGlmZkclKnByb3RlY3RlZEc2JC1GIzYkLUkkcHNpRzYiNiRJInhHRipJInlHRipGLUYt</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d:=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),d):</Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d:=expand(subs(eta(x,y)=y/x^(1/2),d)):</Text-field>
</Input>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d:=subs(y=eta*x^(1/2),d):</Text-field>
</Input>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d:=convert(d,diff);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiZEc2IiomKUkieEdGJCMiIiIiIiMhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYjSSRldGFHRiQtSSIkR0YuNiRGM0YqRik=">KiZJInhHNiIjISIiIiIjLUklZGlmZkclKnByb3RlY3RlZEc2JC1GKTYkLUkiZkdGJDYjSSRldGFHRiRGMUYxIiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">S:=re*d;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiU0c2IiooSSNyZUdGJCIiIilJInhHRiQjRiciIiMhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYjSSRldGFHRiQtSSIkR0YvNiRGNEYrRic=">KihJI3JlRzYiIiIiSSJ4R0YkIyEiIiIiIy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRis2JC1JImZHRiQ2I0kkZXRhR0YkRjNGM0Yl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The second derivative of f is found from the numerical solution.</Text-field>
</Input>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqd3:=sol(0)[4];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklZXFkM0c2Ii8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYjSSRldGFHRiQtSSIkR0YoNiRGLSIiIyQiM0FjJVIpKnlCOk8kISM9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JImZHNiI2I0kkZXRhR0YrRi1GLSQiM0FjJVIpKnlCOk8kISM9</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Hence, the stress Reynolds number relationship becomes:</Text-field>
</Input>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">S:=subs(diff(f(eta),`$`(eta,2))=rhs(eqd3),S);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiU0c2IiwkKigkIjNBYyVSKSp5QjpPJCEjPSIiIkkjcmVHRiRGKilJInhHRiQjRioiIiMhIiJGKg==">LCQqJkkjcmVHNiIiIiJJInhHRiUjISIiIiIjJCIzQWMlUikqeUI6TyQhIz0=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>