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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 4.9  Laminar Flow in a CVD Reactor</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider the laminar flow in a CVD reactor.  The governing equation is</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><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>=D<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">                                                                             u(x,0) = 1                                                       (4.1.13)</Text-field>
<Text-field style="Text" layout="Normal"></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="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==</Equation>(0,z) = 0<Font encoding="UTF-8"> and u(\342\210\236,z) is defined </Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">where v<Font subscript="true" size="10">max</Font> is the average velocity (cm/s), B is the half-width of the reactor (cm) and D is the diffusion coefficient (cm<Font superscript="true" size="10">2</Font>/s).  Next, the transformation <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>  is used to simplify the governing equation as:</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> =  <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>                                           (4.1.14)</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"> The following transformation is used to combine the variable:<Font superscript="true">[7]</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">	 	                                                     <Font encoding="UTF-8"> \316\267=</Font><Equation executable="false" style="2D Math" input-equation="" display="LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JC1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2Jy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiYtSSNtbkdGJDYkUSI5RidGNS1JI21vR0YkNi1RIn5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWS1GLzYlUSJ6RidGMkY1RjUtRiw2JC1GLzYlUSJWRidGMkY1RjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNvLyUpYmV2ZWxsZWRHRjRGNS1GQjYmUSIxRicvJSVzaXplR1EjMTBGJy8lLHN1cGVyc2NyaXB0R1EldHJ1ZUYnRjUtRkY2L1EiXEYnRltwRl5wRjVGSUZLRk1GT0ZRRlNGVUZXRlotRkI2JlEiM0YnRltwRl5wRjVGNUZeb0Zhb0Zkb0Zmbw==">LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JC1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjUtRiw2Jy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiYtSSNtbkdGJDYkUSI5RidGNS1JI21vR0YkNi1RIn5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWS1GLzYlUSJ6RidGMkY1RjUtRiw2JC1GLzYlUSJWRidGMkY1RjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNvLyUpYmV2ZWxsZWRHRjRGNS1GQjYmUSIxRicvJSVzaXplR1EjMTBGJy8lLHN1cGVyc2NyaXB0R1EldHJ1ZUYnRjUtRkY2L1EiXEYnRltwRl5wRjVGSUZLRk1GT0ZRRlNGVUZXRlotRkI2JlEiM0YnRltwRl5wRjVGNUZeb0Zhb0Zkb0Zmbw==</Equation>                                             (4.1.15)</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The variable u in the new coordinate \316\267 is represented by U.  The boundary conditions for U are:                 </Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">                                                                                    U(0)=0	 	                                                        </Text-field>
<Text-field style="Text" layout="Normal">                                                                                                                                                    (4.1.16)</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">                                                                                   U(0)=1</Text-field>
<Text-field style="Text" layout="Normal"> </Text-field>
<Text-field style="Text" layout="Normal">The governing equation for U is then solved with the above boundary conditions to obtain the final solution.  Example 4.9 is solved in Maple 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="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=V*x*diff(u(x,z),z)-diff(u(x,z),x$2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJiooSSJWR0YkIiIiSSJ4R0YkRigtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkidUdGJDYkRilJInpHRiRGMUYoRigtRis2JEYuLUkiJEdGLDYkRikiIiMhIiI=">LCYqKEkiVkc2IiIiIkkieEdGJUYmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiU2JEYnSSJ6R0YlRi9GJkYmLUYpNiQtRik2JEYsRidGJyEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1:=changevar(u(x,z)=U(eta(x,z)),eq):</Text-field>
</Input>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=expand(simplify(subs(eta(x,z)=x/(9*z/V)^(1/3),eq1))):</Text-field>
</Input>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=subs(x=eta*(9*z/V)^(1/3),eq2):</Text-field>
</Input>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=expand(eq2*z):</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=simplify(eq2/(z/V)^(1/3)):</Text-field>
</Input>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=eq2*9/3^(2/3)/V:</Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=convert(eq2,diff):</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation in the combined variable is:</Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=-eq2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqKCIiJCIiIilJJGV0YUdGJCIiI0YoLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlVHRiQ2I0YqRipGKEYoLUYtNiRGMC1JIiRHRi42JEYqRitGKA==">LCYqJkkkZXRhRzYiIiIjLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlVHRiU2I0YkRiQiIiIiIiQtRig2JEYnRiRGLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The boundary conditions for U are entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=U(0)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JIlVHRiQ2IyIiIUYp">Ly1JIlVHNiI2IyIiIUYn</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=U(infinity)=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLy1JIlVHRiQ2I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHIiIi">Ly1JIlVHNiI2I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHIiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">rhs(dsolve({eq2,bc1,bc2},U(eta)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCouSSRldGFHRicjISImIiIjLUkkZXhwR0YkNiMsJCokRioiIiQjISIiRi0iIiIsJiomRipGMy1JK1doaXR0YWtlck1HRiQ2JSNGNiIiJyNGLUYzRjJGNkYzLUY6NiUjIiIoRj1GPkYyIiIlRjZJI19hR0YkIyIiJkYtLUYvNiMsJCokRkRGM0Y0RjUsJiomRkRGMy1GOjYlRjxGPkZKRjZGMy1GOjYlRkFGPkZKRkNGNS9GREkpaW5maW5pdHlHRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Maple cannot find the limit as \316\267\342\206\222\342\210\236.  Alternatively, the governing equation is solved using the first boundary condition (bc1) only:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U:=rhs(dsolve({eq2,bc1},U(eta)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiVUc2IioqKUkkZXRhR0YkIyIiJiIiIyEiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCQqJiMiIiJGKkY1KUYnIiIkRjVGK0Y1LCYqKEY3RjVGNkY1LUkrV2hpdHRha2VyTUdGLjYlI0Y1IiInI0YqRjcqJEY2RjVGNUY1KiYiIiVGNS1GOzYlIyIiKEY+Rj9GQEY1RjVGNUkkX0MyR0YkRjU=">KipJJGV0YUc2IiMhIiYiIiMtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkKiRGIyIiJCMhIiJGJyIiIiwmKiZGI0YwLUkrV2hpdHRha2VyTUdGKjYlI0YzIiInI0YnRjBGL0YzRjAtRjc2JSMiIihGOkY7Ri8iIiVGM0kkX0MyR0YkRjM=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution is a combination of exponential and Whittaker functions.  In the literature this problem is usually left in terms of integrals (Gamma functions).  However, Maple is able to solve the differential equation explicitly.  Next, the constant _C2 is found using the boundary condition bc2.</Text-field>
</Input>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eval(limit(U,eta=infinity));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCoqSSRldGFHRicjISImIiIjLUkkZXhwR0YkNiMsJCokRioiIiQjISIiRi0iIiIsJiomRipGMy1JK1doaXR0YWtlck1HRiQ2JSNGNiIiJyNGLUYzRjJGNkYzLUY6NiUjIiIoRj1GPkYyIiIlRjZJJF9DMkdGJ0Y2L0YqSSlpbmZpbml0eUdGJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Since the limit does not exit, U/_C2 is plotted until \316\267=10.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot(U/_C2,eta=0..10,axes=boxed,thickness=4,title=&quot;Figure Exp. 4.1.18.&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">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</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">An initial guess L = 1 is used to replace x = \342\210\236.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">evalf(subs(eta=1.,U/_C2));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrR1ovSUshIio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiIiIi">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The value of U/C2 is found at x = L:</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">c0:=evalf(subs(eta=1.,U/_C2));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzBHNiIkIitHWi9JSyEiKg==">JCIrR1ovSUshIio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the length L is increased until the U/_C2(x = ) becomes a constant:</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">while err&gt;1e-6 do L:=L+1; c1:=evalf(subs(eta=L,U/_C2));err:=abs(c1-c0);c0:=c1;od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiIw==">IiIj</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzFHNiIkIitiUyI9ZCQhIio=">JCIrYlMiPWQkISIq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIqRiRwPE0hIio=">JCIqRiRwPE0hIio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzBHNiIkIitiUyI9ZCQhIio=">JCIrYlMiPWQkISIq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiJA==">IiIk</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzFHNiIkIitaIT0+ZCQhIio=">JCIrWiE9PmQkISIq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCInIypSNSEiKg==">JCInIypSNSEiKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzBHNiIkIitaIT0+ZCQhIio=">JCIrWiE9PmQkISIq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiJQ==">IiIl</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzFHNiIkIitaIT0+ZCQhIio=">JCIrWiE9PmQkISIq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIiIUYm">JCIiIUYj</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYzBHNiIkIitaIT0+ZCQhIio=">JCIrWiE9PmQkISIq</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The length and the constant have converged to 10 digit accuracy.  The final length is</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The constant _C2 is found using the boundary condition U(L) = 1</Text-field>
</Input>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">_C2:=1/c0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkX0MyRzYiJCIrL2poKnojISM1">JCIrL2poKnojISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution to the transformed coordinate is:</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqKEkkZXRhRzYiIyEiJiIiIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJEYkIiIkIyEiIkYoIiIiLCYqJkYkRjEtSStXaGl0dGFrZXJNR0YrNiUjRjQiIicjRihGMUYwRjRGMS1GODYlIyIiKEY7RjxGMCIiJUY0JCIrL2poKnojISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot(U,eta=0..10.,thickness=4,axes=boxed,title=&quot;Figure Exp. 4.1.19.&quot;,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">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</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution in the original coordinates is obtained as:</Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u:=subs(eta=x/(9*z/V)^(1/3),U);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkidUc2IiwkKiwkIisxIypvbkEhIikiIiIpIiIqI0YqIiIjRiopKihJInhHRiRGKilGLCNGLiIiJEYqKSomSSJ6R0YkRipJIlZHRiQhIiIjRipGNEY5IyIiJkYuRjktSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkKiojRioiIz1GKilGMUY0RipGN0Y5RjhGKkY5RiosJiosRjpGKkZHRipGN0Y5RjhGKi1JK1doaXR0YWtlck1HRj82JSNGKiIiJ0YzLCQqKiNGKkYsRipGR0YqRjdGOUY4RipGKkYqRioqJiIiJUYqLUZLNiUjIiIoRk5GM0ZPRipGKkYqRio=">LCQqKiIiKiMiIiIiIiMqKEkieEc2IkYmRiQjRiciIiQqJkkiekdGKkYmSSJWR0YqISIiI0YwRiwjISImRictSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2IywkKihGKUYsRi5GMEYvRiYjRjAiIz1GJiwmKipGKUYsRi5GMEYvRiYtSStXaGl0dGFrZXJNR0Y2NiUjRiYiIidGKywkRjsjRiZGJEYmI0YmRiwtRkE2JSMiIihGREYrRkUiIiVGJiQiKzEjKm9uQSEiKQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:={V=0.001};</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjwjL0kiVkdGJCQiIiIhIiQ=">PCMvSSJWRzYiJCIiIiEiJA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">A plot was made by specifying a value for V.</Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(subs(pars,u),x=0..10,z=0..10,axes=boxed,title=&quot;Figure Exp. 4.1.20.&quot;,orientation=[120,60],labels=[x,y,&quot;u&quot;]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="three-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0">[encoded data, 13696 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqKiIiKiMiIiIiIiMqKEkieEc2IkYmRiQjRiciIiQqJkkiekdGKkYmSSJWR0YqISIiI0YwRiwjISImRictSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2IywkKihGKUYsRi5GMEYvRiYjRjAiIz1GJiwmKipGKUYsRi5GMEYvRiYtSStXaGl0dGFrZXJNR0Y2NiUjRiYiIidGKywkRjsjRiZGJEYmI0YmRiwtRkE2JSMiIihGREYrRkUiIiVGJiQiKzEjKm9uQSEiKQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>