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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 69 "Obtaining the Velocity Pr
ofile for Pipe Flow of a Non-Newtonian Fluid" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Area \+
in r-direction for a pipe" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A1:=r-
>2*Pi*r*L;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1GR6#%\"rG6\"6$%)ope
ratorG%&arrowGF(,$*(%#PiG\"\"\"9$F/%\"LGF/\"\"#F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 30 "Area in z-direction for a pipe" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "A2:=r->2*Pi*r*dr;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A2GR6#%\"rG6\"6$%)operatorG%&arrowGF(,$*(%#PiG\"\"\"
9$F/%#drGF/\"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Left-ha
nd side of momentum balance due to shear stress" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "LHS:=A1(r)*tau[rz](r)-A1(r+dr)*tau[rz](r+dr);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LHSG,&**%#PiG\"\"\"%\"rGF(%\"LGF(-&
%$tauG6#%#rzG6#F)F(\"\"#**F'\"\"\",&F)F(%#drGF(F(F*F3-F,6#F4F(!\"#" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Right-hand side of momentum balan
ce due to pressure forces" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "RHS:=A
2(r)*pL-A2(r)*p0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RHSG,&**%#PiG
\"\"\"%\"rGF(%#drGF(%#pLGF(\"\"#**F'\"\"\"F)F.F*F.%#p0GF(!\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Take limit as dr->0 to obtain diff
erential equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq1:=limit(LHS
/dr,dr=0)=RHS/dr:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "eq1:=simplify(-eq1/2/Pi/L);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq1G/,&*&%\"rG\"\"\"--%\"DG6#&%$tauG6#%#rzG6#F(F)F)-
F.F2F)*&*&F(\"\"\",&%#pLG!\"\"%#p0GF)F)F6%\"LG!\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 91 "Solve the differential equation to obtain an ex
pression for shear stress as a function of r" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "s1:=dsolve(eq1,tau[rz](r)); assign(s1); tau[rz]:=unap
ply(tau[rz](r),r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G/-&%$tauG6
#%#rzG6#%\"rG,(*&*&F,\"\"\"%#pLGF0\"\"\"%\"LG!\"\"#!\"\"\"\"#*&*&F,F2%
#p0GF0F2F3F4#F0F7*&%$_C1GF2F,F4F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%$tauG6#%#rzGR6#%\"rG6\"6$%)operatorG%&arrowGF+,(*&*&9$\"\"\"%#pLGF3
\"\"\"%\"LG!\"\"#!\"\"\"\"#*&*&F2F5%#p0GF3F5F6F7#F3F:*&%$_C1GF5F2F7F3F
+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Take the constant of in
tegration to be zero, to make sure that the shear stress does not blow
 up at r=0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "_C1:=0; " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$_C1G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 36 "Ellis Model for non-Newtonian fluids" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "eq2:=-D(vz)(r)=(phi[0]+phi[1]*abs(tau[rz](r))^(alpha-
1))*tau[rz](r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,$--%\"DG6#
%#vzG6#%\"rG!\"\"*&,&&%$phiG6#\"\"!\"\"\"*&&F26#F5F5)-%$absG6#,&*&*&F-
F5%#pLGF5\"\"\"%\"LG!\"\"#F.\"\"#*&*&F-FA%#p0GF5FAFBFC#F5FE,&%&alphaGF
5F.F5F5F5F5F=F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "For given conv
entions, the shear stress is positive" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "assume(tau[rz](r),positive);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "eq2:=simplify(eq2):" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 129 "Solve differential equation for velocity profi
le as function of radius, with the boundary condition of zero velocity
 at pipe wall" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "s2:=dsolve(\{eq2,v
z(R)=0\},vz(r)): assign(s2); vz:=unapply(vz(r),r);" }{TEXT -1 0 "" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#vzGR6#%#r|irG6\"6$%)operatorG%&arro
wGF(,**&*(&%$phiG6#\"\"!\"\"\")9$\"\"#\"\"\"%$pL|irGF3F7%#L|irG!\"\"#F
3\"\"%*&*(F/F7F4F7%$p0|irGF3F7F9F:#!\"\"F<*&*(F9F3&F06#F3F3),$*&*&F5F3
,&F8F3F?FAF3F7F9F:#FAF6,&%&alphaGF3F3F3F3F7*&FJ\"\"\"FL\"\"\"F:F6*&*&%
\"RGF7,,*,F/F7FSF3F8F7)F6FMF3FMF3F3**F/F7FSF7F8F7FVF7F3*,F/F7FSF7F?F7F
VF7FMF7FA**F/F7FSF7F?F7FVF7FA*(FDF7),$*&*&FSF7FJF7F7F9F:FAFMF3F9F7!\"%
F3F7*(F9\"\"\"FV\"\"\"FL\"\"\"F:F@F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 66 "Velocity at the wall, check expression for velocity in z-
direction" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(vz(R));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 40 "Solve for maximum velocity at centerline" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "vmax:=simplify(vz(0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&*&%\"RG\"\"\",,**&%$phiG6#\"\"!\"\"\"F&F.%$pL|irGF.
%&alphaGF.F.*(F*F'F&F'F/F'F.**F*F'F&F'%$p0|irGF.F0F'!\"\"*(F*F'F&F'F3F
'F4*(&F+6#F.F.%#L|irGF.),$*&*&F&F',&F/F.F3F4F.F'F9!\"\"#F4\"\"#F0F.!\"
%F.F'*&F9\"\"\",&F0F.F.F.\"\"\"F?#F4\"\"%" }}}}{MARK "15" 0 }
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