Example 11.4-3: Part a: checking on the viscosity approximation: 11.4-18
Steady Flow of a non-isothermal film:
The only component of velocity is the z component. It only changes in the x direction as shown in Fig. 2.2-3
Temperature also only varies in the x direction.
Thus vz(x) and T(x) and all terms on the left of eq. B.9-1 are zero.
Note that Table B.9 only requires that rho and k be constant: not mu.
On the right side we have the first term as the only important one if we can neglect viscous dissipation
and variations of thermal conductivity with temperature.
> restart;
> s:=dsolve({diff(T(x),x,x),T(0)=T0,T(delta)=Td},T(x)): Simplified form of eq B.9-1
> assign(s);
> T:=unapply(T(x),x);
> mu17:=x->mu0*exp(B*(1/T(x)-(1/T0))); eq. 11.4-17.
> mu17(x);
> mu17(0);
> mud:=mu17(delta);
> A:=mud/mu0;
> mu19:=x->mu0*A^(x/delta);
> mu19(0);
> mu19(delta);
> mu19(delta/2);
Data for water from plot and fit of data in Table 1.1-2
> B:=1868.6; mu0:=1.787; T0:=273.15; Td:=373.15;
> mu17(.6*delta);
> mu19(.6*delta); From Table 1.1-1 it should be .4665 at 60C
> mus:=[1.787,1.0019,0.6530,0.4665,0.3548,0.2821]; From Table 1.1-1 mu in cp (or mPa*s).
> Ts:=[0,20,40,60,80,100]; The temperatures in the table in C.
> Tmus:=[[Ts[n]/100,mus[n]]$n=1...6]; Making a list of lists.
> with(plots):
> p1:=plot(Tmus,style=point,symbol=circle):
> p2:=plot([mu17(x*delta),mu19(x*delta)],x=0...1,color=[red,green]):
> display([p1,p2]); The red curve is the value from eq 17, the green one from the approximate of this: eq. 19 and the circles are the data in the Table.
![[Maple Plot]](images/ex1143a1.gif)
>