# Nonisothermal Parallel Plate Flow Problem

1. Nov 6, 2011

### TexanFromTexa

1. The problem statement, all variables and given/known data

The problem wants us to develop an equation for velocity distribution and the mass flow rate of a fluid, flowing between 2 parallel plates:

The distance between the two plates is a.
The top plate is maintained at constant T1; the bottom plate is maintained at constant T2.
The coordinate normal to the plates is X; the coordinate in the direction of flow is Z.
(Thus, the bottom plate is defined as x = 0 and the top plate x = a).

To attempt the problem, we are told to solve for solve for $\tau$$_{xz}$ from the equation of motion, and then identify the velocity distribution and mass flow rate using the first equation below:

2. Relevant equations

$\tau$$_{xz}$ = - $\mu$$\frac{dv_{z}}{dx}$

$\mu$ = $\mu$$_{o}$ (1 + $\beta$ ( T$_{o} - T_{1}$) x } / {T_{o}^2 a}[/itex] )

3. The attempt at a solution

Solving for $\tau$$_{xz}$ from the equation of motion:

$\tau$$_{xz}$ = C1 + x $\frac{dP}{dz}$ = $\mu$ $\frac{dv_{z}}{dx}$

From here, I use no slip boundary conditions at x = 0 and x = a and solve the above equation plugging in for $\mu$.

From this I get a very convoluted expression:

v$_{z}$ = $\mu$$_{o}$ ( $\frac{T_{o}^2 a}{\beta ( T_{o} - T_{1} )}$ + ( $\frac{-a^6 T_{o}^3}{ln(1 + \beta ( T_{o} - T_{1} ) / T_{o}^2) \beta^5 (T_{o} - T_{1})^5}$) ln(1 + \beta ( T_{o} - T_{1} x / (a T_{o}^2) $\frac{dP}{dz}$

However, plugging in T1 = To, should yield the simple equation for isothermal flow. Instead, it = 0.

PS. I accidentally solved the problem the first time through with $\tau$$_{xz}$ = $\frac{1}{\mu}$ $\frac{dvz}{dx}$, and it worked nicely. Is there a good way to justify this? Or am I simply insane.