Nonisothermal Parallel Plate Flow Problem

[TexanFromTexa]

Homework Statement

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:

Homework Equations

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

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

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.

Any advice?

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.

 

[KataKoniK]

Dear fellow scientist,

Thank you for sharing your attempt at solving the problem. I can see that you have put a lot of effort into it. However, I believe there are a few issues with your solution that may be causing the discrepancy.

Firstly, the equation you have used for \tau_{xz} is not correct. It should be \tau_{xz} = -\mu \frac{d v_z}{dx}. This is because the shear stress is proportional to the velocity gradient, not the velocity itself.

Secondly, I am not sure where the constant C1 comes from in your equation. It should not be present in the expression for \tau_{xz}. Additionally, when you plug in the expression for \mu, it should be \mu = \mu_o \left(1 + \beta \frac{T_o - T_1}{T_o^2} x\right), not the expression you have written.

Finally, I am not sure how you got the expression for v_z in terms of x and \frac{dP}{dz}. Could you please explain your steps in more detail?

As for your question about justifying \tau_{xz} = \frac{1}{\mu} \frac{d v_z}{dx}, I believe that this is not a valid justification. The equation you have used is for a Newtonian fluid, whereas the problem statement does not specify the type of fluid. Additionally, the problem asks for an equation for \tau_{xz}, not for \frac{d v_z}{dx}.

I hope this helps in identifying any errors in your solution and finding a correct approach to the problem. Keep up the good work!