Fluid dynamics of entry region in a pipe

[dweeegs]

Homework Statement

Steady, laminar flow of an incompressible Newtonian fluid with constant physical properties. The area of interest in the problem is in the entrance region between two wise, horizontal parallel plates separated by distance 2B. z is in the direction of flow and x is the direction in which vz (velocity in the z-direction) varies. The plates are stationary and the flow is due to an imposed axial pressure gradient. At the entrance, the velocity is uniform in the cross-section. Far downstream the fluid becomes fully developed and a parabolic shape results.

a) What are the non-zero components of the velocity vector and stress tensor in the entrance region? In what direction do the vary?

b) In what direction does the pressure vary? For entrance flow, can dP/dz be replaced by ΔP/L?

c) Write the reduced differential equation(s) that describe the velocity and pressure fields in the entrance region. Clearly state all of the required boundary conditions

Homework Equations

Navier-Stokes equations and the tensor equations (in rectangular coordinates)

The Attempt at a Solution

(a) I said there's velocity in the x and z direction in the entrance region. The velocity profile changes until it is fully developed, which is why I said z-direction. In order to become a parabola, the velocity also has to be moving inwards in the x-direction, also has to do with the 'no slip' condition assumed on the pipe's surface.

I said the velocity in the z direction depends on x and z. This is because it changes to a parabola as the fluid moves in the z-direction (where the velocity in the z direction would still depend on x). I said the velocity in the x direction only depends on z, though I was shaky on that. My reasoning on that was again with the fact that it becomes fully developed as we move in the z-direction, where it eventually disappears. I'm not sure why it doesn't depend in the x-direction, though I threw it out because I couldn't come up with a good reason.

I can do the tensors myself if my thinking from above is confirmed by someone else

(b) I assumed pressure varies in the x and z direction because there are velocities in both directions. I couldn't explain if we could replace dP/dz with ΔP/L

(c) again, I can do myself if my results from (a) are confirmed.

Help! i just need someone to explain if I'm right or wrong.

 

[dweeegs]

I'm going to change my assumption for velocity in the x-direction.

The fluid would be coming from both the top and bottom of the pipe and meet in the middle, where vx=0. Obviously this means there's a change in the x-direction with the x-component of the velocity.

so now vx is a function of x and z.