1. Louisiana Tech University Ruston, LA 71272 Flows With More Than One Dependent Variable - 2D Example Juan M. Lopez Steven A. Jones BIEN 501 Wednesday, April 4, 2008

2. Louisiana Tech University Ruston, LA 71272 Recall - Generalized Newtonian where Recall that: tr stands for “trace,” which is the sum of the diagonal elements. Tr( T )= T ii While the expression looks complicated, it will look much simpler once a given form for  is found.

3. Louisiana Tech University Ruston, LA 71272 Generalized Newtonian where Recall that: tr stands for “trace,” which is the sum of the diagonal elements. Tr( T )= T ii

4. Louisiana Tech University Ruston, LA 71272 Parallel Plate Poiseuille Flow Given: A steady, fully developed, laminar flow of a Newtonian fluid in a rectangular channel of two parallel plates where the width of the channel is much larger than the height, h, between the plates. Find: The velocity profile and shear stress due to the flow. Assumptions: Entrance Effects Neglected No-Slip Condition No vorticity/turbulence

5. Louisiana Tech University Ruston, LA 71272 Additional and Highlighted Important Assumptions The width is very large compared to the height of the plate. No entrance or exit effects. Fully developed flow. THEREFORE… –Velocity can only be dependent on vertical location in the flow (v x ) –(v y ) = (v z ) = 0 –The pressure drop is constant and in the x- direction only.

6. Louisiana Tech University Ruston, LA 71272 Boundary Conditions No Slip Condition Applies –Therefore, at y = -h/2 and y = +h/2, v = 0 The bounding walls in the z direction are often ignored. If we don’t ignore them we also need: –z = -w/2 and z = +w/2, v = 0, where w is the width of the channel. For this problem we include this, and make the width finite to make this dependent on two variables.

7. Louisiana Tech University Ruston, LA 71272 Incompressible Newtonian Stress Tensor Adapted from Table 3.3 in the text. Now, we cancel terms out based on our assumptions. This results in our new tensor:

8. Louisiana Tech University Ruston, LA 71272 Navier-Stokes Equations In Vector Form: Which we expand to component form from table 3.4:

9. Louisiana Tech University Ruston, LA 71272 Reducing Navier-Stokes

10. Louisiana Tech University Ruston, LA 71272 Reducing Navier-Stokes This reduces to: Including our constant pressure drop: Oops! Now we have a nonhomogenous higher-order differential equation that is inseparable. How do we deal with it?

11. Louisiana Tech University Ruston, LA 71272 DiffEq Assumptions We will assume this solution is a combination of simple parallel plate Poiseuille flow plus some perturbation that is dependent on the walls and finite width. Extracting the 1D Poiseuille flow, we can rewrite the equation as:

12. Louisiana Tech University Ruston, LA 71272 DiffEq Solution - Setup We can separate this into two equations, each of which equals zero. Why? –0=0+0

13. Louisiana Tech University Ruston, LA 71272 DiffEq Solution - Poiseuille

14. Louisiana Tech University Ruston, LA 71272 DiffEq Solution - Poiseuille Now we can focus our remaining efforts on the perturbation function.

15. Louisiana Tech University Ruston, LA 71272 Perturbation Function - Reduction We can approach this perturbation function by a separation of variables method, as it is homogeneous.

16. Louisiana Tech University Ruston, LA 71272 Perturbation Function - Separation Because each term is independent of the other term, the ONLY way this can be true is if each of the expressions is equal to a constant. Thus we define a constant as follows: Now we can use our boundary conditions to solve for these constants.

17. Louisiana Tech University Ruston, LA 71272 Perturbation Function – B.C.’s

18. Louisiana Tech University Ruston, LA 71272 Perturbation Function – B.C.’s

19. Louisiana Tech University Ruston, LA 71272 Perturbation Function – B.C.’s

20. Louisiana Tech University Ruston, LA 71272 Perturbation Function – Integration DID YOU CATCH THAT? This is a form of the Fourier Transform. Express a function as a series of sin and cosine terms, and then you can integrate and

21. Louisiana Tech University Ruston, LA 71272 Perturbation Function – Integration The textbook covers a way of calculating the shear stress. However, we have the stress tensor, so we can go to this tensor directly to calculate this from our equation above. You should be able to start spotting the similarities between our velocity equation, above, and the stress tensor on the left.

22. Louisiana Tech University Ruston, LA 71272 Discussion Why would it be useful to run an analysis like this? –Helps select critical design dimensions for a flow channel. –If there is a controlling dimension, we can design a workaround. Where else do you think they run this type of analysis in engineering?

23. Louisiana Tech University Ruston, LA 71272 Announcements Office hours today, let me know if you need them Tutorial lab tonight…will go over more problems and answer questions about the current assignment. New assignment to be posted soon.

24. Louisiana Tech University Ruston, LA 71272 QUESTIONS?