1. Louisiana Tech University Ruston, LA 71272 Lubrication/Thin Film & Peristaltic Flows Juan M. Lopez Lecture 10 BIEN 501 Wednesday, March 28, 2007

2. Louisiana Tech University Ruston, LA 71272 Nondimensionalizing Momentum balance (and mass and energy balances) often written in nondimensional form –What is the advantage of nondimensionalizing our equations in this way? Solutions are more general –Limits of integration are 0-1 regardless of what the characteristic length L 0 is Dimensionless groups result –Reynolds number, Ruark number, Strouhal number, many others for other balance equations –Provide insight into physics of the problem – relative importance of different effects

3. Louisiana Tech University Ruston, LA 71272 Limiting Cases Often can’t find an analytical solution to flow problems Limiting cases – can provide significant insight Dimensionless form of Navier-Stokes equations: What do we call the limiting case when N Re <<1? –Creeping flow How does this affect the Navier-Stokes equations? What are some examples of cases where such flows are important? –Microfluidics, flow in porous media, colloidal dispersions (small L 0 ) –Polymer processing (large  )

4. Louisiana Tech University Ruston, LA 71272 Creeping Flow Creeping Flow approximation is an example of scaling –By comparing the order of magnitude of terms we can make useful simplifications to complex equations –Resulting equations sometimes introduce some inconsistencies Creeping flow examples in text –Cone and Plate Viscometer –Screw Extruder –Flow past a Sphere –Melt spinning Lubrication flows (Thin Draining Films)…

5. Louisiana Tech University Ruston, LA 71272 Reference Videos Creeping FlowBoundary Layer

6. Louisiana Tech University Ruston, LA 71272 Lubrication Flows Liquid flows in long narrow channels and in thin films –Dominated by viscous stresses –Nearly unidirectional –Classic example: steady, 2D ( x,y ) flow in a thin channel or narrow gap between solid objects Pressure gradient much greater in x direction than y direction, therefore treat P as a function of x only x y

7. Louisiana Tech University Ruston, LA 71272 Lubrication Flows Similar to plane Poiseuille flow except: is a function of x instead of constant and v x =v x (x,y) instead of v x =v x (y) only Lubrication approximation also assumes: Additional information from continuity equation –Boundary conditions might be Pressure at two points x 0 and x 1 or v x or  v x  y specified at two values of y

8. Louisiana Tech University Ruston, LA 71272 Limiting Cases Other important limiting case, N Re >>1 –What are these called? Nonviscous or Stokes flows Reduces to

9. Louisiana Tech University Ruston, LA 71272 Lubrication Flows This relates to your textbook derivation in the following way:

10. Louisiana Tech University Ruston, LA 71272 Lubrication Flows Following the derivation, our B.C.s:

11. Louisiana Tech University Ruston, LA 71272 Lubrication Flows Now we find the flowrate:

12. Louisiana Tech University Ruston, LA 71272 Introduction to Liebnitz From your Appendix A.1.H, we see how to differentiate an integral. Our continuity equation is:

13. Louisiana Tech University Ruston, LA 71272 Introduction to Liebnitz There is a problem in the textbook derivation, that becomes apparent if we are to apply Liebnitz accurately. Because the terms on the RHS are dx, the LHS must be dy. This also is required in order to make the Q substitution.

14. Louisiana Tech University Ruston, LA 71272 Example 4.6 and Problem 4.11 We analyze our lubrication flow result on a simple geometry. This is a simplification of synovial fluid lubrication between joints.

15. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface Previously established,

16. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface We solve for the forces in the vertical and horizontal directions. We change the integration variable to h instead of x.

17. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface With W being the depth into the page for the lubrication flow, the normal force is found by:

18. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface Now, we can plug in some numbers: How large must sliding velocities be to support a normal component of weight of 1000 N for a fluid with a viscocity μ = 1 centipoise = 10 -3 Pa-s (water) vs. a fluid with viscosity μ = 10 poise = 1 Pa-s (viscous oil).

19. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface From the results above: First, the low viscosity fluid

20. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface Second, the high viscosity fluid: It becomes readily apparent that the change in viscosity has a great effect upon the feasibility of our system.

21. Louisiana Tech University Ruston, LA 71272 Lubrication – Sliding Surface Discussion: –Do you expect this to be a different reaction if a different fluid is used? –Let’s perform an experiment. Corn Starch Water Shear-Thickening Fluid

22. Louisiana Tech University Ruston, LA 71272 Announcements/Reminders HW 4 has been posted on blackboard. –It is for 2 weeks, so don’t panic! –Extra office hours have been worked in for the homework AND the exam preparation. HW 3 due this week (Friday) Office hours 1 hours shorter today (I have a meeting after lunchtime). We DO have tutorial lab tonight.

23. Louisiana Tech University Ruston, LA 71272 QUESTIONS?