Flow of Thin Fluid Films Math 451 Final Presentation Joe Brachocki Daniel Fong Jamila Hedhli Sanjay Muddam 1

Overview Motivation Experiment Explicit Scheme Experiment vs. Explicit Conclusion

Solitary Waves : Solitary waves exhibit strongly non-parabolic velocity profiles in front of the main hump. Waves of similar size repel monotonically. Discovered by the Scottish engineer named John Scott Russell (1808-1882).

Longwave Expansion Longwave expansion is used on the Linearized Navier-Stokes equations, at order α and α^2, to Observe the velocity.

KDV and K-S equations The Longwave Expansion is further derived/expanded until the KDV and K-S equations are reached. They differ in their assumptions KDV: Dominant Dispersion K-S: Negligible dispersion (small wave amplitude) epsilon -> 0 The characteristics of the solutions allow for checking of existence of solitary waves.

Incompressible Flow Assumption We did not use the K-S equation. For our thin film experiment, we assumed incompressibility of the fluid film. This allowed us to use the following version of the Navier-Stokes equation: From this equation, a non-dimensionless form was derived, similar to what was done during the process of reaching the K-S equation (just described)

K-S and Thin Film Equation Relation The K-S equation, derived from Navier Stokes is with epsilon -> 0. It has similar structure (e.g. 4th order) to the “Flow of Thin Film Equation”, also derived from Navier Stokes: Due to this similarity, we were motivated to do experiments / numerical simulations using the above equation to try and see if it would also show the existence of solitary waves.

Overview Motivation Experiment Explicit Scheme Experiment vs. Explicit Conclusion

Experiment Purpose : To see a solitary wave and compare it to our explicit and implicit scheme Fluid Used: Polydimethylsiloxane (PDMS) Method: The Art of Collecting Fluid Inclination: Regular, Vertically and Inverted Different angles: 90, 66.5, 45, 40, 35, 30, 20 Thickness of the Fluid Describe how to perform the experiment, then mention how did we collect the fluid. Talk about the thickest of the fluid, also talk about why capillary ridge forms. What should we expect from the different angles. SJ, I’m not sure how you want this to be done.

Experiment Equipments Describe how to perform the experiment, then mention how did we collect the fluid. Talk about the thickest of the fluid, also talk about why capillary ridge forms. What should we expect from the different angles. SJ, I’m not sure how you want this to be done.

Fluid Flows on Coated & Uncoated Slides Which Picture is Coated? Our code is based on zero contact angle, that’s why the graphs produced by the code match the uncoated slide better. Uncoated slides lead to zero contact angle while coated slides lead to relatively large contact angle

Fluid Flows Down a 45 Degree Incline (Coated Slide) I picked an angle, I just choose 45 degree incline since the movies are nicer.

Fluid Flows Down a 45 Degree Incline (Uncoated Slide)

Overview Theoretical Experiment Explicit Scheme Experiment vs. Explicit Conclusion

Explicit Scheme I Discretization of 4th Order Term * We assumed zero contact angle in our explicit scheme, therefore the numerical results agree better with the uncoated slide.

Explicit Scheme II Discretization of 2nd Order Term Discretization of the Last Term

Explicit Scheme III Time Discretization Capillary Ridge Constant Physical Time Space & Time Scale Physical Length * The biggest problem with explicit scheme is time step.

Explicit Scheme IV Boundary Condition No-slip BC Initial Condition 1 0.1

Plot of Fluid Flows Down a 45 Degree Incline

Overview Motivation Experiment Explicit Scheme Experiment vs. Explicit Conclusion

Comparison of Experimental and Explicit Result on a 45 Degree Incline (Coated Slide) Fluid height = 0.025 cm

Comparison of Experimental and Explicit Result on a 45 Degree Incline (Uncoated Slide) Fluid height = 0.025 cm

Overview Theoretical Experiment Explicit Scheme Experiment vs. Explicit Conclusion

Unexpected Experimental Result

Unexpected Results … Wave?

Unexpected Results … Wave? Time : 6 seconds

Conclusion We see signs that solitary waves may be present Still Running the Explicit Scheme and need to wait Once we get the implicit scheme working, it will be quicker to run the code. In the experiment, we saw some small humps but couldn’t determine if they were solitary waves. They have small amplitudes and travel at high speeds. Need to have a more accurate view. Saw a small “possible” hump in one movie.