Perturbation in Differential Equations: Solutions and Applications

1. Perturbation: Background • Algebraic • Differential Equations

2. Perturbed equation • Original Equation Perturbation

3. Perturbed equation • Answer can be in the form Perturbation • Change in result (absolute values) vs Change in equation • Simple (Regular) Perturbation

4. Perturbed equation • Original Equation Perturbation • Two roots instead of one • Roots are not close to the original root

5. Answer may NOT be in the form Perturbation • Change in result (absolute values) vs Change in equation • Other root varies from the original root dramatically, as epsilon approaches zero! • Singular perturbation

6. Solution Differential Equations SKIP • Perturbation-1 • Solution • Regular Perturbation

7. Solution • Perturbation-1 • Solution Differential Equations • Regular Perturbation

8. Differential Equations • Another Regular Perturbation • Perturbation-1a SKIP • Solution • Perturbation-1b • Solution

9. Differential Equations • Perturbation-2 • Exact Solution • (eg using Integrating factors method) • Singular Perturbation

10. Singular Perturbation Differential Equations • Can the solution be of the following form? (to satisfy the extra boundary condition?) • No! • Based on the perturbed equation

11. Differential Equations • At the limit • Method to find solution • Transform variables (x,y,e) • Called “Stretching Transformation” • Zooms in the ‘rapidly varying domain’ • Obtain “inner solution”

12. Outer Soln Inner solutions • Method to find solution Differential Equations • Inner solution: • Let e =0 and simplify eqn • 2nd order equation, satisfying only one boundary condition (x=0) • one constant remains arbitrary • Valid only near x=0 • Obtain outer solution, for first order equation, satisfying one Boundary Condition (x=1) • Valid everywhere, except near x=0

13. Outer Soln Inner solution Differential Equations • Method to find solution • Match the two solutions in the segment in between, by choosing the remaining constant • Match the value and the slopes • Close to the exact solution

14. “Real” solution “Real” solution Approx solution Approx solution Numerical Solution (to BL) • Grid generation • Structured grid vs unstructured grid • Uniform vs non-uniform grid • What about placing more grid everywhere? • More grid points near surface • Similar to “stretching transformation”

15. Boundary Layer theory • Situations we have seen so far • Laminar flow in cylinder • Fully developed (entrance effects are negligible) • Steady State • Unsteady State • Again, entrance effects are negligible • Movement of infinite plate, in a semi-infinite medium V0

16. Boundary Layer theory • Semi-infinite plate • Inviscid flow (irrotational) • Will NOT satisfy ‘no slip’ condition at the plate Fluid Velocity V0 • Flow over cylinder (inviscid) • Flow over sphere (2D, viscous flow) (tutorial problem) • Flow over any other shape (while accounting for no-slip condition and not assuming fully developed flow) is treated with “boundary layer theory”

17. Boundary Layer theory • Semi-infinite plate • Away from plate, inviscid solution is valid (and will satisfy the boundary condition). This is “outer solution” • Near the plate, different solution (including viscosity) will be found using ‘stretching transformation’. [Inner Solution] • Inner solution will satisfy the boundary condition (no slip) • Match both solutions to find the other constant Fluid Velocity V0

18. Velocity V0 INF INF INVISCID FLOWASSUMPTION OK HERE Velocity V0 No Slip FRICTION CANNOT BENEGLECTED HERE Velocity 0 0 0 Boundary Layer theory • Solid Boundary

19. INF INF d d 0 0 Boundary Layer theory B L thickness 99% Free Stream Velocity • Solid Boundary What happens to d when you move in x? x B L thickness increases with x Momentum Transfer

20. d d Boundary Layer theory • Draw d vs x • Analytical Expression, for velocity vs (x,y), below BL: • Continuity • Navier Stokes Equation y x B L thickness increases with x

21. Hence 1 Boundary Layer theory • Steady,incompressible, two dimensional (semi infinite plate)

22. N-S Eqn • Consider only X and Y equations (2D assumption) • Steady flow, gravity can be incorporated in Pressure term (or assume gravity is in Z direction, for example) • Vz=0, Vx and Vyare not functions of z

23. N-S Eqn • Obtain “order of magnitude” idea • Can be used to ignore small terms (simplify eqn by removing ‘regular’ perturbations) • Can be used to non-dimensionalize equations • example: • Steady State

24. N-S Eqn • Write the NS-eqn in “usual” form, for steady state

25. d y x L N-S Eqn • What are the relevant scales for the lengths (eg what are L1, L2 in this particular case?

26. ~ means “Order of ” • Note: Some books show it as y x • Similarly N-S Eqn • What are the relevant scales for the velocity? • Vx varies from 0 to Vo (or we can call it VINF)

27. Note: The sign is not important here • Continuity 1 y x N-S Eqn • What are the relevant scales for the derivatives?

28. Thin Boundary Layer assumption d y x L N-S Eqn • What are the relevant scales for the derivatives?

29. From Bernoulli’s eqn d y Claim:as m -->0, x L N-S Eqn • Can we approximate pressure drop? • Assume that pressure drop is similar to inviscid flow

30. Each term is small compared to the equivalent in X-eqn • ==> N-S Eqn • For the Y component of N-S equation

31. Prandtl BL eqn (steady state) Unsteady State

32. d y x L Prandtl BL eqn : flow over Flat plate • No pressure Drop • Steady State • 2D-flow (Stream Function) • Stretching Transformation (near the boundary) • Non-dimensionalize y

33. d y x L Prandtl BL eqn : flow over Flat plate • Another perspective for the choice of d • If we write the BL eqn in stream function • Boundary Conditions

34. d y x L Prandtl BL eqn : flow over Flat plate

35. d y x L Prandtl BL eqn : flow over Flat plate Some books may have -ve sign, or a factor of 2, in the equation, depending on the definition of Stream function and transformations used

36. 1 0 0 3 Prandtl BL eqn : flow over Flat plate • Boundary Conditions: • No solution in ‘usual’ form • Blasius Solution: Series solution, valid for small h • For large h, asymptotic series that matches with the boundary condition • Numerical values tabulated (f,f’,f’’...) • Plot of Vx/VINF vs h • Note: definition of h may be slightly different in various books (usually by a factor of 2)

37. Prandtl BL eqn : flow over Flat plate • Blasius Solution • Valid for high Reynolds Number • Re • Local:( dVr/m) • More useful (convenient): (X V r/m ) (sometimes, this is referred to as “local” Reynolds number) • 105 or more • Not valid very near x= 0 (at the point x=0,y=0) • Another way to express boundary layer thickness • Reynolds number high ==> Boundary layer is thin

38. Prandtl BL eqn : flow over Flat plate • Boundary layer thickness • Drag estimate • Other definitions (for thickness) • Similarity • Effect of pressure variation (Loss of similarity and separation) • Thermal vs momentum Boundary Layer • von Karman method

39. References: • Introduction to Mathematical Fluid Dynamics by Richard E Meyer • Perturbation methods in fluid dynamics, by Van Dyke • BSL • 3W&R • Fluid flow analysis by Sharpe • Introduction to Fluid Mechanics by Fox & McDonald