1. No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics by Frank Bierbrauer

2. Updating Fluid Variables In SPH fluid variables f are updated through interpolation about a given point (x a,y a ) using information from surrounding points (x b,y b ). Each surrounding point is given a weight W ab with respect to the distance between point a and b.

3. Particle Deficiency Near a no-slip boundary there is a particle deficiency Any interpolation carried out in this region will produce an incorrect sum

4. Three Ways to Resolve the Particle Deficiency Problem 1.Insert fixed image particles outside the boundary a distance d I away from the boundary c.f. nearest fluid particle at distance d F 2.Insert fixed virtual particles within the fluid and in a direct line to the fixed image particles –Avoids creation of errors when fluid and image particles are not aligned 3. Co-moving image particles with d I = d F

5. 123123

6. Velocity Update Using Image Particles 1.Fixed image approach: u I = u F +(1+d I /d F )(u W - u F ) 2.Virtual image approach: u I = u V +(1+d I /d V )(u W - u V ) –Virtual velocities u V are created through interpolation

7. Velocity Update Using the Navier- Stokes Equations Update the velocity using the Navier-Stokes equations and a second order finite difference approximation to the velocity derivatives

8. At the no-Slip Wall (W) Navier-Stokes Equations Finite-Difference Approximation at the wall

9. Velocity Update Much of this reduces down as, in general, a no-slip wall has condition u W =(U 0,0). Therefore, at the wall, u t = u x = u xx = v= v x = v xx = 0

10. The Viscoelastic Case The equations are ( = 1,2) where

11. Further Reduction Using giving

12. At the Wall As well as

13. Non-Newtonian (elastic) Stress Only have the velocity condition u W = (U 0,0) as well as y =0

14. Must Solve Need u b and v b and W Need as well as S t and e.g.

15. Density Update Equation

16. Polymeric Stress Update Equations

19. Velocity Update Equations

20. Solution for u b and v b

21. If x = 0, 21 = 1, = S

22. Equivalent Newtonian Update Equations

23. Giving