2. Viscoelasticity

3. While water and air are Newtonian, lots of other common stuff isn’t: –Blood, paint, and many others have nonlinear viscosity (the faster these fluids deform, the less viscous it becomes) –Silly putty, cornstarch in water (elastically resist fast changes, but flow eventually) –Gels, pastes, dough (can hold a shape, but mixes like a fluid --- not quite a solid) –Sand, powder, rubble (actually granular, but in bulk sometimes flow like fluids)

4. Solids Lower down on the list the materials could be seen as solids with plasticity Plasticity = permanent deformation = flow The question becomes: is the flowing part more important than the solid part? If so, might be worth simulating as a fluid with special solid-like properties

5. Regular Viscoelasticity See [Goktekin et al.’04] and [Irving’06] Idea: add another fluid variable, elastic strain  Encodes how much “memory” the fluid has of the state it wants to bounce back to –Percent stretched or sheared in axis directions Include another fluid force, like pressure, proportional to elastic strain gradient Track elastic strain as it moves and rotates with the fluid, make it decay to zero (“creep” - silly putty) or clamp it to some range (like gels and pastes)

6. Granular Materials Granular materials like sand are a little trickier They are visibly not a continuum, rather lots of tiny grains (think rigid bodies) in frictional contact But, if the # of grains is large, in most situations can be approximated as a continuum Flow laws come out of frictional contact laws

7. Mohr-Coulomb Basic continuum model: Mohr-Coulomb (continuum generalization of Coulomb friction) Sand remains rigid (or slightly elastic) if the pressure (normal force) is larger than some constant times the shear stress (tangential force) If it flows, it has a viscous term proportional to pressure (not rate of flow!)

8. Sand vs. Water Hydrostatic (not moving) water pressure increases linearly with depth –the bottom supports weight of all the water above it Sand pressure reaches a maximum: friction transfers load to walls –the bottom only supports some of the weight above it, and the walls the rest See this effect in silo failures and in hourglasses

9. Approximation! That said, for many piling-up cases, the water pressure is a good approximation to sand pressure Idea: compute incompressible flow as if for water, then add friction effects in at the end –Estimate tangential force needed to stop flow in grid cell –If pressure is large compared to that, mark cell as rigid –Rigidify connected groups of rigid cells (find translational and angular velocity) –Apply friction-like viscosity elsewhere

10. Movies Sand models collapsing

11. Thank You! Web site http://www.cs.ubc.ca/~rbridson/fluidsimulation will be updated with notes and code