Fluids

1. Fluids

2. Eulerian View • In a Lagrangian view each body is described at each point in space. • Difficult for a fluid with many particles. • In an Eulerian view the points in space are described. • Bulk properties of density and velocity

3. Compressibility • A change in pressure on a fluid can cause deformation. • Compressibility measures the relationship between volume change and pressure. • Usually expressed as a bulk modulus B • Ideal liquids are incompressible. V p

4. Fluid Change • A change in a property like pressure depends on the view. • In a Lagrangian view the total time derivative depends on position and time. • An Eulerian view is just the partial derivative with time. • Points are fixed

5. Consider a fixed amount of fluid in a volume dV. Cubic, Cartesian geometry Dimensions dx, dy, dz. The change in dV is related to the divergence. Incompressible fluids must have no velocity divergence Volume Change

6. A mass element must remain constant in time. Conservation of mass Combine with divergence relationship. Write in terms of a point in space. Continuity Equation

7. Stress • A stress measures the surface force per unit area. • A normal stress acts normal to a surface. • A shear stress acts parallel to a surface. • A fluid at rest cannot support a shear stress. A A

8. Force in Fluids • Consider a small prism of fluid in a continuous fluid. • Describe the stress P at any point. • Normal area vectors S form a triangle. • The stress function is linear.

9. Represent the stress function by a tensor. Symmetric Specified by 6 components If the only stress is pressure the tensor is diagonal. The total force is found by integration. Stress Tensor

10. Force Density • The force on a closed volume can be found through Gauss’ law. • Use outward unit vectors • A force density due to stress can be defined from the tensor. • Due to differences in stress as a function of position next