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 fixedPoints are fixed

5. Volume Change  Consider a fixed amount of fluid in a volume  V. Cubic, Cartesian geometryCubic, Cartesian geometry Dimensions  x,  y,  z.Dimensions  x,  y,  z.  The change in  V is related to the divergence. Incompressible fluids must have no velocity divergenceIncompressible fluids must have no velocity divergence

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

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

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