Fluids. 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.
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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
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
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
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
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.
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
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.
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.
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