Trajectories

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

Fluid Change  A change in a property like pressure depends on the view.  In the Lagrangian view the total time derivative depends on position and time.  The Eulerian view uses just the partial derivative with time. Points in space are fixedPoints in space are fixed

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

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

Jacobian Tensor  A general coordinate transformation can be expressed as a tensor. Partial derivatives between two systemsPartial derivatives between two systems Jacobian N  N real matrixJacobian N  N real matrix Inverse for nonsingular Jacobians.Inverse for nonsingular Jacobians.  Cartesian coordinate transformations have an additional symmetry. Not generally true for other transformationsNot generally true for other transformations

Transformation Gradient  The components of a gradient of a scalar do not transform like a position vector. Inverse transformation Covariant behavior Position is contravariant  Gradients use a shorthand index notation.

Volume Element  An infinitessimal volume element is defined by coordinates. dV = dx 1 dx 2 dx 3dV = dx 1 dx 2 dx 3  Transform a volume element from other coordinates. components from the transformationcomponents from the transformation  The Jacobian determinant is the ratio of the volume elements. x1x1 x2x2 x3x3

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.

Streamlines  A streamline follows the tangents to fluid velocity. Lagrangian view Dashed lines at left Stream tube follows an area  A streakline (blue) shows the current position of a particle starting at a fixed point.  A pathline (red) tracks an individual particle. Wikimedia image

Rotational Flow  The curl of velocity measures rotation per unit area. Stokes’ theorem  Fluid with zero curl is irrotational. Transform to rotating system with zero curl Defines angular velocity next