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Trajectories

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.

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Trajectories

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  1. Trajectories

  2. 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

  3. 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 fixed

  4. 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

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

  7. 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.

  8. Volume Element • An infinitessimal volume element is defined by coordinates. • dV = dx1dx2dx3 • Transform a volume element from other coordinates. • components from the transformation • The Jacobian determinant is the ratio of the volume elements. x3 x2 x1

  9. 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

  10. 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

  11. 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 Rotational Flow next

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