2. The Continuity Equation (incompressible)

Which flow model is CFD usually based on?

infinitesimal fluid element (PDE) and Eulerian (conservation form)

infinitesimal fluid element

PDE

Eulerian

conservation form

PDE + Eulerian

infinitesimal fluid element fixed in space with the fluid moving through it

continuity equation in the conservation form

net ṁ out of the element = time rate of decrease of mass inside the element

net mass flow rate out of the element

continuity equation in tensor/summation notation

continuity equation in vector notation

div (∇)

divergence operator

physical meaning of ∇

- gradients of velocity components along each direction
- time rate of change of the volume of a moving fluid element, per unit volume

continuity equation in the non-conservation form (vector notation)

substantial derivative

time rate of change following a moving fluid element

local derivative

time rate of change at a fixed point

convective derivative

time rate of change due to the movement of the fluid

incompressible continuity equation

incompressible continuity equation (tensor form)

incompressible continuity equation (vector form)

solenoidal

- no sources or sink
- divergence free

incompressible dilation equation

dilation is zero

dilation

local rate of change of fluid volume per unit volume