LOCAL CONSERVATION EQUATIONS From global conservation of mass: A Apply this to a small fixed volume

Flux of mass in (kg/s) = Flux of mass out (kg/s) = Net Flux of mass in ‘x’ = Net Flux of mass in ‘y’ = Net Flux of mass in ‘z’ =, u, w, v Mass per area per time (kg/(m 2 s)

Net Flux of mass in x, y and z = DIVERGENCE Theorem DIVERGENCE Theorem – relates integral over a volume to the integral over a closed area surrounding the volume DIVERGENCE Theorem Other forms of the DIVERGENCE Theorem θ is any scalar for any tensor

From global mass conservation:Using the DIVERGENCE Theorem

local version of continuity equation

If the density of a fluid parcel is constant Local conservation of mass fluid reacts instantaneously to changes in pressure - incompressible flow

CONSERVATION OF MOMENTUM Momentum Theorem Normal (pressure) and tangential (shear) forces in tensor notation:

Use Divergence Theorem for tensors: to convert: Expanding the second term:

0 Local Momentum Equation Valid for a continuous medium (solid or liquid) For example, for x momentum:

4 equations, 12 unknowns; need to relate variables to each other

Simulation of wind blowing past a building (black square) reveals the vortices that are shed downwind of the building; dark orange represents the highest air speeds, dark blue the lowest. As a result of such vortex formation and shedding, tall buildings can experience large, potentially catastrophic forces.

Conservation of momentum also known as: Cauchy’s equation Relation between stress and strain rate 4 equations, 12 unknowns; need to relate flow field and stress tensor For a fluid at rest, there’s only pressure acting on the fluid, and we can write: p is pressure and δ ij is Kronecker’s delta, which is i = j, and i = j ; The minus sign in front of p is needed for consistency with tensor sign convention σ ij is the “deviatoric” part of the stress tensor parameterizes the diffusive flux of momentum

For an incompressible Newtonian fluid, the deviatoric tensor can be written as: Another way of representing the deviatoric tensor, a more general way, is: And for incompressible flow: Strain rate tensor For instance:

back to the momentum eq.:

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Navier-StokesEquation(s)

Strain rates – strain, or deformation, consists of LINEAR and SHEAR strain Rate of change in length, per unit length is: u u+ (∂u/ ∂x)δx u dt LINEAR or NORMAL STRAIN AB t + t δxδx (u+ (∂u/ ∂x)δx) dt

SHEAR STRAIN u v+ (∂v/ ∂x)δx u dt B δxδx (u+ (∂u/ ∂y)δy) dt δyδy u+ (∂u/ ∂y)δy v v dt (v+ (∂v/ ∂x)δx)dt dαdα dβdβ C A Shear strain is: dα = CA / CB

LINEAR and SHEAR strains can be used to describe fluid deformation In terms of the STRAIN RATE TENSOR: the diagonal terms are the normal strain rates the off-diagonal terms are half the shear strain rates This tensor is symmetric

VORTICITY (Rotation Rate) vs SHEAR STRAIN u v+ (∂v/ ∂x)δx u dt B δxδx (u+ (∂u/ ∂y)δy) dt δyδy u+ (∂u/ ∂y)δy v v dt (v+ (∂v/ ∂x)δx)dt dαdα dβdβ C A Shear strain is: dα = CA / CB Vorticity is: