LOCAL CONSERVATION EQUATIONS

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

2. , w Mass per area per time (kg/(m2 s) , v , u Flux of mass out (kg/s) = Flux of mass in (kg/s) = Net Flux of mass in ‘x’ = Net Flux of mass in ‘y’ = Net Flux of mass in ‘z’ =

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

4. From global mass conservation: Using the DIVERGENCE Theorem

5. local version of continuity equation

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

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

8. Use Divergence Theorem for tensors: to convert: Expanding the second term and combining:

9. 0 Local Momentum Equation Valid for a continuous medium (solid or liquid) For example, for x momentum:

10. 4 equations, 12 unknowns; need to relate variables to each other

11. 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.

12. Conservation of momentum also known as: Cauchy’s momentum equation 4 equations, 12 unknowns; need to relate flow field and stress tensor Relation between stress and strain rate For a fluid at rest, there’s only pressure acting on the fluid, and we can write: p is pressure and δij is Kronecker’s delta, which is 1 @ i = j, and 0 @ i = j ; The minus sign in front of p is needed for consistency with tensor sign convention σij is the “deviatoric” part of the stress tensor σijparameterizes the diffusive flux of momentum

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

14. Strain rates – strain, or deformation, consists of LINEAR and SHEAR strain change in length LINEAR or NORMAL STRAIN (u+ (∂u/ ∂x)δx) dt x @ t @ t + dt u+ (∂u/ ∂x)δx u A’ B’ δx A B Rate of change in length, per unit length (strain rate) is:

15. SHEAR STRAIN (u+ (∂u/ ∂y)δy) dt A C u+ (∂u/ ∂y)δy dα (v+ (∂v/ ∂x)δx)dt B u δx δy dβ v+ (∂v/ ∂x)δx v dt u dt v dα = CA / CB Shear strain rateis:

16. 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

17. VORTICITY (Rotation Rate) vs SHEAR STRAIN (u+ (∂u/ ∂y)δy) dt A C u+ (∂u/ ∂y)δy dα (v+ (∂v/ ∂x)δx)dt B u δx δy dβ v+ (∂v/ ∂x)δx v dt u dt v dα = CA / CB Shear strain is: Rotation rate is:

18. Vorticity is twice rotation rate

19. back to the momentum eq.:

20. 0 0 strain rate tensor

21. back to Cauchy’s momentum eq.: Navier-Stokes Equation(s)