Viscosity Flashcards
How many indices and components does a rank n tensor have?
n indices and 3^n components
Define isotropic (tensor)
A tensor whose components are unchanged under rotation of the coordinate axes
Give the Cauchy momentum equation
ρ Du/Dt = ρf + ∇ · σ^T
Under what condition for the stress tensor is angular momentum conserved?
Symmetric
Define Newtonian fluid
A fluid where the shear stress depends linearly on the velocity gradient.
Define kinematic viscosity
ν = µ/ρ0
Define the no-slip boundary condition
u = 0 on a solid boundary ∂V
How does incompressible viscous fluids (f = −∇U) affect conservation of KE
dE/dt != 0 = −µ INT_V |ω|^2 dV
Describe how viscosity affects vorticity
Viscosity cause vortices to decay and spread out
Give the vorticity equation with viscosity (E)
∂ω/∂t + ∇ ×(ω × u) = ν∆ω
Considering the Reynolds number, what do U and L denote?
U defines a characteristic flow speed |u| and L define a characteristic length scale
Define the Reynolds number
The dimensionless quantity Re = UL/ν
Give the low-Reynolds number limit
Viscosity dominates and we can neglect Du/Dt. The equations reduce to Stoke’s equations.
Define the stress tensor with viscosity. What do we impart on the pressure?
σij = −p δij + dij
p = -1/3 σii
dii = 0
What additional term to the momentum eq gives the incomp Navier-Stokes?
µ/ρ0 ∆u
