The Taxonomy of Fluid Dynamics
The Cauchy Equation
• Valid for any fluid
r
∂ ρV
r
rr
r r
+ ∇ ⋅ ρVV = ρ g + ∇ ⋅ σ ij
•
∂t
The Navier-Stokes Equation
Approximations of the Navier-Stokes equation:
Creeping Flow
•
•
•
•
Linear
Density drops out
Very
low Re
r
r
∇P = µ∇ 2V
Inviscid Flow
• Nonlinear
• Viscous terms drop out
• Bernoulli equation valid
along streamlines
• Eulerrequation
r
DV
r
= −∇P + ρ g
• ρ
Dt
( )
(
)
• Validr for Newtonian fluid only
r
r
DV
r
= −∇P + ρ g + µ∇ 2V
• ρ
Dt
Irrotational Flow
• Linear
• Viscous terms drop out
• Bernoulli equation valid
everywhere
• Superposition valid
• Potential
r r rflow solutions
ζ = ∇ ×V = 0
• r r
V = ∇φ , ∇ 2φ = 0
Boundary Layer
• Nonlinear
• Some viscous terms drop
out
• Very high Re
∂u ∂v
+
=0
∂x ∂y
•
dU
∂u
∂u
∂ 2u
u +v
=U
+ν 2
∂x
∂y
∂y
dx