Navier-Stokes Equations in 2D
u
ρDu/Dt = -∂p/∂x + μ∇²u
Navier-Stokes Equations in 2D
w
ρDw/Dt = -∂p/∂z + μ∇²w - ρg
Mass Conservation in 2D
u and w
∂u/∂x + ∂w/∂z = 0
Viscous Layers
Assumption 1 - Length Scales
x~L and z~H
-and
H<
Thin Layer Reynold’s Number
Re ~ Inertia / Viscous
~ [ρDu/Dt]/[μ∂²u/∂z²]
~ [ρU²/L][μU/H²]
Inertia»_space; Viscous
-gives the shallow water equations
Viscous»_space; Inertia
-viscous thin layer equations
Viscous Layers
Assumption 2 - Reynold’s Number
Re «_space;1
- in the w Navier-Stokes equation inertia is negligible
- and viscous terms are negligible in comparison to pressure
- so the equation reduces to the equation for hydrostatic pressure
The Lubrication Equations
0 = -∂p/∂x + μ∂²u/∂z²
0 = -∂p/∂z - ρg
∂u/∂x + ∂w/∂z = 0
Viscous Layer Boundary Conditions
p=po at z=h
μ∂u/∂z = 0 at z=h
u=0 at z=0
Viscous Layer
Velocity Field Derivatino
- start with hydrostatic pressure
- sub in to the first lubrication equation
- integrate with respect to z, remember constant of integration
- impose boundary condition u=0 at z=0
- rearrange for u
Viscous Layer
Velocity Field Equation
u = ρg/2μ ∂h/∂x (z²-2hz)
Viscous Layer
Non-Linear Diffusion Equation Derivation
-incompressible and 2D so: ∂u/∂x + ∂w/∂z = 0 -depth integrate from 0 to h -apply the kinematic boundary condition at the free surface: w=Dh/Dt at z=h(x,t) -sub in u -rearrange
Viscous Layer
Non-Linear Diffusion Equation
∂h/∂t = ρg/3μ ∂/∂x[h³ ∂h/∂x]
Viscous Layer
Non-Linear Diffusion Equation in Terms of q
∂h/∂t = -∂q/∂x
Viscous Layer
q
q = ∫ u dz
= ∫ ρg/2μ ∂h/∂x (z²-2hz) dz
= - ρg/3μ h³ ∂h/∂x
-where the integrals are from 0 to h
2D Lava Stream off Cliff
Outline
- a 2D stream of lava flows steadily off a cliff at x=0
- the flux is q=Q, constant
- want to determine surface profile h(x)
- steady so ∂h/∂t = 0
2D Lava Stream off Cliff
Steps
- sub ∂h/∂t = 0 and q=Q in to the non-linear diffusion equation
- integrate with respect to x
- for boundary conditions, h=0 at x=0 is a good approximation to determine constant of integration
Release of Fixed Volume of Fluid
Governing Equations
∂h/∂t = ρg/3μ ∂/∂x[h³ ∂h/∂x]
Release of Fixed Volume of Fluid
Boundary Conditions
h=0 at x=xn(t) -where xn(t) is the position of the moving front -and ∫h(x,t)dx = V = constant -where the integral is from 0 to xn(t)
Release of Fixed Volume of Fluid
How to solve?
- numerically
- asymptotic / analytical analysis for t->∞
- we consider the latter using similarity theory
Similarity Solution
Steps
1) scaling analysis
2) motivates new variables
3) recast problem
4) solve, and transform back to original coordinates
Heat Diffusion Example
Boundary Conditions
-heat diffusion from a hot boundary at x=0
-boundary conditions:
θ(0,t) = θo
θ->0 as x->∞
Heat Diffusion Example
Coordinate Transformatinos
(x,t) -> (ζ,τ) -where t=τ ζ=x[ϰt]^(-1/2) -and θ = f θo
