# 5. Viscous Layers Flashcards

1
Q

Navier-Stokes Equations in 2D

u

A

ρDu/Dt = -∂p/∂x + μ∇²u

2
Q

Navier-Stokes Equations in 2D

w

A

ρDw/Dt = -∂p/∂z + μ∇²w - ρg

3
Q

Mass Conservation in 2D

u and w

A

∂u/∂x + ∂w/∂z = 0

4
Q

Viscous Layers

Assumption 1 - Length Scales

A

x~L and z~H
-and
H<

5
Q

Thin Layer Reynold’s Number

A

Re ~ Inertia / Viscous
~ [ρDu/Dt]/[μ∂²u/∂z²]
~ [ρU²/L][μU/H²]

6
Q

Inertia&raquo_space; Viscous

A

-gives the shallow water equations

7
Q

Viscous&raquo_space; Inertia

A

-viscous thin layer equations

8
Q

Viscous Layers

Assumption 2 - Reynold’s Number

A

Re &laquo_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
9
Q

The Lubrication Equations

A

0 = -∂p/∂x + μ∂²u/∂z²
0 = -∂p/∂z - ρg
∂u/∂x + ∂w/∂z = 0

10
Q

Viscous Layer Boundary Conditions

A

p=po at z=h
μ∂u/∂z = 0 at z=h
u=0 at z=0

11
Q

Viscous Layer

Velocity Field Derivatino

A
• 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
12
Q

Viscous Layer

Velocity Field Equation

A

u = ρg/2μ ∂h/∂x (z²-2hz)

13
Q

Viscous Layer

Non-Linear Diffusion Equation Derivation

A
```-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```
14
Q

Viscous Layer

Non-Linear Diffusion Equation

A

∂h/∂t = ρg/3μ ∂/∂x[h³ ∂h/∂x]

15
Q

Viscous Layer

Non-Linear Diffusion Equation in Terms of q

A

∂h/∂t = -∂q/∂x

16
Q

Viscous Layer

q

A

q = ∫ u dz
= ∫ ρg/2μ ∂h/∂x (z²-2hz) dz
= - ρg/3μ h³ ∂h/∂x
-where the integrals are from 0 to h

17
Q

2D Lava Stream off Cliff

Outline

A
• 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
18
Q

2D Lava Stream off Cliff

Steps

A
• 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
19
Q

Release of Fixed Volume of Fluid

Governing Equations

A

∂h/∂t = ρg/3μ ∂/∂x[h³ ∂h/∂x]

20
Q

Release of Fixed Volume of Fluid

Boundary Conditions

A
```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)```
21
Q

Release of Fixed Volume of Fluid

How to solve?

A
• numerically
• asymptotic / analytical analysis for t->∞
• we consider the latter using similarity theory
22
Q

Similarity Solution

Steps

A

1) scaling analysis
2) motivates new variables
3) recast problem
4) solve, and transform back to original coordinates

23
Q

Heat Diffusion Example

Boundary Conditions

A

-heat diffusion from a hot boundary at x=0
-boundary conditions:
θ(0,t) = θo
θ->0 as x->∞

24
Q

Heat Diffusion Example

Coordinate Transformatinos

A
```(x,t) -> (ζ,τ)
-where
t=τ
ζ=x[ϰt]^(-1/2)
-and
θ = f θo```