5. Viscous Layers

Question 1

Navier-Stokes Equations in 2D

u

Answer 1

ρDu/Dt = -∂p/∂x + μ∇²u

Question 2

Navier-Stokes Equations in 2D

w

Answer 2

ρDw/Dt = -∂p/∂z + μ∇²w - ρg

Question 3

Mass Conservation in 2D

u and w

Answer 3

∂u/∂x + ∂w/∂z = 0

Question 4

Viscous Layers

Assumption 1 - Length Scales

Answer 4

x~L and z~H
-and
H<

Question 5

Thin Layer Reynold’s Number

Answer 5

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

Question 6

Inertia&raquo_space; Viscous

Answer 6

-gives the shallow water equations

Question 7

Viscous&raquo_space; Inertia

Answer 7

-viscous thin layer equations

Question 8

Viscous Layers

Assumption 2 - Reynold’s Number

Answer 8

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

Question 9

The Lubrication Equations

Answer 9

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

Question 10

Viscous Layer Boundary Conditions

Answer 10

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

Question 11

Viscous Layer

Velocity Field Derivatino

Answer 11

• 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

Question 12

Viscous Layer

Velocity Field Equation

Answer 12

u = ρg/2μ ∂h/∂x (z²-2hz)

Question 13

Viscous Layer

Non-Linear Diffusion Equation Derivation

Answer 13

-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

Question 14

Viscous Layer

Non-Linear Diffusion Equation

Answer 14

∂h/∂t = ρg/3μ ∂/∂x[h³ ∂h/∂x]

Question 15

Viscous Layer

Non-Linear Diffusion Equation in Terms of q

Answer 15

∂h/∂t = -∂q/∂x

Question 16

Viscous Layer

q

Answer 16

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

Question 17

2D Lava Stream off Cliff

Outline

Answer 17

• 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

Question 18

2D Lava Stream off Cliff

Steps

Answer 18

• 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

Question 19

Release of Fixed Volume of Fluid

Governing Equations

Answer 19

∂h/∂t = ρg/3μ ∂/∂x[h³ ∂h/∂x]

Question 20

Release of Fixed Volume of Fluid

Boundary Conditions

Answer 20

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)

Question 21

Release of Fixed Volume of Fluid

How to solve?

Answer 21

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

Question 22

Similarity Solution

Steps

Answer 22

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

Question 23

Heat Diffusion Example

Boundary Conditions

Answer 23

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

Question 24

Heat Diffusion Example

Coordinate Transformatinos

Answer 24

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