# 5. Viscous Layers Flashcards

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