Navier-Stokes Equations
• A mathematical description of fluid motion under the following assumptions:
– Constant density, ρ – Constant viscosity, µ – Continuity (incompressible flow)

∇ •u = 0
CHEM 520

Derivation of N-S
• Derivation based on force balance about a fluid element: – Net force = sum of forces • Convective force (convective transport of momentum), ∇ • ρ uu • Viscous stress forces (viscous forces), ∇ •τ • Pressure forces, ∇p z • External forces, e.g. gravity, ρ g

∂ ρ u = −∇ • ρ uu − ∇ •τ − ∇p + F ∂t ⇒ ∂ ρ u = −∇ • ρ uu − ∇ •τ − ∇p + ρ g ∂t

y

∆z ∆x ∆y

x

CHEM 520

Derivation of N-S τ = − µ∇ u

• Viscous stress tensor • For Newtonian fluid τ xx = µ − 2 + 2 3 ∇ • u   ∂x  τ yy = µ − 2 + 2 3 ∇ • u  ∂y
 ∂v    ∂w  τ zz = µ − 2 + 2 3 ∇ • u  ∂z    ∂u ∂v  τ xy = τ yx = − µ  +   ∂y ∂x     ∂u

τ   xx τ yx τ zx    τ = τ τ τ  xy yy zy τ   xz τ yz τ zz   

( ) ( )
1-D

( )

3-D

τ yx

 ∂u  = −µ    ∂y   

τ yz = τ zy = − µ   τ zx = τ xz = − µ 

 ∂v ∂w   + ∂z ∂y   

 ∂w ∂u  +   ∂x ∂z 

CHEM 520

Derivation of N-S
Viscous stress (recap) • Stress has unit of pressure. • It describes the local variation in velocity. • Viscous force = Stress x area • Force balance using stress:
– Viscous forces balance (1) convective force, (2) pressure force, and (3) external force. – τ ∆A = u•grad(u) ∆A + P ∆A + F ∆A

τ yx

 ∂u  = −µ    ∂y   

0

0
CHEM 520

Derivation of N-S
From force balance: Constant ρ and µ: Navier-Stokes eqt. Inviscid flow, ∇ • τ = 0 : Euler’s eqt.
D ρ u = −∇ •τ − ∇p + F Dt u r r r D ρ u =−∇•  −µ∇u  −∇p + F       Dt r r r D u = µ∇2u −∇p + u F ρ Dt D ρ u = −∇p + F Dt

CHEM 520

N-S equation summary
Assumptions: – Constant density, ρ – Constant viscosity, µ – Continuity (incompressible flow) Rate of change of