Navier-Stokes Equations Click here for shearing stress relationships.

Key Concepts: The basis for analysis of viscous fluid flow are the differential equations of

motion of the fluid (the Navier-Stokes equations). The elemental forces causing fluid acceleration include pressure forces, viscous forces, and gravitational forces.

In a Nut Shell: The general equations of linear momentum in differential form for an incompressible, viscous liquid (Newtonian model) are known as the Navier-Stokes equations. The forces included on a

differential element of liquid are the pressure force, the body force (gravity), and the shear force.

The convective terms of acceleration (such as u∂u/∂x and others) in the Navier-Stokes equations are nonlinear making them very difficult to solve analytically. However, analytical solutions can be found under certain, restrictive assumptions.

Navier-Stokes Equations using Rectangular Coordinates

ρ(∂u/∂t + u∂u/∂x + v∂u/∂y + w∂u/∂z) = - ∂P/∂x + ρg_x + μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²)

ρ(∂v/∂t + u∂v/∂x + v∂v/∂y + w∂v/∂z) = - ∂P/∂y + ρg_y + μ(∂²v/∂x² + ∂²v/∂y² + ∂²v/∂z²)

ρ(∂w/∂t + u∂w/∂x + v∂w/∂y + w∂w/∂z) = - ∂P/∂z + ρg_z + μ(∂²w/∂x² + ∂²w/∂y² + ∂²w/∂z²)

where ρ = the mass density of the liquid, μ = dynamic viscosity of the liquid

u,v,w = the x,y,z components of velocity of the liquid

∂P/∂x, ∂P/∂y, ∂P/∂z = pressure gradients in the x,y, and z-directions

g_x, g_y, g_z = body force (gravity) per unit mass in x,y, and z-directions

∂u/∂t, ∂v/∂t, ∂w/∂t = transient components of acceleration in x,y, and z-directions

∂u/∂x, ∂u/∂y, ∂u/∂z etc = velocity gradient terms

u∂u/∂x, v∂u/∂y, w∂u/∂z etc = convective components of acceleration

μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) = shear stress terms in the x-direction

μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) = shear stress terms in the y-direction

μ(∂²w/∂x² + ∂²w/∂y² + ∂²w/∂z²) = shear stress terms in the z-direction

For steady flow ∂u/∂t, ∂v/∂t , ∂w/∂t = 0. A similar set of Navier-Stokes equations is available in cylindrical coordinates. Click here for examples.

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