Navier-Stokes equations

From Chemepedia

The Navier-Stokes equations describe the motion of Newtonian fluids of constant density ρ and constant density μ.

Equation of Motion and derivation

The Navier-Stokes equations are derived from specific cases of the equation of Motion. The equation of motion in the material derivative form can be written as


Assuming incompressible Newtonian fluid, we arrive at the Navier-Stokes Equations


where the symbol represents the material derivative. The del operator, , is independent of the coordinate system. Each term in the above equation has the units of a "body force" (force per unit volume). This equation can be interpreted in the context of Newton's second law of motion: the acceleration of a fluid packet (or the material derivative) is equal to the sum of forces acting on the fluid packet, which potentially arise from (1) a pressure differential, (2) a viscous force, and (3) a body force.

For the full derivation see the Equation of motion

Cartesian coordinates

Defining the velocity field as

Cylindrical coordinates

Defining the velocity field as

Spherical coordinates

Defining the velocity field as

Common boundary conditions

  • No slip boundary condition: At the boundary or surface of a wall, the layer of fluid adjacent to the boundary has zero velocity relative to the boundary. If the wall is stationary, this means the velocity of the fluid must be
  • Axisymmetric boundary condition: This is where the geometry is symmetric to the flow profile (such as in pipe flow). Along the axis of symmetric, the gradient in the velocity field must be zero, or
  • Interfacial boundary condition between two fluids: At the boundary of two fluids, both the velocity and shear stress must be continuous, i.e. and