1.9 Equations for Newtonian Fluids

Substitution of equation (1.8.1) into equation (1.6.5) gives the result

When ? and are constant, and for incompressible flows, this simplifies greatly with the help of the continuity condition ? v = 0 to the vector form

where

is called the Laplace operator.

Equation (1.9.2) can be written in component notation as

Either form (1.9.2) or (1.9.3) is referred to as the Navier-Stokes equation.

The last form (1.9.3) is the form most useful for problem solving. Form (1.9.2) is useful for physical understanding and other manipulations of our equations. The Navier-Stokes equations in non-Cartesian coordinate systems are given in the Appendix.