1.10 Boundary Conditions

To obtain a solution of the Navier-Stokes equations that suits a particular problem, it is necessary to add conditions that need to be satisfied on the boundaries of the region of interest. The conditions that are most commonly encountered are the following:

1. The fluid velocity component normal to an impenetrable boundary is always equal to the normal velocity of the boundary. If n is the unit normal to the boundary, then

   
   

   on the boundary. If this condition were not true, fluid would pass through the boundary. This condition must hold true even in the case of vanishing viscosity ( inviscid flows ).

   If the boundary is moving, as in the case of a flow with a free surface or moving body, then, with F( x, t) = 0 as the equation of the bounding surface, (1.10.1) is satisfied if

   
   

   This condition is necessary to establish that F = 0 is a material surface that is, a surface moving with the fluid that always contains the same fluid particles. An important special case of the material surface is the free surface, a constant pressure surface, typically the interface between a liquid and a gas.

2. Stress must be continuous everywhere within the fluid. If stress were not continuous, an infinitesimal layer of fluid with an infinitesimal mass would be acted upon by a finite force, giving rise to infinite acceleration of that layer.

   At interfaces where fluid properties such as density are discontinuous,...