Navier-Strokes Equation

• Generalized equations of motion of a real flow named after the inventors CLMH Navier and GG Stokes are derived from the Newton's second law
• Newton's second law states that the product of mass and acceleration is equal to sum of the external forces acting on a body.
• External forces are of two kinds-
  • one acts throughout the mass of the body ----- body force ( gravitational force, electromagnetic force)
  • another acts on the boundary----------------------   surface force (pressure and frictional force).

Objective - We shall consider a differential fluid element in the flow field (Fig. 24.1).  Evaluate the surface forces acting on the boundary of the rectangular parallelepiped shown below.

Fig. 24.1 Definition of the components of stress and their locations in a differential fluid element

• Let the body force per unit mass be

(24.6)

and surface force per unit volume be

(24.7)

• Consider surface force on the surface AEHD, per unit area,

[Here second subscript x denotes that the surface force is evaluated for the surface whose outward normal is the x axis]

• Surface force on the surface BFGC per unit area is

• Net force on the body due to imbalance of surface forces on the above two surfaces is

(since area of faces AEHD and BFGC is dydz)

(24.8)

• Total force on the body due to net surface forces on all six surfaces is

(24.9)

• And hence, the resultant surface force dF, per unit volume, is

(since Volume= dx dy dz)

(24.10)

• The quantities , and   are vectors which can be resolved into normal stresses denoted by and shearing stresses denoted by as

	(24.11)

The stress system has nine scalar quantities. These nine quantities form a stress tensor.