Page 58 - Computational Fluid Dynamics for Engineers

2.2 Navier-Stokes Equations 43

and write the Navier-Stokes equations for an incompressible three-dimensional
flow as follows:

Continuity equation
du dv dw
(2.2.1)
dx dy dz
x-component of the momentum equation
Du dp t (da xx t da xy da x (2.2.2)
Dt dx V 9x + dy + dz + Qfx
^/-component of the momentum equation

Dv dp da, yx da, yy da<
yz
+ + + Qfy (2.2.3)
Dt dy \ dx dy + dz
z-component of the momentum equation
Dw dp fda zx da zy da z
Q- = + + + + Qfz (2.2.4)
Dt -d~ z \ dx dy dz
dz
where DjDt represents the substantial derivative given by

D
() 9 0 , 3 0 , J( d() _d() (2.2.5)
Dt dt + u- dx dy + w dz dt + F - V (
Equations (2.2.2) to (2.2.4) make use of Newton's second law of motion with
their left-hand sides representing mass acceleration per unit volume and their
right-hand sides representing the sum of net forces per unit volume acting on
the fluid which consists of surface and body forces. Surface forces arise because
of molecular stresses in the fluid (such as pressure, p, which is present in a fluid
at rest and acts normal to a surface) and viscous stresses which act normal to
a surface or tangentially (shear stress). The first term on the right-hand side
of Eqs. (2.2.2)-(2.2.4) denotes the net pressure force per unit volume and the
minus sign arises because, by definition, a positive pressure acts inward. The

Fig. 2.1. Definitions of viscous stress components
applied to the faces of a control volume by the
surrounding fluid. Force components are stress
components multiplied by areas of corresponding
faces.