Page 377 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 361

These are known as the Navier-Stokcs Equations of motion for incompressible Newtonian
fluid. There are four unknown functions v l5 v^v^ and/? in the three equations. The fourth
equation is supplied by the continuity equation A = 0, i.e.,

or, in invariant form,

If all particles have their velocity vectors parallel to a fixed direction, the flow is said to be
a parallel flow or a uni-directional flow. Show that for parallel flows of an incompressible
linearly viscous fluid, the total normal compressive stress at any point on any plane parallel
to and perpendicular to the direction of flow is the pressure/?.
Solution. Let the direction of the flow be the jtj-axis, then

and from the equation of continuity,

Thus, the velocity field for a parallel flow is

For this flow,

thus,

Example 6.7.2

Let z-axis be pointing vertically upward and let