Page 421 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 405

Now suppose that the fluid is perturbed from rest such that

Substituting Eq, (6.27.3) into Eq. (6.27.1),

Since we assumed infinitesimal disturbances, the terms vy'(dv/'/cbty) andp'/p 0 are negligible
and the equations of motion now take the linearized form

In a similar manner, we consider the mass conservation equation

and obtain the linearized equation

Differentiating Eq. (6.27.4) with respect to jt/and Eq. (6.27.5) with respect to ?, we
eliminate the velocity to obtain

We further assume that the flow is barotropic, i.e., the pressure depends explicitly on density
only, so that the pressure/? = p(p}. Expandingp(p} in a Taylor series about the rest value of
pressure p 0, we have

Neglecting higher-order terms

where