Chapter 7, Problem 28P

In an oscillating incompressible flow field the force per unit mass acting on a fluid particle is obtained from Newton's second law in intensive form (see Prob. 7-19),

   $\frac{\stackrel{\to }{F}}{m}=\frac{\partial \stackrel{\to }{V}}{\partial t}+\left(\stackrel{\to }{V}.\stackrel{\to }{\nabla }\right)\stackrel{\to }{V}$

Suppose the characteristic speed and characteristic length for a given flow field are $V_{\infty }$and L, respectively. Also suppose that $\omega$is a characteristic angular frequency (rad/s) of the oscillation (Fig. P7-28). Define the following nondimensionalized variables,
   $t*=\omega t,\stackrel{\to }{x}*=\frac{\stackrel{\to }{x}}{L},\stackrel{\to }{\nabla }*=L\stackrel{\to }{\nabla },\text{and}\text{<mover accent="true"> <mi> V </mi> <mo> → </mo> </mover> *=}\frac{\stackrel{\to }{V}}{V_{\infty }}$

Since there is no given characteristic scale for the force per unit mass acting on a fluid panicle, we assign one, noting that $\left\{\stackrel{\to }{F}/m\right\}=\left\{L/t^2\right\}$ . Namely, we let
   $\left(\stackrel{\to }{F}/m\right)=\frac{1}{{\omega }^{2}L}\stackrel{\to }{F}/m$

Nondimensionalize the equation of motion and identify any established (named) dimensionless parameters that may ippear.