Chapter 7, Problem 22P

Newton's second law is the foundation for the differential equation of conservation of linear momentum (to be discussed in Chap. 9). In terms of the material acceleration following a fluid particle (Fig. P7-23), we write Newton's second law as follows:

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

Or, dividing both sides by the mass m of the fluid particle,
$\frac{\stackrel{\to }{F}}{m}=\frac{\partial \stackrel{\to }{V}}{\partial t}+\left(\stackrel{\to }{V}\cdot \stackrel{\to }{\nabla }\right)\stackrel{\to }{V}$

Write the primary dimensions of each additive term in the (second) equation, and verify that the equation is dimensionally homogeneous. Show all your work.