Chapter 17.5, Problem 61E

Navier-Stokes equation The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the flow in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is

     $\rho \left(\frac{\partial V}{\partial t}+\left(V\cdot \nabla \right)V\right)=-\nabla p+\mu \left(\nabla \cdot \nabla \right)V.$

In this notation, V = (u, v, w) is the three-dimensional velocity field, p is the (scalar) pressure, ρ is the constant density of the fluid, and μ is the constant viscosity. Write out the three component equations of this vector equation. (See Exercise 40 for an interpretation of the operations.)