The derivation of the differential equation \(E^2(E^2\psi)=0\) is part of the steps in deriving Stoke’s law.

We have, in the previous section, derived the Navier-Stokes equation:

The terms and are attributed to inertia forces, while the terms and are due to hydrostatic forces and viscous forces respectively. For a very viscous fluid, the viscosity of the fluid is very high but fluid velocities are very low. Therefore, the non-inertia forces dominate the inertia forces and eq37 is reduced to:

Taking the curl on both sides of eq33,

Since the curl of the gradient of a function is zero and the viscosity of an incompressible fluid is constant,

Substituting eq14 where in the vector identity , we get:

Substitute eq33b in eq33a

Assuming that the fluid is flowing in the direction, eq3 becomes . Noting that the curl of a function in spherical coordinates is:

and substituting eq15 and eq16 in , we get:

###### Question

Show that .

###### Answer

Using the curl of a function in spherical coordinates,

Substituting eq35 in eq34 and working out the algebra, we have:

where .

Comparing eq35a with eq34,

Continuing with the algebra for eq35a, we get: