Find the solution to Stokes flow past a sphere where the far-field velocity satisfies the elongational flow defined as \(v_{x}=\dot{\gamma} x, v_{y}=\) \(\dot{\gamma} y\), and \(v_{z}=-2 \dot{\gamma} z\).

First verify that the continuity equation is satisfied. Then transform the velocity to spherical coordinates to show that the flow is symmetric if \(z\) is chosen as the direction of symmetry.

Now set up the boundary conditions for \(\psi\). In particular, show that only the \(Q_{2}\) term remains for the above far-field velocity. Complete the solution for \(\psi\) by boundary-condition fitting. What is the force on the sphere?