Question1
Adjust the solution process discussed in class for uniform slow viscous flow past a sphere to the case of flow past a circular cylinder.
a. start with the 3D Navier-Stokes equations for viscous flow. Non dimensionalize system, let Re → 0 and formulate the pde problem for the stream function in polar coordinates ψ(r,θ).
b. answer the pde problem using separation of variables by letting ψ(r,θ)=F(r)G(θ). Use the boundary conditions far away from the cylinder to decide a functional form for G(θ) and reduce the pde to an ode for F(r).
c. answer the ode for F(r). Show that the boundary conditions far away from the cylinder cannot be fulfilled.
d. Cancel the fastest growing term of F(r) and determine the velocity field. Is the answer appropriate near the cylinder?
e. The failure of the solution at infinity is known as Stokes' paradox. Give a probable description for why the solution fails.