According to Fung, in Stoke's flow the pressure p satisfies the Laplace equation (nabla2 p = 0) and the velocity components satisfy the biharmonic equations (nabla4 u = 0). Does this mean that the velocity field becomes independent of the value of the dynamic viscosity? Thanks!

Stokes flow is the lead order term in an asymptotic expansion in terms of the Reynolds number R (in the limit R->0). Each succesive term in such an expansion is independant or R (and hence the kinematic viscosity) but the solution is not since it is the sum of these terms each of which has a Reynolds number number factor in from of it.