Stokes flow around a paraboloid of revolution: Analytical treatment in an unbounded domain

Stokes flow describes the motion of a Newtonian, incompressible fluid in regimes where inertial effects are negligible compared to viscous forces. In the case of axisymmetric flow and by employing a stream function ψ, the governing equation reduces to E4ψ=0, where E2 is a second order elliptic partial differential operator and E4=E2○E2. In this work, we derive the general solution of the equation E4ψ=0 in the parabolic coordinate system, which is given as series expansions of specific combinations of mixed-order Bessel and modified Bessel functions of first and second kinds or as a polynomial. This analytical framework is applied to study the Stokes flow around a rigid paraboloid solving a boundary value problem, accordingly. A truncated error analysis is conducted, proving that using at least eight terms of the obtained series expansion, the absolute error is less than 10−5. Sample streamlines are depicted with respect to the order of truncation. The obtained stream function expansion allows for the computation of key hydrodynamic quantities, such as the velocity and the pressure fields, offering insights into the behavior of the particular flow. It may also serve as a foundation for further investigations of problems in a wide range of areas, from chemical engineering to biomedicine.