For the case of a sphere in a uniform far field flow, it is advantageous to use a cylindrical coordinate system . The –axis is through the centre of the sphere and aligned with the mean flow direction, while  is the radius as measured perpendicular to the –axis. The origin is at the sphere centre. Because the flow is axisymmetric around the –axis, it is independent of the azimuth .
In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function , depending on  and :
with  and  the flow velocity components in the  and  direction, respectively. The azimuthal velocity component in the –direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surface of some constant value , is equal to  and is constant.
For this case of an axisymmetric flow, the only non-zero component of the vorticity vector  is the azimuthal –component 
The Laplace operator, applied to the vorticity , becomes in this cylindrical coordinate system with axisymmetry:
From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity  in the –direction and a sphere of radius , the solution is found to be
The solution of velocity in cylindrical coordinates and components follows as:
The solution of vorticity in cylindrical coordinates follows as:
The solution of pressure in cylindrical coordinates follows as: 
The solution of pressure in spherical coordinates follows as: 
The formula of pressure is also called dipole potential analogous to the concept in electrostatics. 
A more general formulation, with arbitrary far-field velocity-vector , in cartesian coordinates  follows with:
In this formulation the non-conservative term represents a kind of so-called Stokeslet. The Stokeslet is the Green's function of the Stokes-Flow-Equations. The conservative term is equal to the dipole gradient field. The formula of vorticity is analogous to the Biot–Savart law  in electromagnetism.
The following formula describes the viscous stress tensor for the special case of Stokes flow. It is needed in the calculation of the force acting on the particle. In Cartesian coordinates the vector-gradient  is identical to the Jacobian matrix. The matrix  represents the identity-matrix. 
The force acting on the sphere is to calculate by surface-integral, where  represents the radial unit-vector of spherical-coordinates: