In this work, we intend to present a solution to a partial differential equation (PDE) model developed in the bipolar coordinate system that finds applications in transport phenomena. The PDE taken is the Navier Stokes equation representing the steady flow of a viscous fluid past a pair of separated fluid spheres (termed as a two-sphere problem). The equation and appropriate boundary conditions are solved using the method of separation of variables, and we derived analytical expressions for the flow variables. We carried out numerical evaluations and depicted the flow configuration and pressure distribution for different sets of model parameters. Our analysis’s conclusions are prospective of finding applications in fluid-flow problems (transport phenomena) in industrial engineering. For instance, we compared the pressure distribution in the two-sphere problem with that of a single sphere problem. The results showed that when the fluid’s viscosity takes one-tenth of its original value, pressure in the vicinity of the sphere in the case of the single-sphere problem also reduced to one-tenth of its original value. However, in the case of two spheres, the decrease in pressure is significant, with the value being up to 15 times the original value. This result and others derived from our study foresee a possible means of pressure regulation around objects in fluid flow problems.