Creeping flow of a sphere nearby a cylinder

We present a three-dimensional solution of a sphere nearby an infinite cylinder at low Reynolds number. We utilize the Lamb’s general solution based on spherical harmonics and develop a framework based on cylindrical harmonics to solve the flow field around the sphere and outside the cylinder, respectively. The solution is solved semi-analytically by considering geometrical parameters, including sphere radius, sphere velocity, separation distance and cylinder radius. The drag force coefficients of the sphere which are dependent on the distance between the cylinder surface and the sphere, as well as the velocity contours in the vicinity of the sphere, are analyzed. We also provide an analytical formula to calculate the drag force. The analytical formula has good quantitative agreement with the semi-analytical solution when the radius of the cylinder is smaller than the sphere. Such analysis can give insights into the details of the complex interaction between the sphere and cylinder.