Abstract
We consider low‐Reynolds‐number axisymmetric flow about two spheres using a novel, biharmonic stream function. This enables us to calculate analytically not only the forces, but also the dipole moments (stresslets and pressure moments) and the associated resistance functions. In this paper the basics properties of axisymmetric flow and the stream function are discussed. Explicit series expansions, obtained by separation in bispherical coordinates, will be presented in a follow‐up paper.
Highlights
The grand resistance and mobility tensors describe the hydrodynamic interaction between rigid bodies suspended in a fluid medium and play an all-important role in colloidal science 1–7
They express the linear relationship between the Cartesian force multipole moments exerted by the particles on the fluid and the gradients of the ambient flow velocity taken at the particle centers
In the special case of just two spherical bodies, owing to O 2 -symmetry about the line connecting the particle centers, the full tensors can be reduced to a set of scalar resistance and mobility functions 8–15
Summary
The grand resistance and mobility tensors describe the hydrodynamic interaction between rigid bodies suspended in a fluid medium and play an all-important role in colloidal science 1–7 They express the linear relationship between the Cartesian force multipole moments exerted by the particles on the fluid and the gradients of the ambient flow velocity taken at the particle centers. Problem I can be reduced to a harmonic equation for the azimuthal velocity with Dirichlet boundary conditions This was solved by separation in bispherical coordinates almost hundred years ago by Jeffery 29 , who calculated, in particular, the torques acting on two spheres that rotate with given angular velocities around the line of centers. We present the theory of the biharmonic stream function for axisymmetric flow, while in the follow-up paper, we derive the series expansions for the forces and dipole moments in bispherical coordinates
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