Vector field solution for Brinkman equation in presence of disconnected spheres

This paper describes the solution of a vector function around disconnected spheres where the fields are governed by a hybrid between Stokes and Helmholtz equations. The governing relation known as Brinkman equation typically appears to represent the spatial variations in Darcy's flow in porous medium as well as in unsteady Stokesian hydrodynamics. The presented analysis provides a general solution technique of the aforementioned field equation assuming inhomogeneous Dirichlet conditions at the surface of the disconnected spheres and a decaying variation at infinity. The methodology relies on the expansions in multiple sets of vector basis functions corresponding to each sphere. The key result in the formulation is the mutual transformations between the basis functions of two such sets. This allows the derivation of the matrix relations coupling the unknown amplitudes with the given inhomogeneous boundary conditions. The presented mathematical theory is validated by complementing numerical calculations. Accordingly, the solution is constructed using the outlined method, and the error in the form of departure from the intended boundary condition is evaluated. This error vanishes very quickly with increasing number of basis solutions demonstrating high accuracy and exponential spectral convergence of the numerical scheme. The versatility of the method is also demonstrated by describing the flows under both steady and unsteady conditions around particles moving in porous and liquid medium, respectively.

Figure 1

Schematic diagram of a two-sphere system and a three-sphere system.

Figure 2

Spectral convergence for normal translation along $z$ with RMS error (left panel) and maximum error (right panel), where $\gamma$ is $\mathbf{i}$ (thin dotted line), 1 (thin solid line), $10\mathbf{i}$ (thick dotted line), and 10 (thick solid line). The rows represent different separations $s$ as displayed.

Figure 3

Spectral convergence for tangential translation along $x$ with RMS error (left panel) and maximum error (right panel) where $\gamma$ is $\mathbf{i}$ (thin dotted line), 1 (thin solid line), $10\mathbf{i}$ (thick dotted line), and 10 (thick solid line). The rows represent different separations $s$ as displayed.

Figure 4

Spectral convergence for tangential rotation along $x$ with RMS error (left panel) and maximum error (right panel), where $\gamma$ is $\mathbf{i}$ (thin dotted line), 1 (thin solid line), $10\mathbf{i}$ (thick dotted line), and 10 (thick solid line). The rows represent different separations $s$ as displayed.

Figure 5

Spectral convergence for normal rotation along $z$ with RMS error (left panel) and maximum error (right panel), where $\gamma$ is $\mathbf{i}$ (thin dotted line), 1 (thin solid line), $10\mathbf{i}$ (thick dotted line), and 10 (thick solid line). The rows represent different separations $s$ as displayed.

Figure 6

Spectral convergence for three-sphere system along $z$ with RMS error (left panel) and maximum error (right panel), where $\gamma$ is $\mathbf{i}$ (thin dotted line), 1 (thin solid line), $10\mathbf{i}$ (thick dotted line), and 10 (thick solid line). The first row and the second row represent two extreme spheres translating with the same phase and opposite phase, respectively.

Figure 7

Streamlines for flow field in a porous medium around two impermeable spheres moving towards each other ($\gamma =1$) with center-to-center separation $s=2.5*a$ and $s=2.25*a$.

Figure 8

Streamlines created by two oscillating spheres with opposite phase, where $k=\sqrt{i}$ and center-to-center equilibrium separation $s=2.5$ (left panel) and $s=2.25$ (right panel). The first row and the second row correspond to real part and imaginary part of the velocity field which represent the instantaneous flow at equilibrium and the extreme positions of the particles, respectively.