TY - JOUR

T1 - Vector field solution for Brinkman equation in presence of disconnected spheres

AU - Liu, Bo

AU - Bhattacharya, S.

N1 - Funding Information:
This work is supported by NSF Grant No. CBET-1805930.
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/10

Y1 - 2020/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85093362677&partnerID=8YFLogxK

U2 - 10.1103/PhysRevFluids.5.104303

DO - 10.1103/PhysRevFluids.5.104303

M3 - Article

AN - SCOPUS:85093362677

VL - 5

JO - Physical Review Fluids

JF - Physical Review Fluids

SN - 2469-990X

IS - 10

M1 - 104303

ER -