# zbMATH — the first resource for mathematics

Boundary integral solutions of coupled Stokes and Darcy flows. (English) Zbl 1188.76232
This work presents an accurate computational method based on the boundary integral formulation for solving boundary value problems for Stokes and Darcy flows. The method is applied to problems where the equations are coupled across the interface through appropriate boundary conditions. First, the adopted technique reformulates singular integrals of the boundary integral formulation for the fluid flows as single and double layer potentials. Then the layer potentials are regularized and discretized using standard quadratures. At the final step, the leading term in the regularization error is eliminated in order to increase the order of accuracy.
Various presented test cases show that the results are consistent with theoretical predictions. In the first example, the authors compute the Stokes flow in a channel driven by a uniform pressure gradient. Assuming the porous lower boundary and imposing the Beavers-Joseph-Saffman slip boundary condition, the authors obtain a velocity profile that is non-zero on the boundary. Using corrected expressions, the slip velocity can be calculated more accurately than with uncorrected expressions. The error as a function of regularization parameter $$\delta$$ reduces in magnitude, and the convergence rate increases from $$O(\delta)$$ to $$O(\delta^{2})$$. Then the authors test the Darcy solution by prescribing an exact velocity on a circle. The kernels in the Darcy velocity have singularities of higher order, and the authors indicate numerical errors in the uncorrected case as large as 20% in velocity and 70% in pressure. The corrections reduce the error by high factors of 100–1000, and the accuracy increases by one order.
The solved equations are steady-state, but once the fluid velocity is known, the position of the Lagrangian particles can be updated in time, thus extending the method to time-dependent problems.

##### MSC:
 76M15 Boundary element methods applied to problems in fluid mechanics 76D07 Stokes and related (Oseen, etc.) flows 76S05 Flows in porous media; filtration; seepage
Full Text:
##### References:
 [1] Ainley, J.; Durkin, S.; Embid, R.; Boindala, P.; Cortez, R., The method of images for regularized stokeslets, J. comput. phys., 227, 4600-4616, (2008) · Zbl 1388.76060 [2] Alvarez, P.J.; Illman, W.A., Bioremediation and natural attenuation: process fundamentals and mathematical models, (2006), Wiley Interscience [3] Beale, J.T., A convergent boundary integral method for three-dimensional water waves, Math. comp., 70, 977-1029, (2001) · Zbl 0980.76053 [4] Beale, J.T., A grid-based boundary integral method for elliptic problems in three dimensions, SIAM J. numer. anal., 42, 599-620, (2001) · Zbl 1159.65368 [5] Beale, J.T.; Lai, M.-C., A method for computing nearly singular integrals, SIAM J. numer. anal., 38, 1902-1925, (2001) · Zbl 0988.65025 [6] Beale, J.T.; Layton, A.T., On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. appl. math. comput. sci., 1, 91-119, (2006) · Zbl 1153.35319 [7] Bear, J.; Verruijt, A., Modeling groundwater flow and pollution, (1987), Riedel Publ. [8] Beavers, G.S.; Joseph, D.D., Boundary conditions at a naturally permeable wall, J. fluid mech., 30, 197-207, (1967) [9] Broday, D.M., Motion of nanobeads proximate to plasma membranes during single particle tracking, Bull. math. biol., 64, 531-563, (2002) · Zbl 1334.92028 [10] Burman, E.; Hansbo, P., A unified stabilized method for stokes’ and darcy’s equations, J. comput. appl. math., 198, 35-51, (2007) · Zbl 1101.76032 [11] Chwang, A.T.; Wu, T.Y.-T., Hydromechanics of low-Reynolds-number flow, part 2, singularity method for Stokes flows, J. fluid mech., 67, 787-815, (1975) · Zbl 0309.76016 [12] Cisneros, L.; Cortez, R.; Dombrowski, C.; Goldstein, R.; Kessler, J., Fluid dynamics of self-propelled micro-organisms, from individuals to concentrated populations, Exp. fluids, 43, 737-753, (2007) [13] Colton, D.; Kress, R., Inverse acoustic and electromagnetic scattering theory, (1998), Springer-Verlag Berlin · Zbl 0893.35138 [14] Cortez, R., The method of regularized stokeslets, SIAM J. sci. comput., 23, 4, 1204-1225, (2001) · Zbl 1064.76080 [15] Cortez, R.; Fauci, L.; Medovikov, A., The method of regularized stokeslets in three dimensions: analysis, validation, and application to helical swimming, Phys. fluids, 17, 1-14, (2005) · Zbl 1187.76105 [16] Daripa, P.; Palaniappan, D., Singularity induced exterior and interior Stokes flows, Phys. fluids, 13, 11, 3134-3154, (2001) · Zbl 1184.76121 [17] Discacciati, M.; Miglio, E.; Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. numer. math., 43, 57-74, (2002) · Zbl 1023.76048 [18] Elasmi, L.; Feuillebois, F., Green function for a Stokes flow near a porous slab, Z. angew. math. mech., 81, 11, 743-752, (2001) · Zbl 1015.76017 [19] Flores, H.; Lobaton, E.; Méndez-Diez, S.; Tlupova, S.; Cortez, R., A study of bacterial flagellar bundling, Bull. math. biol., 67, 137-168, (2005) · Zbl 1334.92029 [20] J. Galvis, M. Sarkis, Balancing domain decomposition methods for mortar coupling Stokes-Darcy systems, Lecture notes in Computational Science and Engineering, 2006. · Zbl 1430.76440 [21] Guest, J.K.; Prevost, J.H., Topology optimization of creeping fluid flows using a darcy – stokes finite element, Int. J. numer. meth. eng., 66, 461-484, (2006) · Zbl 1110.76310 [22] Hanspal, N.; Waghode, A.; Nassehi, V.; Wakeman, R., Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations, Trans. porous med., 64, 73-101, (2006) [23] Higdon, J.J.L., The generation of feeding currents by flagellar motions, J. fluid mech., 94, 305-330, (1979) · Zbl 0423.76100 [24] Higdon, J.J.L., A hydrodynamic analysis of flagellar propulsion, J. fluid mech., 90, 685-711, (1979) · Zbl 0412.76097 [25] Jager, W.; Mikelić, A., On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. appl. math., 60, 1111-1127, (2000) · Zbl 0969.76088 [26] Keller, J.B.; Rubinow, S.I., Slender-body theory for slow viscous flow, J. fluid mech., 75, 705-714, (1976) · Zbl 0377.76036 [27] Kress, R., Linear integral equations, (1999), Springer-Verlag New York [28] Layton, W.J.; Schieweck, F.; Yotov, I., Coupling fluid flow with porous media flow, SIAM J. numer. anal., 40, 6, 2195-2218, (2003) · Zbl 1037.76014 [29] Lighthill, J., Mathematical biofluid dynamics, (1975), SIAM Philadelphia [30] Lighthill, J., Flagellar hydrodynamics, SIAM rev., 18, 161-230, (1976) · Zbl 0366.76099 [31] Lighthill, J., Helical distributions of stokeslets, J. eng. math., 30, 35-78, (1996) · Zbl 0883.76099 [32] Liron, N.; Mochon, S., Stokes flow for a stokeslet between two parallel flat plates, J. eng. math., 10, 287-303, (1976) · Zbl 0377.76030 [33] Mardal, K.A.; Tai, X.-C.; Winther, R., A robust finite element method for darcy – stokes flow, SIAM J. numer. anal., 40, 1605-1631, (2002) · Zbl 1037.65120 [34] Masud, A.; Hughes, T.J.R., A stabilized mixed finite element method for Darcy flow, Comput. meth. appl. mech. eng., 191, 4341-4370, (2002) · Zbl 1015.76047 [35] Mayo, A., The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. numer. anal., 21, 2, 285-299, (1984) · Zbl 1131.65303 [36] Phillips, O.M., Flow and reactions in permeable rocks, (1991), Cambridge University Press Cambridge [37] Pozrikidis, C., Boundary integral and singularity methods for linearized viscous flow, (1992), Cambridge University Press Cambridge · Zbl 0772.76005 [38] Pozrikidis, C., On the transient motion of ordered suspensions of liquid drops, J. fluid mech., 246, 301-320, (1993) · Zbl 0825.76859 [39] Pozrikidis, C., Introduction to theoretical and computational fluid dynamics, (1997), Oxford University Press Oxford · Zbl 0886.76002 [40] Pozrikidis, C.; Farrow, D.A., A model for fluid flow in solid tumors, Ann. biomed. eng., 31, 181-194, (2003) [41] Raja Sekhar, G.P.; Amaranath, T., Stokes flow inside a porous spherical shell, Z. angew. math. phys., 51, 481-490, (2000) · Zbl 0959.76085 [42] Saffman, P.G., On the boundary condition at the surface of a porous medium, Stud. appl. math. L, 2, 93-101, (1971) · Zbl 0271.76080 [43] Shatz, L.F., Singularity method for oblate and prolate spheroids in Stokes and linearized oscillatory flow, Phys. fluids, 16, 3, 664-677, (2004) · Zbl 1186.76472 [44] S. Tlupova, Improved Accuracy of Numerical Solutions of Coupled Stokes and Darcy Flows Based on Boundary Integrals, Ph.D. Thesis, Tulane University, 2007. [45] Truskey, G.A.; Yuan, F.; Katz, D.F., Transport phenomena in biological systems, (2003), Prentice Hall
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.