Journal of Engineering Mathematics

, Volume 114, Issue 1, pp 19–41 | Cite as

Computation of a regularized Brinkmanlet near a plane wall

  • Hoang-Ngan Nguyen
  • Sarah D. Olson
  • Karin LeidermanEmail author


In this study, we derive a solution for Brinkman flow induced by a regularized point force in the presence of a plane wall. The process involves solving two coupled subproblems: one for the forced Brinkman flow in free-space and one for the unforced Brinkman flow in a half-space that enforces the no-slip boundary condition on the plane wall. Both subproblems are solved in the Fourier domain and a specific regularization of the force is designed so that the transformed solutions have Gaussian decay properties in certain limits. Similar to the singularly forced case, however, and as shown previously by others, there is no analytic inverse transform and thus it must be approximated numerically. The Gaussian decay of our solutions make the numerical approximation of the inverse transform more efficient. We provide detailed methodology to compute velocities from collections of regularized point forces and to solve for the forces at collections of points where the Brinkman velocity is prescribed. Because the unsteady Stokes equations can be recast as the Brinkman equations, our technique is applicable when solving for either type of flow, in the presence of a plane wall.


Brinkman equations Plane wall Porous medium Regularization method Unsteady Stokes equations 



Nguyen and Leiderman were supported, in part, by funding under NSF DMS-1413078. Olson was supported, in part, by NSF DMS-1413110 and NSF DMS-1455270. The authors also thank Dr Forest Mannan and Dr Paul Martin for helpful discussions.


  1. 1.
    Auriault J (2009) On the domain of validity of Brinkman’s equation. Transp Porous Med 79:215–223MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brinkman H (1947) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl Sci Res A1:27–34zbMATHGoogle Scholar
  3. 3.
    Childress S (1972) Viscous flow past a random array of spheres. J Chem Phys 56:2527–2539CrossRefGoogle Scholar
  4. 4.
    Durlofsky L, Brady J (1987) Analysis of the Brinkman equation as a model for flow in porous media. Phys Fluids 30(11):3329–3341CrossRefzbMATHGoogle Scholar
  5. 5.
    Howells I (1974) Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J Fluid Mech 64:449–475CrossRefzbMATHGoogle Scholar
  6. 6.
    Spielman L, Goren S (1968) Model for predicting pressure drop and filtration efficiency in fibrous media. Environ Sci Technol 1(4):279–287CrossRefGoogle Scholar
  7. 7.
    Damiano E (1998) The effect of the endothelial-cell glycocalyx on the motion of red blood cells through capillaries. Microvasc Res 55(1):77–91CrossRefGoogle Scholar
  8. 8.
    Damiano E, Duling B, Ley K, Skalak T (1996) Axisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer. J Fluid Mech 314:163–189CrossRefzbMATHGoogle Scholar
  9. 9.
    Leiderman K, Miller L, Fogelson A (2008) The effects of spatial inhomogeneities on flow through the endothelial surface layer. J Theor Bio 252:313–325MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Secomb T, Hsu R, Pries A (1998) A model for red blood cell motion in glycocalyx-lined capillaries. Am J Phys Heart Circ Phys 274(3):H1016–H1022Google Scholar
  11. 11.
    Secomb T, Hsu R, Pries A (2001) Motion of red blood cells in a capillary with an endothelial surface layer: effect of flow velocity. Am J Phys Heart Circ Phys 281(2):H629–H636Google Scholar
  12. 12.
    Tarbell J, Zhong-Dong S (2013) Effect o the glycocalyx layer on transmission of interstitial flow shear stress to embedded cells. Biomech Model Mechanbiol 12:111–121CrossRefGoogle Scholar
  13. 13.
    Weinbaum S, Zhang X, Han Y, Vink H, Cowan S (2003) Mechanotransduction and flow across the endothelial glycocalyx. Proc Natl Acad Sci USA 100(13):7988–7995CrossRefGoogle Scholar
  14. 14.
    Cogan N, Donahue M, Whidden R, De La Fuente D (2013) Pattern formation exhibited by biofilm formation within microfluidic chambers. Biophys J 104(9):1867–1874CrossRefGoogle Scholar
  15. 15.
    Kapellos G, Alexiou T, Payatakes A (2007) Hierarchical simulator of biofilm growth and dynamics in granular porous materials. Adv Water Res 30(6):1648–1667CrossRefGoogle Scholar
  16. 16.
    Leiderman K, Fogelson A (2010) Grow with the flow: a spatial-temporal model of platelet deposition and coagulation under flow. Math Med Biol 28:47–84MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leiderman K, Fogelson A (2012) The influence of intraclot transport on the development of platelet thrombi under flow. Bull Math Biol 75:1255–1283CrossRefzbMATHGoogle Scholar
  18. 18.
    Onasoga-Jarvis A, Leiderman K, Fogelson A, Wang M, Manco-Johnson M, Di Paola J, Neeves K (2013) The effect of factor viii deficiencies and replacement and bypass therapies on thrombus formation under venous flow conditions in microfluidic and computational models. PLoS ONE 8: e78732-1–e78732-12Google Scholar
  19. 19.
    Chu J, Kim M (2001) Two-dimensional oscillatory Stokes flows between two parallel planes. Fluid Dyn Res 29:7–24MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Feng J, Ganatos P, Weinbaum S (1998) Motion of a sphere near planar confining boundaries in a Brinkman medium. J Fluid Mech 375:265–296CrossRefzbMATHGoogle Scholar
  21. 21.
    Green C, Sader J (2005) Small amplitude oscillations of a thin beam immersed in a viscous fluid near a solid surface. Phys Fluids 17(7):073102MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Avudainayagam A, Geetha J (1998) A boundary-integral equation for two-dimensional oscillatory Stokes flow past an arbitrary body. J Eng Math 33(3):251–258MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pozrikidis C (1989) A study of linearized oscillatory flow past particles by the boundary-integral method. J Fluid Mech 202:17–41MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tekasakul P, Tompson RV, Loyalka SK (1998) Rotatory oscillations of arbitrary axi-symmetric bodies in an axi-symmetric viscous flow: numerical solutions. Phys Fluids 10(11):2797–2818CrossRefGoogle Scholar
  25. 25.
    Clarke R, Cox S, Williams P, Jensen O (2006) The drag on a microcantilever oscillating near a wall. Proc R Soc A 62:913–933CrossRefzbMATHGoogle Scholar
  26. 26.
    Clarke R, Jensen O, Billingham J (2008) Three-dimensional elastohydrodynamics of a thin plate oscillating above a wall. Phys Rev E 78:056310MathSciNetCrossRefGoogle Scholar
  27. 27.
    Clarke R, Jensen O, Billingham J, Williams P (2005) Three-dimensional flow due to a microcantilever oscillating near a wall: an unsteady slender-body analysis. J Fluid Mech 545:397–426MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Green CP, Sader JE (2002) Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J Appl Phys 92(10):6262–6274CrossRefGoogle Scholar
  29. 29.
    Green CP, Sader JE (2005) Frequency response of cantilever beams immersed in viscous fluids near a solid surface with applications to the atomic force microscope. J Appl Phys 98(11):114913CrossRefGoogle Scholar
  30. 30.
    Tung RC, Jana A, Raman A (2008) Hydrodynamic loading of microcantilevers oscillating near rigid walls. J Appl Phys 104(11):114905CrossRefGoogle Scholar
  31. 31.
    Tuck E (1969) Calculation of unsteady flows due to small motions of cylinders in a viscous fluid. J Eng Math 3(1):29–44CrossRefzbMATHGoogle Scholar
  32. 32.
    Chu JH, Kim MU (2004) Oscillatory stokes flow due to motions of a circular disk parallel to an infinite plane wall. Fluid Dyn Res 34(2):77–97MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang W, Stone H (1998) Oscillatory motions of circular disks and nearly spherical particles in viscous flows. J Fluid Mech 367:329–358CrossRefzbMATHGoogle Scholar
  34. 34.
    Blake J (1971) A note on the image system for a Stokeslet in a no-slip boundary. Proc Camb Philos Soc 70:303–310CrossRefzbMATHGoogle Scholar
  35. 35.
    Pozrikidis C (1989) A singularity method for unsteady linearized flow. Phys Fluids A 1:1508–1520MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ainley J, Durkin S, Embid R, Boindala P, Cortez R (2008) The method of images for regularized Stokeslets. J Comput Phys 227:4600–4616MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Cortez R, Cummins B, Leiderman K, Varela D (2010) Computation of three-dimensional Brinkman flows using regularized methods. J Comput Phys 229:7609–7624MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Leiderman K, Olson S (2016) Swimming in a two-dimensional Brinkman fluid: computational modeling and regularized solutions. Phys Fluids 28:021902CrossRefGoogle Scholar
  39. 39.
    Nguyen HN, Olson S, Leiderman K (2016) A fast method to compute triply-periodic Brinkman flows. Comput Fluids 133:55–67MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ho N, Leiderman K, Olson S (2016) Swimming speeds of filaments in viscous fluids with resistance. Phys Rev E 93:043108CrossRefGoogle Scholar
  41. 41.
    Olson S, Leiderman K (2015) Effect of fluid resistance on symmetric and asymmetric flagellar waveformst. J Aero Aqua Bio-mech 4:12–17CrossRefGoogle Scholar
  42. 42.
    Leiderman K, Bouzarth E, Cortez R, Layton A (2013) A regularization method for the numerical solution of periodic Stokes flow. J Comput Phys 236:187–202MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Nguyen H, Cortez R (2014) Reduction of the regularization error of the method of regularized Stokeslets for a rigid object immersed in a three-dimensional Stokes flow. Commun Comput Phys 15:126–152MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Trefethen L (2008) Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev 50:67–87MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Ahmadi E, Cortez R, Fujioka H (2017) Boundary integral formulation for flows containing an interface between two porous media. J Fluid Mech 816:71–93MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Cortez R, Fauci L, Medovikov A (2005) The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming. Phys Fluids 17(3):031504MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Cortez R (2001) The method of regularized Stokeslets. SIAM J Sci Comput 23:1204–1225MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Wróbel JK, Cortez R, Varela D, Fauci L (2016) Regularized image system for Stokes flow outside a solid sphere. J Comput Phys 317:165–184MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Hecht G (2009) Intestinal microbes in health and disease. Gastroent 136:1849–1850CrossRefGoogle Scholar
  50. 50.
    Suarez S, Pacey A (2006) Sperm transport in the female reproductive tract. Hum Reprod Update 12:23–37CrossRefGoogle Scholar
  51. 51.
    Olson S, Fauci L (2015) Hydrodynamic interactions of sheets vs. filaments: attraction, synchronization, and alignment. Phys Fluids 27:121901CrossRefzbMATHGoogle Scholar
  52. 52.
    Leal L (2007) Advanced transport phenomena: fluid mechanics and convective transport processes. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  53. 53.
    Pozrikidis C (1992) Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsColorado School of MinesGoldenUSA
  2. 2.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations