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The effect of magnetic field on flow induced-deformation in absorbing porous tissues

1 Department of Mathematics, Capital University of Science and Technology, Islamabad 44000, Pakistan
2 Department of Mathematics, Penn State University—York Campus, York, Pennsylvania 17403-3326, USA

In order to understand the interaction between magnetic field and biological tissues in a physiological system, we present a mathematical model of flow-induced deformation in absorbing porous tissues in the presence of a uniform magnetic field. The tissue is modeled as a deformable porous material in which high cavity pressure drives fluid through the tissue where it is absorbed by capillaries and lymphatics. A biphasic mixture theory is used to develop the model under the assumptions of small solid deformation and strain-dependent linear permeability. A spherical cavity formed during injection of fluid in the tissue is used to find fluid pressure and solid displacement as a function of radial distance and time. The governing nonlinear PDE for fluid pressure is solved numerically using method of lines whereas tissue solid displacement is computed by employing trapezoidal rule. The effect of magnetic parameter on fluid pressure, solid displacement and tissue permeability is illustrated graphically.
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Keywords magnetohydrodynamics (MHD); biological tissue; mixture theory; solid deformation; method of lines

Citation: Aftab Ahmed, Javed I. Siddique. The effect of magnetic field on flow induced-deformation in absorbing porous tissues. Mathematical Biosciences and Engineering, 2019, 16(2): 603-618. doi: 10.3934/mbe.2019029


  • 1. H. Darcy, Les fontaines publiques de la ville de Dijon, Paris: Dalmont, 1856.
  • 2. D. D. Joseph, D. A. Nield and G. Papanicolaou, Nonlinear equation governing flow in a saturated porous medium, Water Res., 18 (1982), 1049–1052.
  • 3. K. Terzaghi, Erdbaumechanik auf bodenphysikalischen Grundlagen, Wien: Deuticke, 1925.
  • 4. M. A. Biot, General theory of three dimensional consolidation, J. Appl. Phys., 12 (1941), 155–164.
  • 5. M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26 (1955), 182–185.
  • 6. A. Fick, Ueber diffusion, Annalen der Physik, 170 (1855), 59–86.
  • 7. C. Truesdell, Sulle basi della thermomeccanica, Atti della Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, 22 (1957), 33–38.
  • 8. R. J. Atkin and R. E. Craine, Continuum theories of mixtures: Basic theory and historical development, Q. J. Mech. Appl. Math., 29 (1976), 209–244.
  • 9. R. J. Atkin and R. E. Craine, Continuum theories of mixtures: Applications, J. I. Math. Appl., 17 (1976), 153–207.
  • 10. K. R. Rajagopal and L. Tao, Mechanics of mixtures, World Scientific, Singapore, 1995.
  • 11. J. I. Siddique, A. Ahmed, A. Aziz and C. M. Khalique, A review of mixture theory for deformable porous media and applications, Appl. Sci., 7 (2017), 1–15.
  • 12. D. E. Kenyon, The theory of an incompressible solid-fluid mixture, Arch. Ration. Mech., 62 (1976), 131–147.
  • 13. D. E. Kenyon, A mathematical model of water flux through aortic tissue, B. Math. Biol., 41 (1979), 79–90.
  • 14. G. Jayaraman, Water transport in the arterial wall: a theoretical study, J. Biomech., 16 (1983), 833–840.
  • 15. R. Jain and G. Jayaraman, A theoretical model for water flux through the arterial wall, J. Biomech. Eng., 109 (1987), 311–317.
  • 16. M. Klanchar and J. M. Tarbell, Modeling water flow through arterial tissue, B. Math. Biol., 49 (1987), 651–669.
  • 17. V. C. Mow and W. M. Lai, Mechanics of animal joints, Annu. Rev. Fluid Mech., 11 (1979), 247–288.
  • 18. W. M. Lai and V. C. Mow, Drag induced compression of articular cartilage during a permeation experiment, Biorheology, 17 (1980), 111–123.
  • 19. M. H. Holmes, Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression, J. Biomech. Eng., 108 (1986), 372–381.
  • 20. A. Ahmed, J. I. Siddique and A. Mahmood, Non-Newtonian flow-induced deformation from pressurized cavities in absorbing porous tissues, Comput. Method. Biomec., 20 (2017), 1464–1473.
  • 21. C. W. J. Oomens, D. H. V. Campen and H. J. Grootenboer, A mixture approach to the mechanics of skin, J. Biomech., 20 (1987), 877–885.
  • 22. T. R. Ford, J. S. Sachs, J. B. Grotberg and M. R. Glucksberg, Mechanics of the perialveolar interstitium of the lung, First World Congress of Biomechanics, La Jolla, 1 (1990), 31.
  • 23. M. H. Friedman, General theory of tissue swelling with application to the corneal stroma, J. Theor. Biol., 30 (1971), 93–109.
  • 24. C. Nicholson, Diffusion from an injected volume of a substance in brain tissue with arbitrary volume fraction and tortuosity, Brain Res., 333 (1985), 325–329.
  • 25. N. T. M. Eldabe, G. Saddeek and K. A. S. Elagamy, Magnetohydrodynamic flow of a biviscosity fluid through porous medium in a layer of deformable material, J. Porous Media, 14 (2011), 273–283.
  • 26. J. I. Siddique and A. Kara, Capillary rise of magnetohydrodynamics liquid into deformable porous material, J. Appl. Fluid Mech., 9 (2016), 2837–2843.
  • 27. A. Naseem, A. Mahmood, J. I. Siddique and L. Zhao, Infiltration of MHD liquid into a deformable porous material, Results Phys., 8 (2018), 71–75.
  • 28. S. Sreenadh, K. V. Prasad, H. Vaidya, E. Sudhakara, G. Krishna and M. Krishnamurthy, MHD Couette Flow of a Jeffrey Fluid over a Deformable Porous Layer, Int. J. Appl. Comput. Math., 3 (2017), 2125–2138.
  • 29. J. Bagwell, J. Klawitter, B. Sauer and A. Weinstein, A study of bone growth into porous polyethylene, Presented at the Sixth Annual Biomaterials Symposium, Clemson University, Clemson, SC, USA, (1974), 20–24.
  • 30. R. M. Pilliar, H. V. Cameron and I. MacNab, Porous surface layered prosthetic devises, Biomed. Eng., 4 (1975), 12–16.
  • 31. M. L. Bansal, Magneto Therapy, Jain Publishers, New Delhi, 1976.
  • 32. P. N. Tandon, A. Chaurasia and T. Gupta, Comput. Math. Appl., 22 (1991), 33–45.
  • 33. S. I. Barry and G. K. Aldis, Flow-induced deformation from Pressurized cavities in absorbing porous tissues, B. Math. Biol., 54 (1992), 977–997.
  • 34. M. H. Holmes, A theoretical analysis for determining the nonlinear hydraulic permeability of a soft tissue from a permeation experiment, B. Math. Biol., 47 (1985), 669–683.
  • 35. S. I. Barry and G. K. Aldis, Radial flow through deformable porous shells, J. Aust. Math. Soc. B, 34 (1993), 333–354.
  • 36. J. S. Hou, M. H. Holmes,W. M. Lai and V. C. Mow, Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications, J. Biomech. Eng., 111 (1989), 78–87.
  • 37. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, New York, Dover, 1972.
  • 38. W. E. Schiesser, The numerical method of lines: Integration of partial differential equations, Academic Press, San Diego, 1991.
  • 39. A. Farina, P. Cocito and G. Boretto, Flow in deformable porous media: Modelling an simulations of compression moulding processes, Math. Comput. Model., 26 (1997), 1–15.
  • 40. S. I. Barry and G. K. Aldis, Fluid flow over a thin deformable porous layer, J. Appl. Math. Phys. (ZAMP), 42 (1991), 633–648.


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