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

Aftab Ahmed; Javed I. Siddique; Aftab Ahmed; Javed I. Siddique

doi:10.3934/mbe.2019029

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 2: 603-618. doi: 10.3934/mbe.2019029

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Research article

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

Aftab Ahmed ^{1
,
,},
Javed I. Siddique ²

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

Received: 07 June 2018 Accepted: 05 October 2018 Published: 11 January 2019

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.
- 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[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 603-618. doi: 10.3934/mbe.2019029

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Abstract

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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