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Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium

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Abstract

We study theoretically and computationally the incompressible, non-conducting, micropolar, biomagnetic (blood) flow and heat transfer through a two-dimensional square porous medium in an (x,y) coordinate system, bound by impermeable walls. The magnetic field acting on the fluid is generated by an electrical current flowing normal to the x–y plane, at a distance l beneath the base side of the square. The flow regime is affected by the magnetization B 0 and a linear relation is used to define the relationship between magnetization and magnetic field intensity. The steady governing equations for x-direction translational (linear) momentum, y-direction translational (linear) momentum, angular momentum (micro-rotation) and energy (heat) conservation are presented. The energy equation incorporates a special term designating the thermal power per unit volume due to the magnetocaloric effect. The governing equations are non-dimensionalized into a dimensionless (ξ,η) coordinate system using a set of similarity transformations. The resulting two point boundary value problem is shown to be represented by five dependent non-dimensional variables, f ξ  (velocity), f η (velocity), g (micro-rotation), E (magnetic field intensity) and θ (temperature) with appropriate boundary conditions at the walls. The thermophysical parameters controlling the flow are the micropolar parameter (R), biomagnetic parameter (N H ), Darcy number (Da), Forchheimer (Fs), magnetic field strength parameter (Mn), Eckert number (Ec) and Prandtl number (Pr). Numerical solutions are obtained using the finite element method and also the finite difference method for Ec=2.476×10−6 and Prandtl number Pr=20, which represent realistic biomagnetic hemodynamic and heat transfer scenarios. Temperatures are shown to be considerably increased with Mn values but depressed by a rise in biomagnetic parameter (N H ) and also a rise in micropolarity (R). Translational velocity components are found to decrease substantially with micropolarity (R), a trend consistent with Newtonian blood flows. Micro-rotation values are shown to increase considerably with a rise in R values but are reduced with a rise in biomagnetic parameter (N H ). Both translational velocities are boosted with a rise in Darcy number as is micro-rotation. Forchheimer number is also shown to decrease translational velocities but increase micro-rotation. Excellent agreement is demonstrated between both numerical solutions. The mathematical model finds applications in blood flow control devices, hemodynamics in porous biomaterials and also biomagnetic flows in highly perfused skeletal tissue.

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Abbreviations

x, y:

coordinates parallele and perpendicular to base of square in Fig. 1

u, v:

x-direction and y-direction translational velocities

κ :

vortex viscosity of biomagnetic, micropolar fluid

k * :

permeability of isotropic porous medium

b * :

Forchheimer inertial (quadratic drag) coefficient

ρ :

density of biomagnetic, micropolar fluid

N :

micro-rotation component (angular velocity of micro-elements)

T :

temperature

ν :

kinematic viscosity

μ 0 :

magnetic property

γ :

micropolar spin-gradient viscosity (gyro-viscosity)

j :

micro-inertia density

c ρ :

specific heat capacity of biomagnetic, micropolar fluid

k f :

thermal conductivity of biomagnetic, micropolar fluid

E * :

magnetic field intensity

T c :

Curie temperature of biomagnetic, micropolar fluid

Da :

Darcy number

Fs :

Forchheimer number

N H :

biomagnetic parameter

R :

dimensionless micropolar vortex viscosity ratio

Rl :

dimensionless micro-inertia density parameter

Mn :

magnetization numberf

ε :

dimensionless temperature ratio

Ec :

Eckert (viscous dissipation) number

Pr :

Prandtl number

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Author notes

  1. O. Anwar Bég

    Present address: Engovation Engineering Science Research, 15 Southmere Avenue, Bradford, BD73NU, UK

Authors and Affiliations

  1. Aerodynamics, Aerosciences Program, BAE Systems, King Faisal Air Academy, Riyadh, Saudi Arabia

    O. Anwar Bég

  2. Applied Mathematic Group, Department of Mathematics, Indian Institute of Technology, Roorkee, India

    R. Bhargava & S. Rawat

  3. 14 Brayston Gardens, Gatley, Cheadle, UK

    Kalim Halim

  4. Engineering Department, Manchester Metropolitan University, Oxford Rd., Manchester, M15GD, UK

    H. S. Takhar

Authors
  1. O. Anwar Bég
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  2. R. Bhargava
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  3. S. Rawat
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  4. Kalim Halim
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  5. H. S. Takhar
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Corresponding author

Correspondence to O. Anwar Bég.

Additional information

Dedicated to Professor Y.C. Fung (1919-), Emeritus Professor of Biomechanics, Bioengineering Department, University of California at San Diego, USA for his seminal contributions to biomechanics and physiological fluid mechanics over four decades and his excellent encouragement to the authors, in particular OAB, with computational biofluid dynamics research.

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Bég, O.A., Bhargava, R., Rawat, S. et al. Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium. Meccanica 43, 391–410 (2008). https://doi.org/10.1007/s11012-007-9102-6

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