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Slip and Hall Current Effects on Jeffrey Fluid Suspension Flow in a Peristaltic Hydromagnetic Blood Micropump

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Abstract

The magnetic properties of blood allow it to be manipulated with an electromagnetic field. Electromagnetic blood flow pumps are a robust technology which provide more elegant and sustainable performance compared with conventional medical pumps. Blood is a complex multi-phase suspension with non-Newtonian characteristics which are significant in microscale transport. Motivated by such applications, in the present article, a mathematical model is developed for magnetohydrodynamic pumping of blood in a deformable channel with peristaltic waves. A Jeffery’s viscoelastic formulation is employed for the rheology of blood. A two-phase fluid-particle (“dusty”) model is utilized to better simulate suspension characteristics (plasma and erythrocytes). Hall current and wall slip effects are incorporated to achieve more realistic representation of actual systems. A two-dimensional asymmetric channel with dissimilar peristaltic wave trains propagating along the walls is considered. The governing conservation equations for mass, fluid and particle momentum are formulated with appropriate boundary conditions. The model is simplified using long wavelength and creeping flow approximations. The model is also transformed from the fixed frame to the wave frame and rendered non-dimensional. Analytical solutions are derived. The resulting boundary value problem is solved analytically, and exact expressions are derived for the fluid velocity, particulate velocity, fluid/particle fluid and particulate volumetric flow rates, axial pressure gradient, pressure rise and skin friction distributions are evaluated in detail. Increasing Hall current parameter reduces bolus growth in the channel, particle-phase velocity and pressure difference in the augmented pumping region, whereas it increases fluid phase velocity, axial pressure gradient and pressure difference in the pumping region. Increasing the hydrodynamic slip parameter accelerates both particulate and fluid phase flow at and close to the channel walls, enhances wall skin friction, boosts pressure difference in the augmented pumping region and increases bolus magnitudes. Increasing viscoelastic parameter (stress relaxation time to retardation time ratio) decelerates the fluid phase flow, accelerates the particle-phase flow, decreases axial pressure gradient, elevates pressure difference in the augmented pumping region, and reduces pressure difference in the pumping region. Increasing drag particulate suspension parameter decelerates the particle-phase velocity, accelerates the fluid phase velocity, strongly elevates axial pressure gradient and reduces pressure difference (across one wavelength) in the augmented pumping region. Increasing particulate volume fraction density enhances bolus magnitudes in both the upper and lower zones of the channel and elevates pressure rise in the augmented pumping region.

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Abbreviations

\(a_{1} ,a_{2}\) :

Wave amplitudes

\(a\) :

Amplitude ratio of the upper wall wave

\(b\) :

Amplitude ratio of the lower wall wave

\(B_{0}\) :

Magnetic field

\(c\) :

Wave speed

\(e\) :

Charge of ions

\(F\) :

Dimensionless time-mean flow in the wave frame

\(h_{1} ,h_{2}\) :

Dimensionless wall deformations

\(m\) :

Hall current parameter

\(M\) :

Hartmann magnetic number

\(n_{\text{e}}\) :

Mass of the electrons

\(N\) :

Suspension parameter

\(\bar{p}\) :

Dimensionless pressure

\(P\) :

Pressure

\(Q_{\text{f}}\) :

Volume flow rate in fluid phase

\(Q_{\text{p}}\) :

Volume flow rate in particulate phase

\(Re\) :

Reynolds number

\(S\) :

Drag coefficient

\(\overline{t}\) :

Dimensionless time

\(t\) :

Time

\(\overline{u}\) :

Non-dimensional axial velocity

\(U_{\text{f}}\) :

Longitudinal velocity in fluid phase

\(U_{\text{p}}\) :

Longitudinal velocity in particulate phase

\(\overline{v}\) :

Dimensionless transverse velocity

\(V_{\text{f}}\) :

Transverse velocity in fluid phase

\(V_{\text{p}}\) :

Transverse velocity in particulate phase

\(X,Y\) :

Coordinate axes

\(\overline{x}\) :

Non-dimensional axial coordinate

\(\overline{y}\) :

Dimensionless transverse coordinate

\(\beta\) :

Dimensionless slip parameter

\(\delta\) :

Dimensionless wave number

\(\rho_{\text{f}}\) :

Density of material constituting fluid

\(\rho_{\text{p}}\) :

Density of particulate phase

\(\sigma\) :

Electrical conductivity

\(\lambda_{1}\) :

Ratio of the relaxation and retardation times

\(\lambda_{2}\) :

Retardation time

\(\lambda\) :

Wavelength

\(\mu_{\text{s}}\) :

Viscosity coefficient

\(\phi\) :

Phase difference

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Authors and Affiliations

  1. Department of Mathematics, Lovely Professional University, Jalandhar, Punjab, 144411, India

    K. Ramesh

  2. Department of Mathematics, National Institute of Technology Uttarakhand, Srinagar, Uttarakhand, 246174, India

    D. Tripathi

  3. Fluid Mechanics and Propulsion, Aeronautical and Mechanical Engineering, University of Salford, Newton Building, G77, The Crescent, Salford, M54WT, England, UK

    O. Anwar Bég

  4. Materials, Corrosion and Structures, Aeronautical and Mechanical Engineering, University of Salford, Newton Building, G79, The Crescent, Salford, M54WT, England, UK

    A. Kadir

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  1. K. Ramesh
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Correspondence to D. Tripathi.

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Ramesh, K., Tripathi, D., Bég, O.A. et al. Slip and Hall Current Effects on Jeffrey Fluid Suspension Flow in a Peristaltic Hydromagnetic Blood Micropump. Iran J Sci Technol Trans Mech Eng 43, 675–692 (2019). https://doi.org/10.1007/s40997-018-0230-5

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