Influence of MHD on micropolar fluid flow past a sphere implanted in porous media

Abstract

This work is devoted to find the impact of the magnetic field on the motion of incompressible micropolar fluid past a sphere implanted in Brinkman’s porous media. The flow is considered to be uniform, at a distance away from the sphere. The boundary conditions applicable at the interface are vanishing of velocity components (no-slip) and vanishing of microrotation (no-spin). Expressions for stream function and microrotation are presented in terms of modified Bessel functions. An explicit expression for drag force exerted on the sphere under the magnetic effect is calculated. Representation of coefficient of drag and tangential velocity for varying physical parameters including Hartmann number, permeability, micropolarity parameter and radius is exhibited pictorially in graphical form. The results reveal the influence of the pertinent parameters on the characteristics of flow and thus provide a way to alter the flow rate. Some special cases of flow past the impermeable sphere are validated with existing results.

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Abbreviations

\(D_N\) :

Coefficient of drag

\(m_{i\,j}\) :

Couple stress tensor

\(F_\mathrm{D}\) :

Drag force

\(\mathbf {J}\) :

Electric current density

\(\mathbf {F}\) :

Lorentz force

\(\mathbf {H}=H_o\mathbf { e}_r\) :

Magnetic field intensity

\(R_{em}\) :

Magnetic Reynolds number

k :

Permeability of the porous media

p :

Pressure

a :

Radius of sphere

\(E^2\) :

Stokesian stream operator

U :

Uniform velocity

\(\mathbf {q}\) :

Velocity vector

\(\epsilon _{i\,j\,m}\) :

Alternating tensor

\(\mu\) :

Coefficient of viscosity of classical viscous fluid

\(\sigma\) :

Electrical conductivity of fluid

(\(\alpha _o\), \(\beta _o\), \(\gamma _o\)):

Gyroviscosity coefficients of micropolar fluid

\(\alpha\) :

Hartmann number

\(\delta _{i\,j}\) :

Kronecker delta

\(\mu _h\) :

Magnetic permeability of fluid

\(\chi\) :

Micropolarity parameter

\(\nu\) :

Microrotation vector

\(\eta ^{-2}=k_1\) :

Non-dimensional permeability parameter

\(\epsilon\) :

Porosity of the porous media

(r, \(\theta\), \(\phi\)):

Spherical polar coordinate system

\(\psi\) :

Stream function

\(\tau _{i\,j}\) :

Stress tensor

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Appendix

Appendix

$$\begin{aligned}&\upsilon _1={n}^{2}-{\eta }^{2}, \\&\quad \upsilon _2={\eta }^{2}+{\alpha }^{2}, \\&\quad {\upsilon _3}={{n}^{2}}-2 {{\alpha }^{2}}, \\&\quad {\upsilon _4}=2 {{\eta }^{2}}+{{\alpha }^{2}}, \\&\quad {\upsilon _5}=2 {{\eta }^{4}}-\left( {{n}^{2}}-2 {{\alpha }^{2}}\right) \, {{\eta }^{2}}-{{n}^{2}}\, {{\alpha }^{2}}, \\&\quad {\upsilon _6}={{\eta }^{4}}+{{\alpha }^{2}}\, \left( 2 {{\eta }^{2}}+{{\alpha }^{2}}\right) , \\&\quad {\varDelta }_1={{\eta }^{2}}\, \left( \upsilon _1 +{{ \lambda _2 }^{2}}\right) \, {{\chi }^{2}}+\left( {{ \lambda _2 }^{2}}\, \upsilon _4 - \upsilon _5 \right) \chi - \upsilon _6 +{{ \lambda _2 }^{2}}\, \upsilon _2 , \\&\quad {\varDelta }_2={-{\eta }^{2}}\, \left( \upsilon _1 +{{ \lambda _1 }^{2}}\right) \, {{\chi }^{2}}+\left( \upsilon _5 -{{ \lambda _1 }^{2}}\, \upsilon _4 \right) \chi + \upsilon _6 -{{ \lambda _1 }^{2}}\, \upsilon _2 , \\&\quad {\varDelta }_3=\left( {{\eta }^{2}}\left( \upsilon _1 +{{ \lambda _2 }^{2}}\right) +2 {{ \lambda _1 }^{2}}\, \left( {{ \lambda _2 }^{2}}-{{ \lambda _1 }^{2}}\right) \right) \, {{\chi }^{2}}+ \\&\quad \left( - \upsilon _5 +{{ \lambda _2 }^{2}}\, \left( \upsilon _4 +4 {{ \lambda _1 }^{2}}\right) -4 {{ \lambda _1 }^{4}}\right) \chi - \upsilon _6 +{{ \lambda _2 }^{2}}\, \left( \upsilon _2 +2 {{ \lambda _1 }^{2}}\right) -2 {{ \lambda _1 }^{4}}, \\&\quad {\varDelta }_4=-{{\eta }^{2}}\, \left( \upsilon _1 +{{ \lambda _1 }^{2}}\right) \, {{\chi }^{2}}+\left( \upsilon _5 -{{ \lambda _1 }^{2}}\, \upsilon _4 \right) \chi +{{\alpha }^{2}}\, \upsilon _4 -{{ \lambda _1 }^{2}}\, \upsilon _2 +{{\eta }^{4}}. \end{aligned}$$

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Madasu, K., Bucha, T. Influence of MHD on micropolar fluid flow past a sphere implanted in porous media. Indian J Phys (2020). https://doi.org/10.1007/s12648-020-01759-7

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Keywords

  • Sphere
  • Micropolar fluid
  • Magnetic field
  • Hartmann number
  • Brinkman’s equation
  • Drag force

PACS Nos.

  • 47.15.G-
  • 47.56.+r
  • 47.65.-d