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Magnetoviscous effects on revolving ferrofluid due to an immersed rotating rod in the presence of gravitational force

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Abstract

In the present investigation, the effects of the rotating magnetic field, magnetization body force, and magnetic torque on revolving ferrofluid due to an immersed rotating rod are considered. The impact of the rotation of the rod and gravitational force are taken into deliberation in the present physical model. A similarity transformation is used to transform the governing partial differential equations into a set of nonlinear-coupled differential equations in dimensionless form. The numerical solution of the transformed nonlinear-coupled differential equations is obtained using COMSOL Multiphysics software in a framework of the finite element method. Results for radial, tangential, and axial velocity distributions are presented for different values of ferromagnetic interaction number, effective magnetization number, the volume concentration of particles, gravitational number, and speed of the rotation of the immersed rod. Coefficients of skin friction on the surface of the disk and wall of the rod are also computed for different values of considered physical parameters. Present results show that magnetic torque creates an additional resistance on the velocity distributions.

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Abbreviations

\( a \) :

Ratio parameter

E :

Dimensionless component of radial velocity

F :

Dimensionless component of tangential velocity

G :

Dimensionless component of axial velocity

\( \varvec{g} \) :

Acceleration due to gravity \( \left( {{\text{m}}\;{\text{s}}^{ - 2} } \right) \)

\( \xi \) :

Effective magnetization parameter

\( G_{0} \) :

Magnetic field gradient \( \left( {{\text{A}}\;{\text{m}}^{ - 2} } \right) \)

\( g_{1} \) :

Gravitational number

\( h \) :

Height of the immersed rod (m)

\( \varvec{H} \) :

Magnetic field intensity \( \left( {{\text{A m}}^{ - 1} } \right) \)

\( I \) :

Sum of the particles moment of inertia \( \left( {{\text{kg m}}^{2} } \right) \)

\( l \) :

Characteristic length (m)

\( \varvec{M} \) :

Magnetization \( \left( {{\text{A m}}^{ - 1} } \right) \)

\( m \) :

Magnetic moment \( \left( {{\text{A m}}^{2} } \right) \)

\( M_{1} \) :

Ferromagnetic interaction number

\( p \) :

Fluid pressure \( \left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 2} } \right) \)

\( r \) :

Radial direction (m)

\( r_{0} \) :

The radius of the immersed rod (m)

\( t \) :

Time (s)

\( \varvec{v} \) :

Velocity of ferrofluid \( \left( {{\text{m s}}^{ - 1} } \right) \)

\( v_{r} \) :

Radial velocity (m/s)

\( v_{\theta } \) :

Tangential velocity (m/s)

\( v_{z} \) :

Axial velocity (m/s)

\( z \) :

Axial direction (m)

\( \rho \) :

Fluid density \( \left( {{\text{kg m}}^{ - 3} } \right) \)

\( \mu_{0} \) :

The magnetic permeability of free space \( \left( {{\text{H m}}^{ - 1} } \right) \)

\( \mu \) :

Reference viscosity of the fluid \( \left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 1} } \right) \)

\( \nabla \) :

Gradient operator \( \left( {{\text{m}}^{ - 1} } \right) \)

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

Rotational relaxation time \( \left( {{\text{s}}^{ - 1} } \right) \)

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

Angular velocity of particles \( \left( {{\text{rad s}}^{ - 1} } \right) \)

\( \varOmega \) :

Vorticity \( \left( {{\text{rad s}}^{ - 1} } \right) \)

\( \omega_{0} \) :

Angular velocity of the immersed rod \( \left( {{\text{rad s}}^{ - 1} } \right) \)

\( \omega \) :

Angular velocity of volume fluid \( \left( {{\text{rad s}}^{ - 1} } \right) \)

\( \theta \) :

Tangential direction \( \left( {\text{rad}} \right) \)

\( \varphi \) :

Volume concentration

\( \nu \) :

Kinematic viscosity without magnetic field \( \left( {{\text{m}}^{2} /{\text{s}}} \right) \)

\( \alpha \) :

Dimensionless distance parameter

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  1. Department of Mathematics, School of Engineering, University of Petroleum and Energy Studies (UPES), Energy Acres Building, Bidholi, Dehradun, Uttarakhand, 248007, India

    Anupam Bhandari

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Bhandari, A. Magnetoviscous effects on revolving ferrofluid due to an immersed rotating rod in the presence of gravitational force. Indian J Phys 95, 2759–2767 (2021). https://doi.org/10.1007/s12648-020-01915-z

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