Magnetoviscous effects on revolving ferrofluid due to an immersed rotating rod in the presence of gravitational force
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ORIGINAL PAPER
Magnetoviscous effects on revolving ferrofluid due to an immersed rotating rod in the presence of gravitational force A Bhandari* Department of Mathematics, School of Engineering, University of Petroleum and Energy Studies (UPES), Energy Acres Building, Bidholi, Dehradun, Uttarakhand 248007, India Received: 09 January 2020 / Accepted: 24 August 2020
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. Keywords: Ferrofluid; Magnetic field; Rotating rod; Velocity; Gravitational number List a E F G g n G0 g1 h H I l M m M1 p r r0 t v
of symbols Ratio parameter Dimensionless component of radial velocity Dimensionless component of tangential velocity Dimensionless component of axial velocity Acceleration due to gravity ðm s2 Þ Effective magnetization parameter Magnetic field gradient ðA m2 Þ Gravitational number Height of the immersed rod (m) Magnetic field intensity A m1 Sum of the particles moment of inertia kg m2 Characteristic length (m) Magnetization A m1 Magnetic moment A m2 Ferromagnetic interaction number Fluid pressure kg m1 s2 Radial direction (m) The radius of the immersed rod (m) Time (s) Velocity of ferrofluid ðm s1 Þ
vr vh vz z q l0 l r ss xp X x0 x h u m a
Radial velocity (m/s) Tangential velocity (m/s) Axial velocity (m/s) Axial direction (m) Fluid density kg m3 The magnetic permeability of free space H m1 Reference viscosity of the fluid kg m1 s1 Gradient operator ðm1 Þ Rotational relaxation time ðs1 Þ Angular velocity of particles rad s1 Vorticity rad s1 Angular velocity of the immersed rod rad s1 Angular velocity of volume fluid rad s1 Tangential direction ðradÞ Volume concentration Kinematic viscosity without magnetic field ðm2 =sÞ Dimensionless distance parameter
*Corresponding author, E-mail: [email protected]
2020 IACS
A Bhandari
1. Introduction Ferrofluids are colloidal
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