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Effect of variable viscosity and thermal conductivity on water-carrying iron (iii) oxide ferrofluid flow between two rotating disks

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Abstract

The paper represents the potential significance of magnetic field-dependent viscosity, temperature-dependent viscosity, and variable conductivity on water-carrying iron (iii) oxide ferrofluid flow between two parallel stretchable rotating disks under the influence of a stationary magnetic field. This problem develops the understanding of the swirling flow of ferrofluid in the presence of magnetization force. The influence of variable viscosity and variable conductivity in the swirling flow of ferrofluid is useful in sealing the rotating shaft and heat transfer enhancement applications. We use similarity transformation to reduce the governing equations into non-dimensional nonlinear differential equations. The transformed non-dimensional boundary layer equations are solved numerically using finite element procedure in COMSOL Multiphysics. Under the influence of the magnetic field, the magnetic torque acting in the flow and enhancement in the volume concentration of iron (iii) oxide nanoparticles both enhance the viscosity of ferrofluid. Increasing temperature-dependent viscosity parameters reduce the viscosity of ferrofluid. However, variable conductivity parameter increases the temperature in the flow. The magnetic torque reduces the radial and axial velocity distributions and magnetization force enhances the velocities. Friction on the disk and local heat transfer mainly depends on the rotation speed and stretching of the disks, and magnetization force.

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Abbreviations

\(b\) :

Temperature-dependent viscosity parameter

\(c_{s}\) :

Dimensionless stress on the surface of the lower disk

\(c_{w}\) :

Dimensionless stress on the wall of the lower disk

\(d\) :

Vertical distance between disks (m)

\(d_{m}\) :

Diameter of magnetic core

(m)

\(Ec\) :

Eckert number

\(s\) :

Thickness of surfactant layer (m)

\(\left( {c_{p} } \right)_{nf}\) :

Specific heat of ferrofluid of at constant pressure \(\left( {{\text{J kg}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(\left( {c_{p} } \right)_{f}\) :

Specific heat of carrier liquid at constant pressure \(\left( {{\text{J kg}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(\left( {c_{p} } \right)_{s}\) :

Specific heat of nanoparticles at constant pressure \(\left( {{\text{J kg}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(f\) :

Dimensionless axial velocity

\(f^{\prime}\) :

Dimensionless radial velocity

\(Ec\) :

Eckert number

\(g\) :

Dimensionless tangential velocity

\(H\) :

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

\(k_{nf}\) :

Thermal conductivity of ferrofluid

\(\left( {{\text{Wm}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(k_{f}\) :

Thermal conductivity of carrier liquid ferrofluid \(\left( {{\text{Wm}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(k_{s}\) :

Thermal conductivity of nanoparticles \(\left( {{\text{Wm}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(k\left( T \right)\) :

Temperature-dependent thermal conductivity of nanoparticles \(\left( {{\text{Wm}}^{ - 1} {\text{K}}^{ - 1} } \right)\)

\(K_{a}\) :

Pyromagnetic coefficient

\(M\) :

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

\(Nu\) :

Local Nusselt number

\(P\) :

Dimensionless pressure

\(p\) :

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

\(q_{w}\) :

Wall heat flux \(\left( {{\text{Wm}}^{ - 1} {\text{L}}^{ - 1} } \right)\)

\({\text{Pr}}\) :

Prandtl number

\(r\) :

Radial direction (m)

\(s_{1}\) :

Stretching rate of lower disk \(\left( {{\text{rad s}}^{ - 1} } \right)\)

\(s_{2}\) :

Stretching rate of upper disk \(\left( {{\text{rad s}}^{ - 1} } \right)\)

\(S_{1}\) :

Dimensionless stretching parameter for lower disk

\(S_{2}\) :

Dimensionless stretching parameter for upper disk

\(T\) :

Temperature \(\left( K \right)\)

\(T_{a}\) :

Temperature of the lower disk \(\left( K \right)\)

\(T_{b}\) :

Temperature of the upper disk \(\left( K \right)\)

\({\text{u}}\) :

Radial velocity (m/s)

\(v\) :

Tangential velocity (m/s)

\(w\) :

Axial velocity \(\left( {m/s} \right)\)

\(z\) :

Axial direction (m)

\(Re\) :

Reynolds number

\(\xi\) :

Dimensionless magnetization

\(\xi_{0}\) :

Strength of the applied magnetic field \(\left( {{\text{A }}m^{ - 1} } \right)\)

\(\rho_{nf}\) :

Density of ferrofluid

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

\(\rho_{f}\) :

Density of carrier liquid

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

\(\rho_{s}\) :

Density of nanoparticles

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

\(\mu_{0}\) :

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

\(\mu_{nf}\) :

Dynamic viscosity of ferrofluid \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\mu_{nf} \left( {H = 0} \right)\) :

Dynamic viscosity of ferrofluid in the absence of magnetic field \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\mu_{nf} \left( {H \ne 0} \right)\) :

Dynamic viscosity of ferrofluid in the presence of magnetic field \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\mu_{nf} \left( T \right)\) :

Temperature-dependent dynamic viscosity \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\mu_{f}\) :

Dynamic viscosity of carrier liquid \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\omega\) :

Rotation parameter

\(\omega_{1}\) :

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

\(\omega_{2}\) :

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

\(\varphi\) :

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

\(\varphi_{1}\) :

Volume concentration of nanoparticles

\(\varphi_{c}\) :

Critical volume fraction

\(\tilde{\varphi }\) :

Volume fraction of magnetic materials

\(\alpha_{nf}\) :

Thermal diffusivity \(\left( {{\text{m}}^{2} {\text{s}}^{ - 1} } \right)\)

\(\alpha\) :

Coefficient of thermal expansion

\(\epsilon\) :

Variable thermal conductivity parameter

\(\left( {\beta , \beta_{1} , \beta_{2} } \right)\) :

Ferromagnetic interaction numbers

\(\theta\) :

Dimensionless temperature

\(\lambda\) :

Dimensionless pressure gradient

\(\eta\) :

Dimensionless vertical distance

\(\tau_{s}\) :

Stress on the surface \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 2} } \right)\)

\(\tau_{w}\) :

Stress on the wall \(\left( {{\text{kg m}}^{ - 1} {\text{s}}^{ - 2} } \right)\)

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

    Anupam Bhandari

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Bhandari, A. Effect of variable viscosity and thermal conductivity on water-carrying iron (iii) oxide ferrofluid flow between two rotating disks. Indian J Phys 96, 3221–3238 (2022). https://doi.org/10.1007/s12648-022-02281-8

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