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Optimization of heat transfer properties on ferrofluid flow over a stretching sheet in the presence of static magnetic field

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Abstract

The main emphasis of the present research is to investigate the composite effects of magnetization force and rotational viscosity on two-dimensional ferrohydrodynamic non-conducting nanofluid flow over a stretching sheet under the influence of the stationary magnetic field. Microrotation of magnetic fluid and rotation of nanoparticles are also considered. Shliomis model is used in the problem formulation, and then the similarity transformation is applied to transform partial differential equations into a set of nonlinear-coupled ordinary differential equations in dimensionless form. Transformed nonlinear-coupled differential equations are solved through the finite element method using COMSOL Multiphysics under the mathematical modeling. Results for velocity distribution, temperature distribution, concentration distribution and angular velocity distribution are obtained after considering the effects of Maxwell parameter, ferromagnetic interaction number, thermal Grashof number, solutal Grashof number, Brownian motion parameter, thermophoresis number, chemical reaction parameter, radiation absorption coefficient, heat generation/absorption parameter, Prandtl number and Schmidt number in the flow. It has been observed that magnetic energy transforms into kinetic energy, thermal boundary layer and concentration boundary layer in the presence of considered physical parameters.

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Abbreviations

A :

Positive constant

a :

Distance

c :

Stretching rate (s−1)

\(c_{\mathrm{p}}\) :

Specific heat at constant pressure \(\left( {{\mathrm{J}}\,{\mathrm{kg}}^{ - 1} \,{\mathrm{K}}^{ - 1} } \right)\)

C :

Concentration of the fluid inside the boundary layer (kg m−3)

\(C_{\infty }\) :

Concentration of the fluid outside the boundary layer (kg m−3)

\(C_{\mathrm{w}}\) :

Species concentration at the wall of the temperature (kg m−3)

\(C_{\mathrm{fx}}\) :

Skin friction coefficient

\(D_{\mathrm{B}}\) :

Brownian diffusion coefficient

\(D_{\mathrm{T}}\) :

Thermophoresis coefficient

f :

Dimensionless stream function

Gr:

Thermal Grashof number

Gm:

Solutal Grashof number

g :

Dimensionless angular velocity

\(g_{0}\) :

Acceleration due to gravity (ms−2)

I :

Sum of the particles moment of inertia (kg m2)

H :

Magnetic field intensity (A m−1)

j :

Micro-inertia per unit mass (m2)

k :

Thermal conductivity (W m−1 K−1)

K :

Material parameter

\(K^{\text{a}}\) :

Pyromagnetic coefficient

\(K_{1}\) :

Chemical reaction parameter

l :

Characteristic length

M :

Magnetization (A m−1)

\(m_{1}\) :

Effective magnetic parameter

Nb :

Brownian motion parameter

Nt :

Thermophoresis parameter

\({\mathrm{Nu}}_{\mathrm{x}}\) :

Nusselt number

Pr:

Prandtl number

\(Q_{0}\) :

Heat absorption coefficient

\(Q_{1}^{\prime }\) :

Radiation absorption coefficient

\(Q_{1}\) :

Radiation absorption coefficient

\(q_{\mathrm{w}}\) :

Heat transfer rate (W)

\(\mathrm{Re}_{\mathrm{x}}\) :

Local Reynolds number

S :

Suction parameter

\({\mathrm{Sh}}_{\mathrm{x}}\) :

Sherwood number

Sc:

Schmidt number

T :

Temperature (K)

\(T_{\mathrm{w}}\) :

Wall temperature (K)

\(T_{\mathrm{c}}\) :

Curie temperature (K)

\(T_{\infty }\) :

Temperature outside the boundary layer (K)

\(u\) :

Velocity component along the sheet (ms−1)

\(v\) :

Velocity component normal to the sheet (ms−1)

\(x\) :

Coordinate along sheet (m)

\(y\) :

Coordinate normal to the sheet (m)

\(\alpha_{1}\) :

Dimensionless distance from origin to dipole

\(\beta\) :

Ferromagnetic interaction number

\(\gamma_{1}\) :

Maxwell parameter

\(\gamma\) :

Chemical reaction parameter

\(\gamma_{0}\) :

Magnetic field strength (A m−1)

\(\gamma_{\mathrm{F}}\) :

Spin gradient viscosity

\(\lambda\) :

Viscous dissipation parameter

\(\lambda_{1}\) :

Relaxation time

\(\varphi\) :

Dimensionless concentration

\(\varPhi_{1}\) :

Volume fraction

\(\varepsilon\) :

Dimensionless curie temperature

\(\eta\) :

Dimensionless coordinate

\(\theta\) :

Dimensionless temperature

\(\lambda\) :

Viscous dissipation parameter

\(\mu\) :

Dynamic viscosity \(\left( {{\mathrm{kg}}\,{\mathrm{m}}^{ - 1} \,{\mathrm{s}}^{ - 1} } \right)\)

\(\nu\) :

Kinematic viscosity \(\left( {{\mathrm{m}}^{ - 2} \,{\mathrm{s}}^{ - 1} } \right)\)

\(\mu_{0}\) :

Permeability

\(\rho\) :

Density \(\left( {{\mathrm{kg}}\,{\mathrm{m}}^{ - 3} } \right)\)

\(\xi\) :

Dimensionless coordinate

\(\chi\) :

Heat absorption parameter

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

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

\(\varTheta\) :

Magnetic potential (A m−1)

\(\psi\) :

Stream function \(\left( {{\mathrm{m}}^{2} \,{\mathrm{s}}^{ - 1} } \right)\)

\(\left( {\rho c_{\mathrm{p}} } \right)_{\mathrm{p}}\) :

Heat capacitance of nanoparticles

\(\varOmega\) :

Angular velocity (rad s−1)

\(\varOmega_{\mathrm{p}}\) :

Angular velocity of particles (rad s−1)

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

    Anupam Bhandari & Akmal Husain

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Bhandari, A., Husain, A. Optimization of heat transfer properties on ferrofluid flow over a stretching sheet in the presence of static magnetic field. J Therm Anal Calorim 144, 1253–1270 (2021). https://doi.org/10.1007/s10973-020-09636-5

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