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Physical Aspects of MHD Nonlinear Radiative Heat Flux in Flow of Thixotropic Nanomaterial

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Abstract

Here magnetohydrodynamic nanomaterial flow of thixotropic fluid is addressed. Electrically conducting fluid is considered. Stagnation-point flow toward variable thicked surface is addressed. Soret and Dufour effects are retained. Formulation is based on Brownian and thermophoresis diffusions. In addition, nonlinear thermal heat flux and convective boundary conditions are also taken into account. The formulated expressions are converted into ordinary ones by appropriate transformations. The resulting system is solved computationally through homotopy algorithm. Convergence of derived solutions is ensured explicitly. Behavior of various thermophysical parameters on temperature, nanoparticle concentration and velocity is graphically analyzed. Heat transfer rate, Sherwood number and skin friction coefficient are discussed through different flow variables. A comparative analysis between the present investigation and published literature has been presented in limiting cases. The obtained outcomes show that velocity of liquid particles decays via magnetic variable, while opposite behavior is examined for higher non-Newtonian parameter. Furthermore, nanoparticle concentration and temperature are enhanced for radiation and thermophoresis variables.

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Abbreviations

\(\hat{u},\hat{v}\) :

Velocity components

\(x,y\) :

Space coordinates

\(\mu\) :

Dynamic viscosity

\(u_{0} ,u_{\infty }\) :

Reference velocities

\(B_{0}\) :

Constant magnetic field

\(\alpha\) :

Thermal conductivity at surface

\(\rho\) :

Fluid density

\(R_{a} ,R_{b}\) :

Material constant

\(\sigma\) :

Electrical conductivity of fluid

\(b\) :

Dimensional constant

\(n\) :

Power law index

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

Susceptibility of concentration

\(K_{\text{T}}\) :

Thermal diffusion ratio

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

Specific heat at constant pressure

\(\tau\) :

Heat capacity ratio

\(k\) :

Thermal conductivity

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

Specific heat of fluid

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

Brownian diffusion coefficient

\(\sigma^{*}\) :

Stefan–Boltzmann constant

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

Thermophoretic diffusion coefficient

\(\hat{T}\) :

Surface heat flux

\(\hat{T}_{\infty }\) :

Ambient temperature

\(\hat{C}\) :

Concentration of fluid

\(\hat{C}_{\infty }\) :

Ambient concentration

\(\hat{u}_{\infty }\) :

Stretching velocity

\(D_{\text{m}}\) :

Mass diffusivity

\(k^{*}\) :

Rosseland mean absorption coefficient

\(\hat{\phi }\) :

Dimensionless concentration

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

Surface heat flux

\(q_{\text{m}}\) :

Surface mass flux

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

Surface shear stress

\(\xi ,\varvec{\eta}\) :

Dimensionless space variables

\(K_{a} ,K_{b}\) :

Non-Newtonian parameters

\(\varsigma\) :

Ratio of velocities

\(\lambda\) :

Variable thickness index

\(R\) :

Radiation parameter

\(M\) :

Magnetic parameter

\(Bi_{i}\) :

Thermal Biot number

\(Bi_{2}\) :

Concentration Biot number

\(Nb\) :

Brownian motion variable

\(Nt\) :

Thermophoresis variable

\(\theta_{w}\) :

Temperature difference parameter

\(Sr\) :

Soret number

\(Df\) :

Dufour number

\(Pr\) :

Prandtl number

\(Sc\) :

Schmidt number

\(C_{fx}\) :

Skin friction coefficient

\(Nu\) :

Nusselt number

\(Sh\) :

Sherwood number

\(Re_{x}\) :

Local Reynolds number

\(\hat{f}\) :

Dimensionless velocity

\(\hbar_{{\hat{f}}} ,\hbar_{{\hat{\theta }}} ,\hbar_{{\hat{\phi }}}\) :

Nonzero auxiliary parameters

\({\text{\pounds}}_{1} ,{\text{\pounds}}_{2} ,{\text{\pounds}}_{3}\) :

Linear operator

\(q_{r}\) :

Radiative heat flux

\(\hat{\theta }\) :

Dimensionless temperature

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Authors and Affiliations

  1. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

    Tasawar Hayat, Madiha Rashid & M. Ijaz Khan

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Tasawar Hayat & Ahmed Alsaedi

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  1. Tasawar Hayat
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  2. Madiha Rashid
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Correspondence to M. Ijaz Khan.

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Hayat, T., Rashid, M., Khan, M.I. et al. Physical Aspects of MHD Nonlinear Radiative Heat Flux in Flow of Thixotropic Nanomaterial. Iran J Sci Technol Trans Sci 43, 2043–2054 (2019). https://doi.org/10.1007/s40995-019-00688-3

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