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Nonlinear thermal radiation and heat source effects on unsteady electrical MHD motion of nanofluid past a stretching surface with binary chemical reaction

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Abstract

An analysis is carried out for the free convection of two-dimensional time-dependent electrically conducting nanofluid past a linearly permeable expanding surface. Additionally, the impact of external heat source, nonlinear thermal radiation, and the dissipative heat energy associated with Joule heating is considered in the current flow phenomena. As a novelty, the behavior of binary chemical reactions enhances the study as well. The relevant governing equations are transformed to ODEs with the suitable choice of similarity transformations and these set of equations are handled by the shooting-based Runge–Kutta fourth-order method. From this analysis, it has been established that the effect of suction and magnetic field slows down the movement of the fluids, but due to the higher electric field strength, it enhances the outcome of the strength of the magnetic field and the electric field tends to increases the viscosity. Furthermore, it has been determined that the increase in radiative heat and heat source leads to the development of nanofluid temperature, and the Nusselt number enhances with the enhancement of the thermal radiation parameter.

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Abbreviations

\(B_{0} ,B\) :

Magnetic and applied magnetic field

\(C_{0} ,C\) :

Reference and fluid concentration

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

Coefficient of skin friction

\(C_{\infty }\) :

Concentration of ambient fluid

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

Brownian and thermophoresis diffusion

\(E\) :

Parameter of activation energy

Ea:

Activation energy

Ec::

Eckert number

\(E_{0} ,E_{1}\) :

Electric field factor and parameter

\(E^{ * }\) :

Applied electric field

f′:

Non-dimensional velocity

Gr:

Grashof number

\(K\) :

Chemical reaction parameter

\(k_{0} ,k\) :

Constant and thermal conductivity

\(k_{{\text{r}}}\) :

Variable reaction rate

k f :

Base fluid and nanoparticle thermal conductivity

\(k^{ * }\) :

Mean radiation absorption coefficient

\(M\) :

Parameter of magnetic field

\(N\) :

Buoyancy ratio parameter

\({\text{Nb}},{\text{Nt}}\) :

Brownian flow and thermophoresis parameter

\({\text{Nu}}_{x} ,\Pr\) :

Local Nusselt and Prandtl number

\(Q\) :

Heat source parameter

\(Q^{ * }\) :

Coefficient of heat generation

\(q_{{\text{m}}} ,q_{{\text{r}}} ,q_{{\text{w}}}\) :

Flux of wall mass, radiative heat wall heat

R d :

Parameter of radiation

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

Reynolds number

Sc:

Schmidt number

\(s\) :

Suction parameter

Shx :

Sherwood number

\(T,\,T_{0}\) :

Fluid and reference temperature

\(T_{{\text{w}}} ,T_{\infty }\) :

Wall surface and ambient temperature

\(u,\,v\) :

x and y directionfluid velocity

u w(x):

Extending velocity

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

Transfer of wall concentration

\(\alpha\) :

Thermal diffusivity

\(\sigma^{ * }\) :

Constant of Stefan–Boltzmann

\(\sigma\) :

Electrical conductivity

\(\delta\) :

Instability parameter

\(\delta_{1}\) :

Temperature difference parameter

\(\eta\) :

Non-dimensional similarity variable

\(\lambda\) :

Mixed convection parameter

\(\mu ,\nu\) :

Fluid dynamic and kinematic viscosity

\(\rho_{{\text{f}}} ,\rho_{{\text{p}}}\) :

Fluid and particle density

\((\rho c)_{{\text{f}}} ,(\rho c)_{{\text{p}}}\) :

Fluid and nanoparticle heat capacity

\(\psi\) :

Stream function

\(\theta\) :

Dimensionless temperature

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

Temperature ratio parameter

\(\varphi\) :

Non-dimensional concentration

\(\tau\) :

Heat capacity ratio

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

Shear stress of the sheet

\(\infty ,{\text{w}}\) :

Free stream and wall/surface condition

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology Arunachal Pradesh, Jote, 791113, India

    R. P. Sharma

  2. Department of Mathematics, JECRC University, Jaipur, Rajasthan, 303905, India

    Om Prakash

  3. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines) Dhanbad, Dhanbad, Jharkhand, 826006, India

    Om Prakash & P. S. Rao

  4. Faculty of Mechanical and Industrial Engineering, Quchan University of Technology, 94771-67335, Quchan, Iran

    I. Rashidi & F. Karimi

  5. Department of Mathematics, S ‘O’ A Deemed to be University, Khandagiri, Bhubaneswar, Odisha, 751030, India

    S. R. Mishra

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Sharma, R.P., Prakash, O., Rashidi, I. et al. Nonlinear thermal radiation and heat source effects on unsteady electrical MHD motion of nanofluid past a stretching surface with binary chemical reaction. Eur. Phys. J. Plus 137, 297 (2022). https://doi.org/10.1140/epjp/s13360-022-02359-6

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