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Theoretical Exploration of Thermal Transportation with Lorentz Force for Fourth-Grade Fluid Model Obeying Peristaltic Mechanism

  • Research Article-Mechanical Engineering
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Abstract

The heat transfer phenomenon plays an imperative role in several biological and industrial processes, like the oil production industries, catalytic reactors, energy losses in several thermal systems, energy storage, papers manufacture and heat exchanger systems. Therefore, the present analysis investigates the heat transfer phenomena on peristaltic transportation of the hydromagnetic flow of non-Newtonian fourth-grade fluid in a tapered asymmetric channel filled with porous media. Moreover, the impacts of heat source/sink and thermal radiation in modeling are retained. The tapered asymmetry in the channel is generated by undertaking the peristaltic wave train inflicted on the non-uniform walls to have altered phases and amplitudes. The analysis is originated by adopting suppositions of long wavelength \(\left( {\delta < < 1} \right)\), small Deborah number \(\left( {\Gamma \to 0} \right)\) and low Reynolds number \(\left( {\text{R} \to 0} \right)\). A regular perturbation technique is utilized to acquire the series outcomes for the axial velocity, pressure gradient and streamlines distribution, while an analytical outcome has been acquired for the thermal profile. The pressure rise at each wavelength on the channel walls has been numerically computed. Impacts of arising parameters in the analysis are surveyed graphically. Outcomes divulge that magnitude of the axial velocity raises with a rise of Darcy number. In contrast, it diminishes with a raise of the magnetic parameter near the center of the channel. Furthermore, the calculated outcomes are found in admirable agreement with the outcomes acquired by the finite element technique and previously published results.

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Abbreviations

\(A_{1}\) :

First Rivlin–Ericksen tensor \(\left[ - \right]\)

\(\bar{X},\,\,\bar{Y}\) :

Space coordinates in laboratory frame \(\left[ m \right]\)

\(\bar{P},p\) :

Pressure in the laboratory and wave frame \(\left[ {kg/s^{2} } \right],\left[ - \right]\)

\(\bar{U},\bar{V}\) :

Velocity components in laboratory frame \(\left[ {ms^{{ - 1}} } \right]\)

\(c\) :

Wave speed \(\left[ {ms^{{ - 1}} } \right]\)

\(t\) :

Time \(\left[ s \right]\)

\(u,v\) :

Velocity components in wave frame \(\left[ {ms^{{ - 1}} } \right]\)

\(Q_{0}\) :

Heat generation parameter \(\left[ {m^{2} s^{{ - 3}} } \right]\)

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

Temperature at upper and lower walls \(\left( K \right)\)

\(d\) :

Channel half width \(\left[ m \right]\)

\(a_{1} ,\,\,a_{2}\) :

Wave amplitude at upper and lower walls \(\left[ m \right]\)

\(B\) :

Heat source/sink parameter \(\left[ { = {{Q_{0} d^{2} } \mathord{\left/ {\vphantom {{Q_{0} d^{2} } {k\left( {T_{1} - T_{0} } \right)}}} \right. \kern-\nulldelimiterspace} {k\left( {T_{1} - T_{0} } \right)}}} \right],\left[ - \right]\)

\(Da\) :

Darcy number \(\left[ { = {K \mathord{\left/ {\vphantom {K {d^{2} }}} \right. \kern-\nulldelimiterspace} {d^{2} }}} \right],\left[ - \right]\)

\(\Pr\) :

Prandtl number \(\left[ { = {{\mu C_{p} } \mathord{\left/ {\vphantom {{\mu C_{p} } k}} \right. \kern-\nulldelimiterspace} k}} \right],\left[ - \right]\)

\(m\) :

Non-uniform parameter of the channel \(\left[ - \right]\)

\(\bar{T}\) :

Temperature in laboratory frame \(\left( K \right)\)

\(R\) :

Radiation parameter \(\left[ { = {{16T_{\infty }^{3} \sigma ^{ * } } \mathord{\left/ {\vphantom {{16T_{\infty }^{3} \sigma ^{ * } } {3k^{ * } k}}} \right. \kern-\nulldelimiterspace} {3k^{ * } k}}} \right],\left[ - \right]\)

\(Q\) :

Time average of the flow

\(M\) :

Magnetic parameter \(\left[ { = \sqrt {{\sigma \mathord{\left/ {\vphantom {\sigma \mu }} \right. \kern-\nulldelimiterspace} \mu }} B_{0} d} \right],\left[ - \right]\)

\(C_{p}\) :

Specific heat \(\left[ {Jkg^{{ - 1}} K^{{ - 1}} } \right]\)

\(g\) :

Acceleration due to gravity \(\left[ {LT^{{ - 2}} } \right]\)

\(\Delta p\) :

Pressure rise \(\left[ - \right]\)

\(\text{R}\) :

Reynold’s number \(\left[ { = {{\rho cd_{1} } \mathord{\left/ {\vphantom {{\rho cd_{1} } \mu }} \right. \kern-\nulldelimiterspace} \mu }} \right]\)

\(B_{0}\) :

Strength of magnetic field \(\left[ T \right]\)

\(\bar{x},\,\,\bar{y}\) :

Space coordinates in wave frame \(\left[ m \right]\)

\(A_{n}\) :

Rivlin–Ericksen tensor for \(n > 1\)\(\left[ - \right]\)

\(G_{r}\) :

Grashof number \(\left[ { = {{\rho g\beta _{T} d^{2} \left( {T_{1} - T_{0} } \right)} \mathord{\left/ {\vphantom {{\rho g\beta _{T} d^{2} \left( {T_{1} - T_{0} } \right)} {\mu c}}} \right. \kern-\nulldelimiterspace} {\mu c}}} \right],\left[ - \right]\)

\(q_{r}\) :

Radiative thermal heat flux \(\left[ {Wm^{{ - 2}} } \right]\)

\(k^{ * }\) :

Mean absorption coefficient \(\left[ {KW^{{ - 1}} } \right]\)

\(\beta _{T}\) :

Thermal expansion coefficient \(\left[ {K^{{ - 1}} } \right]\)

\(\theta\) :

Time mean flow rate in fixed frame \(\left[ - \right]\)

\(\phi\) :

Phase difference \(\left[ {{\text{rad}}} \right]\)

\(\Gamma\) :

Deborah number \(\left[ - \right]\)

\(\lambda\) :

Wavelength [m]

\(\psi\) :

Stream function \(\left[ {m^{2} s^{{ - 1}} } \right]\)

\(\sigma\) :

Electrical conductivity \(\left[ {sm^{{ - 1}} } \right]\)

\(\mu\) :

Viscosity \(\left[ {kgm^{{ - 1}} s^{{ - 1}} } \right]\)

\(\delta\) :

Wavenumber \(\left[ { = {{2\pi d} \mathord{\left/ {\vphantom {{2\pi d} \lambda }} \right. \kern-\nulldelimiterspace} \lambda }\,} \right],\left[ - \right]\)

\(\rho\) :

Fluid density \(\left[ {kgm^{{ - 3}} } \right]\)

\(k\) :

Thermal conductivity \(\left[ {Wm^{{ - 1}} K^{{ - 1}} } \right]\)

\(K\) :

Permeability parameter \(\left[ - \right]\)

\(\gamma\) :

Dimensionless temperature \(= \left[ {{{\bar{T} - T_{0} } \mathord{\left/ {\vphantom {{\bar{T} - T_{0} } {T_{1} - T_{0} }}} \right. \kern-\nulldelimiterspace} {T_{1} - T_{0} }}} \right],\left[ - \right]\)

\(\sigma ^{ * }\) :

Stephen Boltzmann constant \(\left[ {Wm^{{ - 2}} K^{{ - 4}} } \right]\)

NN:

Non-Newtonian

FGF:

Fourth-grade fluid

MHD:

Magnetohydrodynamics

FEM:

Finite element method

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Acknowledgements

We are thankful to the reviewers for their encouraging comments and constructive suggestions to improve the quality of the manuscript. The authors appreciate the financial support from HEC Pakistan through NRPU-6295.

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Authors and Affiliations

  1. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

    M. Y. Rafiq & Zaheer Abbas

  2. Department of Computer Sciences, Bahria University, Islamabad, 44000, Pakistan

    Jafar Hasnain

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  1. M. Y. Rafiq
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Correspondence to Zaheer Abbas or Jafar Hasnain.

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Rafiq, M.Y., Abbas, Z. & Hasnain, J. Theoretical Exploration of Thermal Transportation with Lorentz Force for Fourth-Grade Fluid Model Obeying Peristaltic Mechanism. Arab J Sci Eng 46, 12391–12404 (2021). https://doi.org/10.1007/s13369-021-05877-0

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