^{1}

^{*}

^{1}

^{2}

^{3}

^{3}

^{1}

^{2}

^{3}

Edited by: Dumitru Baleanu, University of Craiova, Romania

Reviewed by: Amin Jajarmi, University of Bojnord, Iran; Kazuharu Bamba, Fukushima University, Japan

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The present study concentrates on squeezed unsteady magneto-hydrodynamic flow of Jeffrey fluid confined between two infinite parallel plates. Description of heat transfer process is disclosed through melting and thermal radiation effects. Features of viscous dissipation are also incorporated. Characteristics of mass transport are explored with chemical reaction. Modulated non-linear partial differential equations are reduced by implementing appropriate transformations. Approximate convergent solutions are calculated through analytical technique. Characteristics of velocity, concentration and fluid temperature are illustrated graphically and also discussed comprehensively physically. The skin friction co-efficient and Nusselt number are sketched and discussed through graphs. It is noticed that horizontal velocity component and temperature of Jeffrey liquid are dominant for greater melting parameter. Moreover, temperature field decays for dominant thermal radiation parameter.

Recently, scientists and researchers have shown intent to analyze the features of non-Newtonian fluid flow imposed by squeezed surfaces, regarding their wide-spread build up and trade area in many industrial and biological processes, such as in the polymer industry, compression, injection shaping, liquid-metal lubrication, formation of paper sheets and thin fiber, molding of plastic sheets and metal and squeezed film, through which power is transmitted. Many researchers discussed the Jeffrey fluid model corresponding to various physical aspects. But this model is not studied widely with the squeezing phenomenon. At beginning, Stefan [

Characteristics of heat transfer have increased noteworthy interest of recent explorers and engineers because of various technical processes in the area of engineering, physiological and industrial processes, including glass blowing and glass fiber productions, wire coating, paper making, metal spinning, expulsion of plastic sheets by aerodynamic, drawing of plastic films and unceasing casting. However, little attention is paid to disclose the features of heat transport through melting phenomenon. It has widespread applications, such as the making of semiconductors, melting permafrost and hardening of magma flow, heat exchanger coils coated by freeze soil in grounded pump, sewage treatment via freeze process, melting of soil, welding processes, and casting of manufacturing processes. Behavior of double-diffusive convection in stagnant Maxwell liquid flow considering melting phenomenon has been explored by Hayat et al. [

A literature survey indicates that researchers have disclosed the properties of squeezed Newtonian and non-Newtonian fluid flow with different boundary conditions of heat transfer widely. But such analysis of squeezing flow with melting heat transport has not explored as of yet. So, the main strategic concern is to fill this void. Thus, the present attempt is concerned to demonstrate the melting mechanism in squeezed Jeffrey fluid flow through two parallel plates. Behavior of heat with mass transport is modeled and exposed considering thermal radiation, viscous dissipation and chemical reaction. The transformed coupled non-linear differential system is evaluated via a homotopy technique [

Consider unsteady, incompressible squeezed Jeffrey fluid flow between horizontal two plates. The bottom plate is located at

Here, velocity components are _{0} represents magnetic field, _{1} denotes electric conductivity, _{2} is the co-efficient of thermal conductivity, specific heat is _{p}, _{1} is absorption coefficient, σ^{*} denotes the Stefan-Boltzmann constant, _{0}/1 − γ_{0}, λ_{1} is relaxation time and λ_{2} is ratio among relaxation and retardation time.

Physical situation of the problem.

The boundary conditions are as follow

Transformations are taken as follows

Pressure term is eliminated from equations (2) and (3), governing equations in view of equation (7) take the forms:

The subjected boundary conditions are

where squeezing parameter is _{q}, Deborah number is β, magnetic parameter is _{r}, Eckert number is

These dimensionless quantities are given by

Viscous dissipation effect vanishes when _{1} = 0, Jeffrey fluid converts to viscous fluid. Moreover, the plates illustrate inward movement for _{q} > 0, and for _{q} < 0, away movement is observed.

Defining skin friction co-efficient, heat and mass transfer rate as follows

In view of dimensionless forms, one may write.

The homotopic procedure was initially purposed by Liao [_{0}, _{0}, _{0}) and concerned linear operators

with

where _{i}(

Clearly, convergent homotopic solutions incorporate the auxiliary non-zero parameters ℏ_{f}, ℏ_{θ} and ℏ_{ϕ}. Said parameters significantly provide way to regulate the convergence. In order to prefer the reasonable estimation of ℏ_{f}, ℏ_{θ} and ℏ_{ϕ}, the _{f}, ℏ_{θ} and ℏ_{ϕ}, are −1.7 ≤ ℏ_{f} ≤ −0.1 and −1.7 ≤ ℏ_{θ} ≤ 0.0 and −1.8 ≤ ℏ_{ϕ} ≤ − 0.1.

Regions of convergence for

The main attraction here is to discuss the stimulus of emerging parameters in flow fields (i.e., velocity, temperature, concentration) under consideration. _{q}) on the vertical and horizontal velocity fields. It is seen that increase in _{q} guarantees an increase in both velocity distributions, respectively. Physically, it is justified that higher squeezing parameter corresponded to larger deformation in fluid particles. Hence, the growing behavior of velocity is noted. _{1} on the horizontal component of velocity. It is delineated that horizontal velocity component decays for growing λ_{1}. Physically, greater values of λ_{1} correspond to higher relaxation time. So, perturbed fluids require more time to take their original shape. Therefore, the velocity profile decreased. The variation in velocity distribution for diverse values of the Hartman number is exhibited in _{q} _{r} on the Nussetl number are illustrated in _{q}, _{1}. It is revealed that the outcome gives favorable agreement.

Behavior of _{q} on

Behavior of λ_{1} on

Behavior of

Behavior of

Behavior of _{q} on θ(η).

Behavior of

Behavior of _{r} on θ(η).

Behavior of

Behavior of

Behavior of σ on ϕ(η).

Behavior of _{q} &

Behavior of _{r} &

Comparison of skin friction co-efficient (_{1}.

^{1/2} |
^{1/2} |
||||
---|---|---|---|---|---|

0.0 | 1.0 | 0.1 | 0.5 | 4.646286 | 4.646286 |

0.5 | 3.513473 | 3.513473 | |||

0.1 | 2.059355 | 2.059355 | |||

1.0 | 0.0 | 0.1 | 0.5 | 2.079543 | 2.079543 |

0.5 | 2.074376 | 2.074376 | |||

1.0 | 2.059355 | 2.059355 | |||

1.0 | 1.0 | 0.0 | 0.5 | 1.697800 | 1.697800 |

0.5 | 2.059355 | 2.059355 | |||

0.2 | 2.393890 | 2.393890 | |||

1.0 | 1.0 | 0.1 | 0.0 | 2.133399 | 2.133399 |

0.5 | 2.059355 | 2.059355 | |||

1.0 | 1.990723 | 1.990723 |

Here, we have investigated heat and mass transport phenomenon effects on squeezed Jeffrey liquid flow between two parallel walls. Heat and mass transport are featured with melting heat transport and first order chemical reaction, respectively. The conclusions are as follows:

Diverse values of relaxation to retardation parameter λ_{1} decay the horizontal velocity profile.

Higher values of melting parameter

Enhancing values of Eckert number

Concentration distribution shows decreasing behavior with increment in chemical reaction parameter σ.

It is expected that the present attempt relates the physical modeling under different flow conditions. It includes dynamical systems, lubrication mechanism, heating, cooling and energy estimation processes, etc. Further, as a future direction, the current investigation can tackle the different dynamical system [

All datasets generated for this study are included in the article/supplementary material.

SA collected the data and wrote the paper. MF made the analysis of the paper. MR, BA, and SR contributed in paper revision.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.