Corresponding Author Email (anber.saleem@tdtu.edu.vn)
Abstract: In this paper, heat transfer of linearly stretched curved surfaces of unsteady magnetized micropolar fluid flow is discussed. Impacts of velocity and thermal slip are considered on the linear curved stretching surface. The mathematical model is assembled under the flow suppositions derived from the classic Navier Stoke equations. This model is reduced to a system of coupled nonlinear differential equations by means of boundary layer approximations. Differential equations become dimensionless when the similarity transformations are applied. The dimensionless system is solved through numerical techniques. The involved dimensionless parameters effects a range of parameters, including the unsteady parameter, magnetic parameter, velocity slip parameter, curvature parameter, micropolar parameter, reciprocal magnetic Prandtl number, dimensionless parameter, Biot number and Prandtl number, all of which are studied in relation to the Nusselt number, skin fraction, velocity profile, temperature profile, micropolar profile and magnetized profile. We provide a robust discussion, and evidence our findings graphically and in tables. Key outcomes of this work include findings such as, the unsteady parameter enhances as thermal and momentum boundary layer decreases. Also, the skin friction rises for increasing curvature parameters, but Nusselt number declines when the curvature parameters rises.
Keywords: unsteady flow, micropolar fluid, numerical scheme, magnetized fluid, slip effects.
Introduction
Incompressible viscous flow over a porous channel was presented by Berman [1]. The perturbation method has been applied using the normal wall velocity to be equal. This work was extended by Sellars [2] in which he highlighted the effects of high suction Reynolds number for laminar flow at the porous wall. Later, Wah [3] extended the previous ideas and presented a model of incompressible viscous flow over uniform porous channel. Terrill [4] discussed the incompressible viscous flow over uniform porous channel for large injection. Techniques including quasi linearization, differentiation and parametric approximations were applied to investigate the effects of a micropolar fluid at porous wall channels, which were numerically studied by Sastry and Rao [5]. Srinivasacharya et al. [6] analyzed the time dependent micropolar fluid flow in between parallel plates. They applied the perturbation method using the suction Reynolds number to find results on flow behaviour. The micropolar time dependent fluid flow under stagnation region at plane surface was investigated by Xu et al. [7]. They applied the series solutions to investigate the flow at the surface. Elbashbeshy et al. [8] applied MHD to the micropolar time dependent Maxwell fluid flow at a stretching surface. Devakar and Raje [9] have been discussing numerical outcomes of immiscible unsteady micropolar fluid flow in a horizontal channel. Recently, authors [10 – 12] have been working on the unsteady micropolar fluid flow with various effects.
During recent years, significant advancement has been made in the examinations of non-Newtonian liquids spurred on by their applications in manufacturing and engineering. The rheological attributes of such liquids are exceptionally valuable in depicting the remarkable properties to every day liquids such as, ketchup, shampoo, cosmetic products, mud, paints, so forth. Classical Navier-Stokes equations cannot explain some of their characteristics. For example, the classical Navier-Stokes equations are unable to accurately model micro-rotation, spin-inertia, body torque and coupled stress, all of which play a key role in the behaviour and movement of fluids such as colloidal suspension, animal blood, liquid crystals, polymeric liquids and liquids holding small amounts of polymeric fluids. Eringen [13] presented pioneering work on Micropolar fluid theory. He analyzed and discussed the jumped conditions and constitutive equations of the micro fluent media and basic field equations [14]. Shukla and Isa [15] studied the generalized Reynolds equations of micropolar lubricants for a one dimensional slider bearing. They highlighted the solid-particle additives in their solution. Lockwood et al. [16] have discussed the electrohydrodynamic contact and lyotropic molten crystals in viscometric flow. Khonsari and Brewe [17] studied the micropolar fluid using the finite journal bearings lubricated on the surface. Micropolar fluid flow under the stagnation point region over a stretching surface investigated by Nazar et al. [18]. Ishak et al. [19] analyzed the micropolar fluid flow at a vertical porous surface under stagnation point. Hayat et al. [20] discussed the axisymmetric micropolar fluid with unsteady stretching sheet analytically. Nadeem et al. [21] explored the fluid flow of a micropolar liquid in rotating horizontal parallel plates with a view to find the solution numerically and analytically. Subhani and Nadeem [22] numerically explored hybrid nanoparticles based on micropolar liquids. The flow of a micropolar liquid at a stretching surface was studied and highlighted their influence using various assumptions [23-24].
Boundary layer flow with uniform free stream at a fixed flat plate has been discussed by Blasius [25]. The numerical method applied on the his work has been discussed by Hogarth [26]. As opposed to the Blasius [25] work, Sakiadis [27] presented the boundary layer flow of a fluid induced in a quiescent ambient fluid by a moving plate. Tsou et al. [28] discussed analytically and experimentally flow of boundary layer on the continuous rotating surface. The results of Sakiadis [27] are certified by Tsou et al. [28]. Crane [29] extended the Tsou et al. [28] work on the stretching plate. Crane [29] protracted this model to an enlarging plate with a stretching velocity in a quiescent fluid that contrasts by means of the distance as of a fixed point and offered exact analytical results. Furthermore, Miklavčič and Wang [30] highlighted the fluid flow over a shrinking sheet. Fang [31] has deliberated the power law velocity over a shrinking sheet using the exact solutions for physical parameters. Akbar et al. [32] explored results of a tangent hyperbolic fluid at stretching surface and discussed this phenomena numerically. Hussain et al. [33] explored the effects of a micropolar fluid at a stretching sheet. Halim et al. [34] have discussed the slipped stretched surface with the nanomaterial flow of a Maxwell fluid. Alblawi et al. [35] worked on the curved surface over an exponential stretching plane numerically. Most of authors have studied the flow over stretching surface with various assumptions [36-40]. This includes a large range of recent work including that by Shah et. Al. [41] on Darcy-Forchheimer flow of Cu – and Ag – nanofluids, and also further work by Ahmad et. al. [42] who presented the Cattaneo-Christov Heat Flux Model when considering entropy generation between rotating disks.
The heat transfer of linear stretching curved surfaces with unsteady magnetized micropolar fluid flow is discussed in this analysis. This mathematical model assembled under the flow suppositions by means of Navier Stokes equations. The model further reduced in the differential equations by means of boundary layer approximations. Differential equations become dimensionlesss when the similarity transformations are applied. The dimensionless system solved through the numerical techniques. Parameters discussed in this study are mentioned above, and we believe that the novel solutions provided in this paper hold significant applications to engineering and industry in general.
Formulations
In this study we present a mathematical model using the Navier Stokes equations under the flow expectations on the curved stretching surface as shown in Fig. 1. The geometry of the curved surface is such that (\(r,\ s\)) are radial apparatuses, \(s\) is the arc length and \(r\) is normal to tangent.