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.