Effect of Brownian Diffusion on Squeezing Elastico-Viscous Nanofluid Flow with Cattaneo-Christov Heat Flux Model in a Channel with Double Slip Effect

The present study deals with the analysis of heat transfer of the unsteady Maxwell nanofluid flow in a squeezed rotating channel of a porous extensile surface subject to the velocity and thermal slip effects incorporating the theory of heat flow intensity of Cattaneo-Christov model for the expression of the energy distribution in preference to the classical Fourier’s law. A set of transformations is occupied to renovate the current model in a system of nonlinear ordinary differential equations that are numerically decoded with the help of MATLAB integrated function bvp4c. The effects of various flow control parameters are investigated for the momentum, temperature and diffusion profiles, as well as for the wall shearing stress and the heat and mass transfer. The results are finally described from the material point of view. A comparison of heat flux models of Cattaneo-Christov and Fourier is also performed. An important result from the present work is that the squeezing parameter is strong enough in the middle of the channel to retard the fluid flow.


Introduction
The heat transfer phenomenon is of great concern because of its impact on industrial applications, including cooling of space and nuclear reactors, heat con-DOI: 10.4236/am.2020. 114021 278 Applied Mathematics duction in tissues pasteurization of milk, magnetic targeting of drugs, etc.
Fourier [1] proposed a heat flux model, named as heat conduction law that produces a parabolic energy equation that advocates an instantaneous change in the temperature of considered system at the beginning of any process. Cattaneo [2] introduced thermal relaxation time so as to produce the hyperbolic energy equation which permitted the heat transport through the transmission of thermal waves with finite speed. The theory of Cattaneo was further improved by Christov [3] who replaced the time derivative in the Cattaneo's model by the Oldroyd's upper-convective derivative [4] that preserved the material-invariant formulation and that became prominent as Cattaneo-Christov heat flux. Ciarletta and Straughan [5] analyzed the stability and uniqueness of the solution of the energy equation for Cattaneo-Christov heat flux model. Thermal relaxation time can be interpreted physically as the time needed for accumulating the thermal energy essential for generating heat flux [6] [7]. The inclusion of the thermal inertial in heat prorogation has effects in the heat transport in nano-material, nanofluids and many areas of ballistics and astrophysics [8] [9] [10].
The practice of adding polymers to mineral oils, known as multi-grade oils, has become recognized since the middle of 1990s [11] [12] [13]. These additions force the resulting lubricants to become non-Newtonian and viscoelastic exerting shear-rate dependent viscosity [14] [15]. The highly non-linear relationship between shear stress and strain rate of non-Newtonian fluids cannot be demon-  [27].
Nanofluids are the new-generation heat transfer fluids that contain higher thermal conductivity at very low particle concentrations than the conventional fluids. This idea of nanofluid was first developed by Choi [28]. Recent research-  [31]. As a part of these researches, Buongiorno [32] composed a mathematical model to study the convective heat transfer in nanofluids taking two important effects, namely the Brownian and thermophoresis diffusions into account.
While Stefan [33] carried out this pioneering work and basic formulations on flow phenomena, so far the analysis of the compression flow process is receiving considerable attention by the researchers because of its purposes in the fields of biomechanics and chemical engineering [34]. Reciprocating engine bearing performance, injection and compression molding, polymer processing, and modeling of lubrication system are realistic applications of squeezing flows [35] [36]. The boundary velocity, proportional to the shearing stress at the solid surface, is playing an important role in boundary value problems. For viscoelastic fluids, the slip condition is considerably important [37]. This feature has many applications in medical science, for example, polishing artificial heart valves [38]. There are several situations that include polymer fluids with high weight molecules, heavy suspensions, and lubrication problems flowing through multiple interfaces. Navier [39] initially proposed the general boundary condition which illustrates the fluid slip at the surface.
So far, few attempts have been made to study the transfer of heat and mass through a three-dimensional compression flow in a rotating channel, and therefore, the objective of the current work is to analyze the effect of thermal relaxation factor on the flow flux of time dependent Maxwell viscoelastic nanofluid that is squeezed in rotating parallel plates with porous stretched surface incorporating Cattaneo-Christov heat flux model.

Mathematical Model
The governing model equations consisting of conservation of mass, momentum, energy and concentration are given by the Cauchy stress tensor, T is the temperature of the fluid, q is the heat flux, Φ is the viscous dissipation term that describes the conversion of mechanical energy to heat. Also, Q S represents the heat sources, s J is the sum of Brownian and thermophoresis diffusions, ρ and p c are the density and specific heat respectively.
The elastico-viscous behavior of fluid will be realized if elastic stress is applied to the fluid, and the resulting strain will be time dependent characterized by relaxation time. The constitutive equation considering time dependent stress relaxation is [34] pI τ = − + S (5) The extra stress tensor S satisfies the upper convected Maxwell model given

Cattaneo-Christov model is proposed by adding thermal relaxation time in
Fourier's Law, also called the modified Fourier heat conduction law, presented by [7] ( ) here, κ is the thermal conductivity and T λ is the thermal relaxation time parameter for the heat flux where 0 T λ = simplifies the expression (7) to classical Fourier's law.
Buongiorno [32] disclosed the combination of Brownian and thermophoresis diffusions given by To demonstrate the physical model of present analysis, it is considered that the flow is laminar, unsteady and three dimensional. An incompressible This plate is stretched with a velocity ( ) A uniform magnetic field of density 0 B is applied to along y-direction and the external electric field is assumed zero.
The boundary conditions of the present physical models are ( ) Now in order to find the approximate solutions of the model it is essential to make the model equations dimensionless using the following non-dimensional variables [35] [36]: Using the above transformations, the Equation (1) The dimensionless boundary conditions are 1, , 0

Numerical Methods
Equations (11)  N . The Brownian motion parameter illustrates a significant variation in temperature profiles, while compared to concentration profiles. These outcomes express a similar result remarkable with the work of Reddy et al. [40]. These figures also reveals that the temperature of the fluid is lifted and the concentration is reduced for thermal relaxation parameter T β .     There is also declining effect of S β on the secondary velocity, displayed in    Table 3

Conclusions
The present paper is to study the effect of thermal relaxation factor on the flow of Maxwell nanofluid squeezing in the parallel rotating plates with porous stretched surface incorporating Cattaneo-Christov heat flux model. The major DOI: 10.4236/am.2020.114021 288 Applied Mathematics outcomes drawn from the study of the present model can be summarized as follows: 1) The thermal boundary layer thickness rises for the Brownian motion parameter, squeezing parameter, stretching parameter and velocity slip parameter.
2) The thermal boundary layer thickness decreases for the thermal slip parameter.
3) The hydrodynamic boundary layer thickness is reduced for the squeezing parameter, Maxwell parameter and stretching parameter.
4) The velocity distributions are higher for the velocity slip parameter.
5) The concentration is elevated for Brownian motion parameter and squeezing parameter.
In conclusion of the current study, it can be argued that the squeezing parameter and the stretching parameter that have the velocity control phenomena, can improve the heat transfer in the nanofluid. This study will provide a great opportunity to develop the cooling performance of mechanical system like automotive radiators and nuclear reactors.