HEAT TRANSFER ANALYSIS FOR THE UNSTEADY UCM FLUID FLOW WITH HALL EFFECTS: THE TWO-PARAMETER LIE TRANSFORMATIONS

This methodology presented the unsteady three-dimensional laminar flow since Hall effects inducing the cross flow in z̃-axis. The boundary layer and the low magnetic Reynolds number approximations are used to simplify the system of equations derived from the constitutive laws. The upper-convected Maxwell (UCM) fluid model used for Hall effects with unsteady heat transfer, which passed through the infinite stretching sheet. This flow model has intensified with the effects of magnetohydrodynamic (MHD), thermal radiation and heat generation-absorption. Here, we selected the two-parameter Lie scaling transformations to convert the highly non-linear partial differential equations (PDEs) to the ordinary differential equations (ODEs) which are studied numerically using the MATLAB bvp4c method. The main parameters are: Deborah number Deu, Hartmann number Mu, Hall effects parameter mu, Prandtl number Pr, thermal radiation parameter δu and heat generation-absorption Qu. Hall effects reduced the transport rate in the x̃-axis but increased the transport rate in the z̃-axis. On the other hand, the Hall parameter is extravagant to transport the internal energy of the system.


INTRODUCTION
The advent of non-Newtonian fluid dynamics is based on the Newton's law of viscosity theory, which is not appropriate for the analysis of all the rheological properties of the fluid. The non-Newtonian fluids were not examined by a single constitutive equation. There are many constitutive relationships for the discovery of the numerous properties of non-Newtonian fluids. In addition , non-Newtonian fluids are categorized into differential form, rate form and integral type. For reality, the major applications of non-Newtonian fluids in the fields of material production, the chemical industry and bioengineering can be described Huang et al. (2019). Examples include mercury amalgams in dental surgery, liquid metals, plastic extrusions, paper coating process, lubricants and different types of oils Ndlovu and Moitsheki (2018); Oosthuizen and Manna (2019). Stress relaxation effects are the main characteristics of the model of the rate type. Through this view, James Clerk Maxwell introduced a model that did not have the capacity to shear dependent viscosity and instead investigated the fluid elasticity.
Due to its large applications, Rajagopal and Gupta (1984) presented an accurate analysis of the non-Newtonian fluid that passed through an infinite porous plate. Sadeghy et al. (2005) demonstrated the flow of UCM to the Sakiadis. Jamil and Fetecau (2010) analyzed the UCM model between helical-flow coaxial cylinders. Mixed convection is used by Abbas et al. (2010) for the Maxwell fluid with stagnation point flow and stretch-ing sheet. Furthermore, the exact solutions are provided by Zheng et al. (2011) by choosing the Maxwell fluid model with an oscillatory channel. The UCM fluid model is used by Hayat et al. (2012) for moving surface with thermal transfer analysis. To extend this form of flow Abel et al. (2012) proposed a heat transfer study of the MHD flow for the UCM fluid model. In addition, Awais et al. (2014) advocated three-dimensional fluid flow for the UCM model. The natural convection used by Zhao et al. (2016) for the fractional Maxwell fluid model, which moves over the vertical plate. Nonlinear thermal radiation, non-uniform heat generationabsorption and fluid particle suspension effects are discussed by Gireesha et al. (2018)  . The MHD study is essential in order to improve the transport rate and the transport of internal energy within the system. The MHD effect plays a vital role in a variety of fields, such as astrophysics, solar storms, stellar dynamics, solar structures and galaxy studies Vijaya and Ramana Reddy (2020). In this view, the Hall effect occurred as a result of the Lorentz force's action. In the case of a conductor, a current is passed and a magnetic field is applied perpendicularly Dharmaiah et al. (2020). As a result, the electrical field potential of the system is developed in both directions of the current and the magnetic field. The current study of the Hall, first discussed by Edwin Hall and later provoked Hall et al. (1879) as a new action of magnetic currents. To extend this theory, many writers have presented a vast literature to identify the physics behind the MHD and the Hall effects ( Sato (1961), Jana and Datta (1977), Mandal and K. Mandal (1983), Nagy and Demendy (1995)). The Hall's effects on the flow of internal energy have become more interesting for researchers as it has various applications in all fields of science. It is Raju et al. (2011) which combines the effects of unstable MHD, Hall effects, continuous suction, oscillation and porous effects between two stretching sheets. Free convection is performed by Maripala and Naikoti (2015) with Hall effects and MHD flow for stretching sheets with viscous dissipation. The MHD free convection fluid flow of Seth et al. (2016) with Hall, thermal radiation and heat abortion effects is considered to be a moving vertical plate. Again, Sreedevi et al. (2016) studied the thermal, Hall and absorption effects of free convective flow through the stretching surface. Also, the exponential effects on the stretching sheet advocated by Srinivasacharya and Jagadeeshwar (2017) with the Hall and Joule heating effects. In addition, MHD fluid flow with specific effects, such as Hall, thermal radiation and heat generation-absorption, has been experienced by several researchers , Krishna and Jyothi (2018)

GEOMETRY OF THE PROBLEM
The primary flow along the x-axis as represented in (Figure 1) with vertical stretching sheet in upward direction and the secondary flow is along the z-axis. y-axis is normal to the stretching sheet which is lie in x zplane. The velocity vector is V = ( u( t, x, y, z), v( t, x, y, z), w( t, x, y, z)). The Hy is the transverse magnetic field which is applied in parallel direction of y-axis. Initially, for t = 0 the surface and the fluid are at rest. The transport of the internal energy is maintained at T ( t, x, y, z) = Tu. But, for t > 0 the surface is going to start move in its own plane with the velocity u( t, x, y, z) = x t and the transfer of the internal energy is fixed with Choose the magnetic Reynolds number is very small then the induced magnetic field will become zero. Due to assumption of the infinite length of theỹ-axis, there is no flow variations in this direction. The imposition of the Hall current generated due to the strong magnetic field in the direction of theỹ-axis in Figure 1. After ignoring the ion slips, thermo electric, viscous and electrical dissipation effects, we used the constitutive laws, generalized Ohm's law and Maxwell's equations ( Harris (1977), Tie-Gang et al. (2009)) with Hall effects: and J + ωeτe Hy and for UCM, the Cauchy stress tensor is and S is defined as where λ indicates the relaxation time, µ indicates the dynamics viscosity and A1 indicates the first Rivlin-Ericksen tensor which is defined as Now, S indicated as a two rank tensor in Eq. (6), the relation can be written as where T indicates the transpose vector and the momentum equation for UCM is applying 1 + λ D Dt on both side of Eq. (9) then we get using the above relation in Eq. (11), we arrived at After neglecting the pressure gradient term in Eq. (12) then the result is For Eq. (3), the term J + ωeτe Hy (J × H) indicates the electric field in the moving frame. The law of conservation of current resulted as Eq. (14) (3), we arrived at the following results where mu = ωeτe indicates the Hall parameter and after solving Eqs.
(16)-(17) simultaneously, we get Now the system of boundary layer equations in terms of x and z are and thermal boundary layer equation is x, y, z) = − ν t , u( t, x, y, z) = uu = x t , w( t, x, y, z) = 0, T ( t, x, y, z) = T∞ + Tu − T∞ x t 2 , at y = 0 v( t, x, y, z) → 0, u( t, x, y, z) → 0, w( t, x, y, z) → 0, T ( t, x, y, z) → T∞, as y → ∞ (25) The unknowns quantities in Eqs. (20)-(25) promoted as: t indicates the unsteadiness, u indicates the axial velocity in x-direction, w indicates the axial velocity in z-direction, ρ fluid density, k indicates the thermal conductivity, CP indicates the specific heat at constant pressure, T indicates fluid temperature, Tu indicates the referenced fluid temperature and T∞ indicates the free stream fluid temperature. Moreover, Uu indicates the reference velocity and Lu indicates the reference length and by the use of these two quantities, we have the following system Now, the system of Eqs. (21)-(25) will be in dimensionless form after using Eq. (26) as ∂u ∂x

TWO-PARAMETER LIE SCALING TRANSFORMATIONS
There are numerous methods to solve the ODEs in literature. But here we used the special class of the Lie scaling group of transformations. The translation method discovered by Sophus Lie and the method was the result of the Invariance of differential equations in the continuous group of symmetries. The applications of this theory are: topology, invariant theory, classical mechanics, relativity, differential geometry, and many more.  )) This flow model has three independent variables, so we need to extend the Lie scaling theory from one-parameter to two-parameter Lie scaling transformations. As, three independent variables are transformed to one independent variable. For this model, the two-parameter Lie scaling transformations described as: where C, D, Ωi, Πi(i = 1, 2, 3, 4, 5, 6) are arbitrary constants. The PDEs (27)-(30) according to the boundary conditions (31) and (32) are solved by using two-parameter scaling transformations as Eq. (33). As it is a complex problem and difficult to get the exact solution so we represent the numerical solution after getting the ODEs. Eq. (33) made the system with the following converted boundary conditions We arrived at the following results

Absolute Invariants
Eqs. (33) and (43) promoted to the following result: and the remaining absolute invariants are: where ξ is the similarity independent variable.

Reduced governing system
Introducing the stream function ψ(x, y, t) as Eqs. (45)-(47), we arrived at the value of ψ(x, y, t) where we get the value of hu after comparing, the system of Eqs. (34)-(35) in terms of absolute invariants Eq. (46) The system of Eqs. (34)-(38) in terms of absolute invariants Eq. (46) where with boundary conditions in terms of absolute invariants

Verification
To verify the result, Eq. (51) reduce to Mukhopadhyay and Bhattacharyya (2012), if we choose M = 1 and β = 0 after setting Deu = Mu = 0 and (h(ξ) → f (ξ), t → ∞). This approached to the validation of the system which has been solved by two-parameter scaling transformations.

ANALYSIS
The ODEs (51) and transport of internal energy . It is the fact that the Lorentz force reduced the primary flow, while the secondary flow trend can be seen in the reverse. The Lorentz force increases the temperature of the particles and the transport of heat increases when the Hartmann number increases. For the Hall parameter mu, Figures 3, 6 and 9 are plotted. The Hall parameter increased the primary flow at the center of the channel, but there is no flow at the end of the channel. Secondary flow parabolically resulted in Hall parameter and increased behavior throughout the channel. Physically, the increase in the Hall parameter resulted in a decrease in the capacitance of the fluid flow components. However, the transport of internal energy decreased due to the Hall effects. For the Deborah number Deu, Figures 4, 7 and 10 are presented. Maxwell's fluid properties increased the primary and secondary flow, but the reversing behavior is seen through the channel in the case of internal energy transport. There is the value of the maximum Deu = 0.1 , which is the right value for showing non-Newtonian fluid effects for this model since it behaved like a viscous fluid for the low Deborah level. Thermal radiation graphed in Figure 11. The transfer of internal energy in the channel was increased due to thermal radiation. The heat generation-absorption and the Prandtl number are listed in Figures 12-13. It can be observed that the internal energy transfer is increased due to heat generation-absorption and Prandtl.

CONCLUSIONS
The present research is the contribution of the Hall effects to the timedependent UCM. This fluid flow has passed through the infinite length of the stretching layer with thermal radiation and heat generation-absorption influences. The governing system of equations is converted from PDEs to ODEs with the help of two-parameter transformations. Then solved by the MATLAB bvp4c and presented via the graphs. The following results were described in the graphs as: 1. Primary velocity f u (ξ) having decreased behavior with magnetic field parameter Mu increasing. Secondary velocity gu(ξ) parabolically increased not only at the center of the wall, but also away from the center when an increase occurred in Mu. The same behavior can be observed for the temperature field θu(ξ).
2. Hall parameter mu raised the primary velocity f u (ξ) at the center of the channel, but there is no result on the boundary wall. Hall parameter mu increased the secondary velocity gu(ξ) not only at the center of the channel, but also at the boundary of the wall. Hall parameter mu reduced the transport of internal energy θu(ξ) through the channel.
3. Primary velocity f u (ξ) increased at the center of the channel when the Deborah number Deu increased. Secondary velocity gu(ξ) parabolically increased at the center of the channel, but there is no result at the boundary wall when the Deborah number Deu is increased. Deborah number Deu decreased the transport of internal energy θu(ξ).
4. Thermal radiation and heat generation-absorption parameters increased the transport of the internal energy θu(ξ) of the fluid model.

ACKNOWLEDGEMENTS
To the FHMT Editorial Board, I am very grateful. This work can not be published successfully without their encouragement and effort. I am also grateful to the reviewers who have done a very good job.