Rheology of hydro-magnetic polymeric material with heat generation/absorption and chemical reaction

This article is published under the Creative Commons CC-BY-ND License (http://creativecommons.org/licenses/by-nd/4.0/). This license permits use, distribution and reproduction, commercial and non-commercial, provided that the original work is properly cited and is not changed in anyway. Abstract: In the present work, the mathematical modelling, transformation and analysis are presented to investigate the magnetohydrodynamic (MHD) fl ow of a polymeric liquid, i.e., Maxwell fl uidic model, over an absorptive wall encapsulated in a medium having porosity. The system model in the form of partial diff erential equations (PDEs) for fl uid fl ow is converted to a corresponding ordinary diff erential equations (ODEs)-based system through introduction of dimensionless quantities with the help of appropriate similarity transformation procedures. The velocity, temperature and concentration profi les of the system are calculated for variations in diff erent prominent physical parameters for upper convected Maxwell (UCM) model including combined infl uences of energy and transfer of mass together with chemical reception and heat generation or absorption as well as Sherwood number, Nusselt number and Deborah number. Lie group analysis was carried out for the nonlinear governing equations and the absolute invariants were found properly. Numerical, as well as analytical computing techniques with the homotopy analysis method (HAM) were utilised to verify and validate the correctness of the approximate solutions for the coupled system. Analysis of the results with exhaustive interpretation for each numerical and graphical illustration further endorsed the eff ectiveness and worth of the solvers. The trend of fl uid fl ow in terms of streamlines are also presented to show the diff erence elaboratively between hydromagnetic and hydrodynamic fl ow situations.


INTRODUCTION
It is generally acknowledged that non-Newtonian fl uid behaviour (categorised by a nonlinear relation between viscosity and strain) can be witnessed in several ordinary and real life fl uids including blood, lubricants, paints, foodstuff (including ketchup, cream, cooking oil, butter, etc.), polymers and chemicals. Distant from condensing and retreating properties, some other rheological eff ects have been observed such as tangential and normal stresses, relaxations and retardation time. Under these features, researchers have proposed fl uid models in the past few years. A survey of literature shows that the 1 st , 2 nd , 3 rd and 4 th grade diff erential type fl uids have been discussed widely while the slightly non-Newtonian Jeff ery fl uid, Maxwell fl uid, Oldroyd and Burgers fl uids are not exploited well as these models are suitable to forecast the relaxation and retardation properties. Harris (1977) proposed mathematical calculations of Maxwell fl uid. Several mathematicians and physicists inspected the Maxwell model with diff erent geometries and Zierep and Fetecau (2007) studied dynamic equilibrium equations of 3 Rayleigh-Stokes. They have considered the Stokes problems and discussed the shear stress and dissipation properties. Stability of Maxwell equations have already been examined by Tan and Masuoka (2007). The authors computed the eigenvalue problem,

December 2020
Journal of the National Science Foundation of Sri Lanka 48 (4) and Rayleigh eff ects, wave eff ect and frequency for over-stability have been found. Renardy and Renardy (2009) investigated linear solidity of homogeneous fl ow of the non-Newtonian (upper convected Maxwell) fl uid. Kumari and Nath (2009) computed the non-Newtonian upper-convected Maxwell (UCM) fl uid past a stagnation point with inclusion of magnetohydrodynamic (MHD). Authors have employed the order analysis to simplify the equations and utilised the fi nite diff erence technique for the solution computations. Mass transfer in UCM fl uid fl ow has been analysed by Hayat et al. (2011a). Mathematical modelling for the MHD case has been performed. A correction for MHD Maxwell fl uid model equation is presented for the fi rst time in the literature and chemical reaction for the generative and destructive case has been investigated. Eff ect of 3D fl ow for UCM fl uidic system over a bi-directionally surface with stretching mechanism has been exploited by Hayat et al. (2011b). Mathematical modelling for the three-dimensional case for any rate type fl uid has been presented. Deborah eff ect has been analysed for the axisymmetric, two-and threedimensional fl ow situations. Unsteady 3D non-Newtonian UCM fl uid fl ow past a smooth linear surface based on linearly stretching in two directions has been presented by Awais et al. (2014). They have modelled the fl ow of Maxwell fl uid for the unsteady case in three-dimensions. Similarity transforms are utilised for simplifi cation and analytical solver homotopy analysis method (HAM), has been invoked for solution construction. Generally in recent past, extensive research work has been conducted to investigate the physical behaviour of fl uidic problems (Shahid et al., 2018;Dehghani et al., 2019;Fiza et al., 2019;Khan et al., 2019;Kumar et al., 2019;Yasmin et al., 2019;Yousif et al., 2019). The objective of current investigation is to examine the Lie group analysis of UCM fl uid fl ow past an absorptive surface with impact of heat and mass transfer. Generative and destructive chemical receptions along with heat generation or absorption eff ects are taken into consideration. Outcomes of the velocity, concentration and temperature distribution are computed, and visual representations of residual errors are displayed to show the effi cacy of the results. Tabular description has also been performed to show the convergence of the numeric solutions and to present the analogy of obtained results with the already available results in the limiting sense.
A graphical presentation has also been given which describes the presentation of sundry measures on velocity, concentration and temperature.

METHODOLOGY
Methodology is presented in two parts; problem formulation with the help of mathematical formulation is represented in the fi rst part, while Lie group analysis is represented in the second part.

Problem formulation
The properties of polymeric liquid with electrical conduction past a fl at absorptive surface are assumed. Equal forces but in opposite directions are incorporated along with the horizontal axis in such a way that the surface is stretched. The magnetic eff ect is along the vertical axis. As shown in Figure 1 the temperature as well as concentration of the fl uid at stretched surface with their ambient values are T w , C w , T ∞ and C ∞ respectively. The system model for the proposed fl uid fl ow is given as follows: ( ... (4) Equation (1) represents law of conservation of mass, while equation (2) is the law of conservation of momentum. Equation (3) represents the law of conservation of energy and equation (4) is the mass transfer equation.

Solution methodology
The set of equations (29-31) along with the boundary condition given in equation (32)  In the case of HAM, the unique way for choosing the initial approximation is as follows: Any function that satisfi es the boundary condition can be utilised as initial guess in HAM, but surely there are number of functions which satisfy boundary conditions. In that situation we have to choose the initial guess which gives rapid convergence. The initial guess can be found easily with the help of initial operator and boundary conditions. A MATHEMATICA code for HAM has been developed in order to enumerate the solutions of system model shown in equations (66-68) along with associated boundary conditions mentioned in equation (69).  Figures 2 and 3 show the error of the given analysis. Figure 2 shows the absolute error for K with velocity profi le while Figure 3 depicts absolute error for M with velocity profi le. It is clear from Figures 2 and 3 that error in velocity profi le for various values of K and M is quite negligible and it exist in the range of 10-07 to 10-09.

Journal of the National Science Foundation of Sri Lanka 48(4) December 2020
It is evident from these graphs that the error in current analysis is very much negligible. Tables 2 and 3 provide a comparative performance of the present fi ndings with the already available data. It is seen that the present solution has an acceptable accord with published data (Afi fy & Elgazery, 2012). For hydro-magnetic situation the fl uid is subjected to a magnetic fi eld, which results in an increase in retarding eff ects that opposes the fl ow. In order word eff ects of magnetic fi eld can be utilised as an agent in the fl uid to control the fl ow very accurately by varying the magnetic fi eld intensity. This behaviour is portrayed in both graphs.
Graphical results (Figures 6-12) are portrayed for diff erent sundry parameters including stagnation point parameter A, Deborah number β, magnetic fi eld parameter M, injection/suction parameter S and heat source/sink eff ect hs and generative/destructive parameter γ on the system. It is elucidated from Figure 6 that with an increment in A (0 ˂ A ˂1) , boundary layer momentum and f ' increase and for A =1.0 , the thickness of boundary layer vanishes because the stretching sheet and free stream velocities are equal at A =1.0 . Moreover for A ˃1.0 the free stream is larger as compared to stretched velocity, which results in an area with high viscosity impact. The Deborah number β's signifi cant impact on f ' is depicted in Figure 7. Large value of the Deborah number results in the acceleration of momentum boundary layer. It is observed that the layer of boundary momentum and velocity decrease for larger values of Deborah number β . Figures 8 and 9 present the infl uence Journal of the National Science Foundation of Sri Lanka 48(4) December 2020 Figure 10: Impact of hs on temperature distribution Figure 11: Impact of generative chemical reaction of concentration profi le Figure 12: Eff ects of calamitous chemical reaction of concentration of suction/injection parameter S for A ˂ 1 and A ˃ 1, respectively. For A ˂ 1(the stretching velocity dominants over free stream velocity) the decrement in velocity fi eld is observed with an with an increment in S whereas A ˃ 1 (mean free stream in dominant) results in an increment in fl uidic system velocity profi le for greater values of S. Moreover, it is also elucidated that A = 1, which results in vanishing the momentum boundary layer. Figure 10 is provided for the signifi cant impact of hs on temperature. The variation of hs from −2 ˂ hs ˂ 2 is considered and it is seen that the temperature profi le increased with an increment in hs from −2 to 2. Figures 11 and 12 show the properties of destructive/ generative reaction γ on φ . It is observed that diff usion fi eld decreases for destructive (γ ˃ 0) and enhances for generative reaction (γ ˂ 0). hs is the internal heat generation/ absorption phenomenon. For hs < 0, we have the internal heat absorption eff ect whereas for hs > 0 we can predict the internal heat generation phenomenon. For hs = 0, no heat generation/ absorption can be studied.
The role of stochastic numerical computing paradigm (Khan et al., 2015;Mehmood et al., 2018;Awan et al., 2018, Sabir et al., 2020, Ahmad et al., 2020, Umer et al., 2019 cannot be denied as an alternate, accurate, reliable and robust computational framework to be exploited in future for solving fl uidic systems representing the model for rheology of hydro-magnetic polymeric material with heat generation/absorption and chemical reaction.

CONCLUSIONS
The current investigation provides an understanding for MHD fl ow of UCM fl uid along with the absorptive surface heated from below including chemical reaction and heat generation or absorption. Lie group analysis is performed and comparison tables are constructed to validate the results available in the literature. The fi nal outcomes of the investigations are listed as follows: • Increase in Deborah number decelerates the fl uid fl ow and the boundary layer thickness. • Magnetic fi eld controls the velocity of the fl uid very effi ciently. • Increment in heat generation/absorption parameter boosts up the temperature of the fl uid. • Generative and destructive reactions have confl icting eff ects. • Sherwood number enhances with γ whereas Nusselt number decreases with hs.
December 2020 Journal of the National Science Foundation of Sri Lanka 48 (4)