Analysis of buoyancy driven flow of a reactive heat generating third grade fluid in a parallel channel having convective boundary conditions
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Abstract
The study investigated the impacts of buoyancy force and internal heat source on a reactive third grade fluid flow within parallel channel. The results obtained for the coupled equations regulating the flow of fluid which are strongly nonlinear are secured by applying Adomian decomposition method (modified). The significant effects of buoyancy force and heat source are noticed to increase the speed transfer of heat within the flow regime. Also, the prevalent outcome of thermal energy at the lower and upper plates reduces with rising values of buoyancy force and reduces with heat source. The present result is compared with the earlier published article where the influence of buoyancy force and heat source were not accounted for.
Keywords
Buoyancy force Thirdgrade fluid Heat source Convective cooling Adomian decomposition method (modified)1 Introduction
Studies involving nonNewtonian fluids recently have been on tremendous increase due to its numerous and significant applications in industries, geology, petrochemical engineering, technology, extraction of crude oil, to mention a few. Example of such fluids as illustrated in [1] are molten plastics, polymers, pulps, foods, mud and most biological fluids with higher molecular weight components. As yet, tremendous efforts are concluded on both Newtonian and nonNewtonian fluid models. One can assess the recent developments in this direction in [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. These types of fluids are categorized as thirdgrade that belongs to a relevant category of nonNewtonian fluid alongside the complex behaviour and conditions that cannot easily be described, hence, so many considerable researches were carried out to formulate the fluid behaviour. However, according to the studies investigated in [16, 17], the thirdgrade design is simplified and efficient of halsening and demonstrating the regular stress effect of shear thinning/thickening. In addition to that, of recent, [18, 19] studied the Magnetohydrodynamic (MHD) mixed convection flow of third grade fluid aimed at an exponentially stretching sheet where the broad, flat layer of the material on a surface is convectively heated and at the same time examined Magnetohydrodynamic (MHD) stretched flow of third grade Nano liquid with convective surface condition. Among relevant information on third grade fluid design with different physical aspects from researchers are extensively presented in [20, 21, 22, 23, 24, 25]
Meanwhile, investigation in [20] revealed the significant effect of buoyancy on fluid motion that is subjected to gravitational forces and variations of measure in the mass of matter contained by specified volume which can occur when there is change in fluid temperature. In order to show the appreciable effect of buoyancy force [20], further explored the compound influence of buoyancy force and asymmetric convective cooling on unsteady MHD flow within the channel and heat transfer in a reactive third grade fluid. Additionally, [26] studied the impacts of thermal buoyancy on the boundary layer flow over an upright plate with convective surface boundary conditions. Moreover, [27] presented the analysis of first and second law of thermodynamics on fluid flow and heat transfer inside a vertical channel with respect to the compound actions of buoyancy force, constant pressure gradient and magnetic field strength. Other related surveys showing the impact of buoyancy force on fluid motion with other physical parameters can be found in existing literature, to mention a few [28, 29, 30, 31].
However, from engineering and industrial applications, the effect of heat transfer on a fluid flow within parallel plates cannot be totally neglected as recently discussed in [32]. In support of that, the estimation of heat transfer depends on temperature which raises the interaction of moving fluid and thus the conduct of the internal energy of the flow system is presented in [33]. Other studies showing the effects of internal heat on the fluid flow include [34] where investigation of free convective flow of fluid with heat source/sink between vertical parallel porous plates. Recently, [35] proposed the general understanding of thermal transfer of magneto hydrodynamic nonNewtonian fluid flow over a dispersing sheet occurring together with exponential heat source. Also, in addition to that, [36] put forward the influence of heat flux design initiated by Cattaneo–Christov on the flow across a wedge and a cone of which the effect of nonuniform wall temperatures are also examined. Other investigations on the impact of heat source are extensively discussed in [37, 38, 39].
Hence, the present study is to extend the investigation in [40] by examining the impact of buoyancy force on a reactive fluid of thirdgrade flowing steadily inbetween upper and lower plates, subject to the effect of internal heat generation with symmetrical convective cooling the walls which was not accounted for in their study. The significant effect of heat source in the fluid motion and heat transfer cannot be overlooked as discussed in [41], hence; the study investigates the influence of heat generation on the fluid motion and thermal energy in the flow regime. The problem is strongly nonlinear with paired differential equations regulating the momentum and energy distributions obtained by means of employing the use of a rapidly convergent modified Adomian decomposition method. The intended modification method is seen to be more reliable as compared with the standard Adomian decomposition method. The major studies on this method are presented extensively in [42, 43] as the series converge with less iteration. Finally, the effects of the Grashof number (\(G_r\)) which is a function of buoyancy force and the internal heat generation that is a linear relation of temperature are thereby presented.
2 Mathematical formulation
3 Method of solution
4 Results and discussion
This portion extensively revealed the impact of buoyancy on a reactive third grade fluid inbetween parallel plates with convective cooling the walls under the influence of internal heat source combined with other essential flow properties are thereby presented and discussed. It is worthy to note that our result shall be equivalent to that of [40] when the heat source parameter (\(\alpha\)) and buoyancy effect parameter known as Grashof number (\(G_r\)) are both equal to 0.
Comparison of numerical results of the velocity distribution
\(\epsilon = \gamma = 0.1, \lambda = 0.5, Bi =10, m = 1, \alpha = G_r = 0\)  

y  W(y)PM [40]  W(y)mADM  \(\hbox {Absolute error}\) 
\(\,1\)  0  \(4.510281037534 \times 10^{17}\)  \(4.510281037534 \times 10^{17}\) 
\(\,0.75\)  0.19851653823852540  0.19851653823852541  \(2.77556 \times 10^{17}\) 
\(\,0.50\)  0.34465078125000004  0.34465078125000004  0 
\(\,0.25\)  0.43574060440063480  0.43574060440063480  0 
0  0.46680000000000000  0.46680000000000005  \(5.55112 \times 10^{17}\) 
0.25  0.43574060440063480  0.43574060440063480  0 
0.50  0.34465078125000004  0.34465078125000004  0 
0.75  0.19851653823852540  0.19851653823852541  \(2.77556 \times 10^{17}\) 
1  0  \(4.510281037534 \times 10^{17}\)  \(4.510281037534 \times 10^{17}\) 
5 Conclusion

The fluid motion increases with rising values of \((G_r)\) and \((\alpha )\).

The maximum temperature is obtained at the centreline with increasing rates of \((G_r)\) and \((\alpha )\).

The least rate of entropy generation occur around the significance zone and increases to the uttermost around the plate surfaces with respect to \((G_r)\) and \((\alpha )\).

The controlled impact of thermal energy irreversibility at the lower and upper plates reduces the impact of \((G_r)\) and \((\alpha )\)
Notes
Compliance with ethical standards
Conflict of interest
The authors hereby declare that there is no conflict of interests as regards the publication of this investigation.
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