Theoretical analysis of modified eyring powell fluid flow
Graphical abstract
Introduction
Fluids are classified into two broad categories as Newtonian and non-Newtonian fluids. The Newtonian fluids obey a linear relationship between the shear stress and shear strain rate while non-Newtonian fluids do not obey the linear relationship [1], [2], [3]. The class of non-Newtonian fluids is further divided into time-dependent fluids (thixotropic and rheopectic), time-independent (shear-thinning and shear-thickening) and viscoelastic fluids. Thixotropic fluid is the one whose viscosity decreases with time while rheopectic fluid is that whose viscosity increases with time. shear-thickening fluid (also called dilatant fluid) increases in viscosity as shear stress increases while shear-thinning fluid (also called pseudoplastic fluid) decreases in viscosity as shear stress increases. Casson fluid, Carreau fluid, Jeffrey fluids, Willaiamson fluid, Eyring-Powell fluid etc. are examples of non-Newtonian fluids. Apart from air, water and oil, most of the fluids in natural, engineering, industrial and other practical processes are non-Newtonian fluids. Technological applications of fluids include application of shear-thinning fluids in dropless paints, shear-thickening fluid in active dampers to dampen sudden acceleration [4], hybrid fluids in body armours [5], extraction of crude oil [6], and biomedical applications [7]. Consequently, it is important to have a representation for stress tensor to carry out any theoretical analysis of a fluid. One primary advantage of Eyring-Powell fluid model is that it was derived from the kinetic theory of liquids rather that empirical relation (from which most of the non-Newtonian fluid models are derived). It was initially modelled to capture fluids that experience creep (a permanent reordering of fluid atoms due to application of high stress, just below the yield stress, to the fluid) [8]. Such fluids include Colloidal glasses [9], cosmetics and paints [10], polyatomic gasses and viscoelastic fluid [11] and clay slips and greases [12].
Sirohi et al. [13] studied the laminar incompressible flow of Eyring-Powell fluid near a suddenly accelerated plate and solved the equations under three different conditions; orthogonal collocation, satisfaction of asymptotic boundary conditions and transformation of boundary value problem to initial value problem and recorded that velocity increases with an increase in Eyring-Powell parameter. Javed et al. [14] extended the work of Sirohi et al. [13] to study the variation of velocity and skin friction coefficient in a boundary layer flow of Eyring-Powell over a stretching sheet. The results indicate that velocity increases with increasing Eyring-Powell parameter. Malik et al. [15] further investigated the flow of Eyring-Powell fluid due to a stretching cylinder with variable viscosity. Two viscosity models were used, namely; Reynold’s model and Vogel’s model. The results show that velocity profile decreases for increasing Eyring-Powell parameters. Agbaje et al. [16] investigated an unsteady boundary layer flow of an incompressible Eyring-Powell nanofluid over shrinking sheet. Velocity profile is found to reduce with the Eyring-Powell fluid parameters. Babu et al. [17] analysed the heat and mass transfer in MHD Eyring-Powell nanofluid flow in porous cone, considering suction and injection. The velocity profile is found to increase with increasing values of the material parameters. Akinshilo and Olaye [18] studied the flow of Eyring-Powell fluid in a pipe and discovered that flow velocity increases with increasing Eyring-Powell parameter. Zhang et al. [19] studied the flow of Eyring-Powell fluid in a permeable surface and Riaz et al. [20] studied the role of hybrid nanoparticles in thermal of Eyring-Powell fluid. Oke and Mutuku [21] studied Eyring-Powell fluid flow over a rotating non-uniform surface and the results show that velocity increases with increasing Eyring-Powell parameter. Following the results from the above cited articles, some authors have reported increase in velocity with Eyring-Powell parameter while others reported decrease in velocity with increasing Eyring-Powell parameter [29], [30].
The seemingly contradicting results on the effect Eyring-Powell parameter on the flow velocity arises often because of the shear-thinning properties of the Eyring-Powell fluid. It is a usual practice to consider Eyring-Powell fluid as a fluid that exhibits either Newtonian fluid properties or shear-thinning fluid property under different conditions. Meanwhile, by keenly following the above cited works, it is observed that Eyring-Powell can also exhibit shear-thickening fluid properties. This explains why there are discrepancies in the results reported on the influence of the material parameter on flow velocity. In this paper, the shear-thickening property of the Eyring-Powell fluid is properly exploited. A new class of fluid, named Modified Eyring-Powell (MEP), is proposed in this study. It is a modified form of the Eyring-Powell fluid which under proper conditions, exhibits Newtonian, shear-thinning or shear-thickening properties. The fluid is decorated with a parameter that makes the shear-thinning and shear-thickening features adjustable. Effects of the material parameter (also called deformation parameter in this paper) and Prandtl number are investigated on the flow dynamics of MEP fluid.
Section snippets
Formulation of Governing Equations
The power law describes relationship between the yield strength and strain rate [22]. This relationship identifies the need to break a single strong bond holding atoms of an element together. Meanwhile, experimental data observed in clay slips and greases show that there are two types of bonds to be broken and this led to the relaxation theory [12], [23], [24]. The relaxation theory sums up the forces required to break a strong bond (governed by the power law) and a weak bond (governed by the
Results and Discussion
The flow of Modified Eyring-Powell over a linearly stretching flat plate shown in figure (1) is studied. The effects of Prandtl number and deformation parameters are depicted as graphs and tables.
Prandtl number is the measure of momentum diffusivity to thermal diffusivity. Increasing Prandtl number means a reduction in thermal diffusivity and thus it results in a reduction in temperature profile. Figure (2) illustrates the reduction in flow temperature as Prandtl number increases. Effects of
Conclusion
A new class of fluid called the Modified Eyring-Powell (MEP) fluid is introduced in this paper. The MEP fluid exhibits the shear-thinning and shear-thickening properties under certain conditions dictated by the deformation parameter A theoretical analysis of the heat transfer rate is carried out on the flow of MEP on a linearly stretching flat plate and the results are depicted as graphs and tables. The following results are obtained in the analysis;
- 1.
Reduction in flow temperature as Prandtl
Declaration of Competing Interest
The authors have declared that no competing interests.
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2022, Journal of the Taiwan Institute of Chemical EngineersCitation Excerpt :First, the Eyring-Powell model is founded on the kinetic theory rather than empirical formulas. Second, it reverts to Newtonian behavior for the low and high shear rates [26]. Hayat et al. [27] considered 3D Eyring Powell fluid flow over a surface with radiative effects.