Scaling group transformation on fluid flow with variable stream conditions

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Abstract

Thermophoresis particle deposition with chemical reaction on Magnetohydrodynamic flow of an electrically conducting fluid over a porous stretching sheet in the presence of a uniform transverse magnetic field with variable stream conditions is investigated using scaling group transformation. Starting from Navier–Stokes equations and using scaling group transformations, the governing equations are obtained in the form of differential equations. The fluid viscosity is assumed to vary as a linear function of temperature. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. Impact of thermophoresis particle deposition in the presence of chemical reaction plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.

Highlights

Thermophoresis particle deposition in the presence of chemical reaction has a substantial effect on the flow field. ► Thermophoresis is expected to alter the concentration boundary layer significantly. ► Present investigation plays a predominant role in the applications of science and technology.

Introduction

Scaling group transformation analysis, also called symmetry analysis, was developed by Sophius Lie to find point transformations that map a given differential equation to itself. This method unifies almost all known exact integration techniques for both ordinary and partial differential equations [1]. Group analysis is the only rigorous mathematical method to find all symmetries of a given differential equation and no ad hoc assumptions or a prior knowledge of the equation under investigation is needed. The boundary layer equations are especially interesting from a physical point of view because they have the capacity to admit a large number of invariant solutions, i.e., basically analytic solutions. In the present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differential equation. Prandtl's boundary layer equations admit more and different symmetry groups. Symmetry groups or simply symmetries are invariant transformations, which do not alter the structural form of the equation under investigation [2]. Newton's law of viscosity states that shear stress is proportional to velocity gradient. Fluids that obey this law are known as Newtonian fluids. Amongst Newtonian fluids we can cite water, benzene, ethyl alcohol, hexane and most solutions of simple molecules. There are numerous fluids that violate Newton's law of viscosity. The non-linear character of the partial differential equations governing the motion of a fluid produces difficulties in solving the equations. In the field of fluid mechanics, most of the researchers try to obtain the similarity solutions in such cases. In case of scaling group of transformations, the group-invariant solutions are nothing but the well known similarity solutions [3]. A special form of Lie-group of transformations, known as scaling group transformation, is used in this paper to find out the full set of symmetries of the problem and then to study which of them are appropriate to provide group-invariant or more specifically similarity solutions.

Thermophoresis is the term describing the fact that small micron sized particles suspended in a non-isothermal gas will acquire a velocity in the direction of decreasing temperature. The gas molecules coming from the hot side of the particles have a greater velocity than those coming from the cold side. The faster moving molecules collide with the particles more forcefully. This difference in momentum leads to the particle developing a velocity in the direction of the cooler temperature. The velocity acquired by the particles is called the thermophoretic velocity and the force experienced by the suspended particles due to the temperature gradient is known as the thermophoretic force. The magnitudes of the thermophoretic force and velocity are proportional to the temperature gradient and depend on many factors like thermal conductivity of aerosol particles and carrier gas. They also depend on the thermophoretic coefficient, the heat capacity of the gas and the Knudsen number. Corrosion of heat exchanger, which reduces heat transfer coefficient, and fouling of gas turbine blades are the examples of this phenomenon. Thermophoresis principle is utilized to manufacture graded index silicon dioxide and germanium dioxide optical fiber preforms used in the field of communications. Thermophoresis phenomenon has many practical applications in removing small particles from gas streams, in determining exhaust gas particle trajectories from combustion devices and in studying the particulate material deposition on turbine blades. It has been found that thermophoresis is the dominant mass transfer mechanism in the modified chemical vapor deposition (MCVD) process as currently used in the fabrication of optical fiber preforms. Thermophoretic deposition of radioactive particles is considered to be one of the important factors causing accidents in nuclear reactors. The detailed study regarding practical applications of thermophoretic phenomenon can be found in Refs. [4], [5]. A paper by Epstein et al. [6] deals with the thermophoretic deposition in natural convection flow from a vertical plate. But the analysis was considered for the cold surface. Garg and Jayaraj [7] analyzed numerically the thermophoretic deposition of small particles due to impingement of a laminar slot jet on an inclined plate using an implicit finite difference scheme. The analysis was done for the cold, hot and adiabatic plate conditions. The study of thermophoresis particle deposition on a vertical plate was extended to porous medium by Chamkha and Pop [8]. Thermophoresis particle deposition under different situations can be seen in Refs. [9], [10], [11], [12], [13], [14]. Thermophoretic deposition of radioactive particles is considered to be one of the important factors causing accidents in nuclear reactors. Goldsmith and May [15] first studied the thermophoretic transport involved in a simple one-dimensional flow for the measurement of the thermophoretic velocity. Thermophoresis in natural convection with variable properties for a laminar flow over a cold vertical flat plate has been studied by Jayaraj et al. [16]. Selim et al. [17] analyzed the effect of surface mass flux on mixed convective flow past a heated vertical flat permeable plate with thermophoresis. Geelhoed et al. [18] presented a particle separator that works on micro scales and that can be implemented on-chip. Recently, Ismoen Muhaimin et al. [19] investigated the effect of thermophoresis particle deposition and variable viscosity on non-Darcy MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction.

Electronic components [20] placed in an enclosure are cooled by natural convection by providing sufficient number of vents to enable the cooled air to enter and the heated air to leave the case. When natural convection cooling is not adequate, air is blown without taking into account the exact need. In this process all kinds of contaminants that are present in the air, such as lint, dust moisture and even oil are deposited on the surface. These contaminants can pile up on the components and plug up narrow passage ways, causing over-heating. It should be remembered that the dust that settles on the electronic components acts as an insulation layer that makes it very difficult for the heat generated in the components to escape. In order to minimize the deposition of contaminants on the surface, the volume flow rate of the air into the surface is to be controlled. This can be achieved by calculating accurately the heat and mass transfer at the surface taking into account the effect of thermophoresis.

Based on the concept of high-mass transfer with a blowing parameter a simple approach for evaluating the effect of wall suction and thermophoresis on aerosol particle deposition from a laminar flow over a flat plate has been discussed by Tsai [21]. The situation of primary interest in this problem is a cooled wall immersed in a hot fluid stream, which can be considered to be a model problem for deposition of aerosol particles from a condensing superheated vapor. Such a wall will capture some of small particles by the mechanisms of Brownian diffusion, convection with suction or blowing and thermophoresis. The technological problems include particle deposition onto a surface from a condensing vapor–gas mixture, a semi-conductor wafer in the electronic industry. Alam et al. [22] have discussed the effect of variable suction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermal radiation. Mixing and recirculation of local fluid streams occur as the fluid moves through tortuous paths in packed beds. This hydrodynamic mixing of fluid at pore level causes thermal and solutal dispersion in porous medium. This becomes more considerable for moderate and fast flows. The book by Nield and Bejan [23] provides more discussions and applications of convective transport in porous media. Detailed discussion and literature survey are available in Ref. [24].

We are particularly interested in cases in which diffusion and chemical reaction occur at roughly the same speed. When diffusion is much faster than chemical reaction, then only chemical factors influence the chemical reaction rate; when diffusion is not much faster than reaction, the diffusion and kinetics interact to produce very different effects. The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reaction. Due to the fast growth of electronic technology, effective cooling of electronic equipment has become warranted and cooling of electronic equipment ranges from individual transistors to main frame computers and from energy suppliers to telephone switch boards and thermal diffusion effect has been utilized for isotopes separation in the mixture between gases with very light molecular weight (hydrogen and helium) and medium molecular weight.

There are two types of reactions such as (i) homogeneous reaction and (ii) heterogeneous reaction. A homogeneous reaction occurs uniformly throughout the given phase, whereas heterogeneous reaction takes place in a restricted region or within the boundary of a phase. The effect of a chemical reaction depends on whether the reaction is heterogeneous or homogeneous. A chemical reaction is said to be first-order, if the rate of reaction is directly proportional to concentration itself. In many industrial processes involving flow and mass transfer over a moving surface, the diffusing species can be generated or absorbed due to some kind of chemical reaction with the ambient fluid, which can greatly affect the flow and hence the properties and quality of the final product. These processes take place in numerous industrial applications, such as the polymer production and the manufacturing of ceramics or glassware. Thus we are particularly interested in cases in which diffusion of the species and chemical reaction occur at roughly the same speed in analyzing the mass transfer phenomenon. The flow of a fluid past a vertical stretching sheet is of fundamental importance since this type of flow constitutes a general and wide class of flows in which the free stream velocity is proportional to the power of the length coordinate measured from the stagnation point. Kandasamy et al. [25] studied the effects of chemical reaction, heat and mass transfer along a vertical stretching sheet with heat source and concentration in the presence of suction or injection. Kandasamy et al. [26] analyzed the thermophoresis and chemical reaction effects on non-Darcy mixed convective heat and mass transfer past a porous wedge with variable viscosity in the presence of suction or injection. Mingchun et al. [27] analyzed the effect of non-thermal equilibrium model of the coupled heat and mass transfer in strong endothermic chemical reaction system of porous media. The recent advances in understanding physics of flows and computational flow modeling (CFM) can make tremendous contributions in chemical engineering. Elperin and Fominykh [28] discussed the effect of exact analytical solution of a convective diffusion from a wedge to a flow with a first-order chemical reaction at the surface. Chamkha [29] investigated the effects of MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Marle [30] analyzed the impact of macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media.

The temperature-dependent fluid viscosity with thermophoresis particle deposition becomes more significant when the concentration gradients and temperature gradients are high. Also, the inertial, dispersion, chemical reaction and suction/injection effects have a significant contribution to convective transport in porous medium. Certainly, the combined effect of these parameters will have large impact on heat and mass transfer rates. In all of the above mentioned studies, fluid viscosity was assumed to be constant. However, it is known that the physical properties of fluid may change significantly with temperature. For lubricating fluids, heat generated by the internal friction and the corresponding rise in temperature affect the viscosity of the fluid and so the fluid viscosity can no longer be assumed constant. The increase of temperature leads to a local increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and so the heat transfer rate at the wall is also affected. Therefore, to predict the flow behavior accurately it is necessary to take into account the viscosity variation for incompressible fluids. Gary et al. [31] and Mehta and Sood [32] showed that, when this effect is included the flow characteristics may change substantially compared to the constant viscosity assumption. Mukhopadhyay et al. [33] investigated the MHD boundary layer flow with variable fluid viscosity over a heated stretching sheet. Mukhopadhyay and Layek [34] studied the effects of thermal radiation and variable fluid viscosity on free convective flow and heat transfer past a porous stretching surface.

The magneto-hydrodynamics of an electrically conducting fluid is encountered in many problems in geophysics, astrophysics, engineering applications and other industrial areas. Hydromagnetic free convection flow has a great significance for the applications in the fields of steller and planetary magnetospheres and aeronautics. Engineers employ magneto-hydrodynamics principles in the design of heat exchangers, pumps, in space vehicle propulsion, thermal protection, control and re-entry and in creating novel power generating systems. However, hydromagnetic flow and heat transfer problems have become more important industrially. In many metallurgical processes involve the cooling of many continuous strips or filaments by drawing them through an electrically conducting fluid subject to a magnetic field, the rate of cooling can be controlled and final product of desired characteristics can be achieved. Another important application of hydromagnetic to metallurgy lies in the purification of molten metals from non-metallic inclusions by the application of a magnetic field. Chakrabarti and Gupta [35] investigated hydromagnetic flow, heat and mass transfer over a stretching sheet. Vajravelu and Hadjinicolaou [36] studied the flow and heat transfer characteristic in an electrically conducting fluid near an isothermal stretching sheet. Sharma and Mathur [37] investigated steady laminar free convection flow of an electrically conducting fluid along a porous hot vertical infinite plate in the presence of heat source or sink. The Navier–Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newton's Law of motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy and diffusion equations. Usually, the term Navier–Stokes equations is used to refer to all of these equations. The steady flow of a Navier/Stokes fluid due to the stretching of a sheet so that the velocity at a particular coordinate is proportional to its location has been studied in much detail within the context of the classical linearly viscous Navier/Stokes fluid and non-Newtonian fluids (see Refs. [38], [39] and many other related papers), and it serves to illustrate the development of boundary layers in such fluids.

Using scaling transformation, three-dimensional, unsteady, laminar boundary layer equations of non-Newtonian fluids were studied by Yurusoy and Pakdemirli [40], [41]. Using scaling group transformation viz. Lie group transformation, they obtained two different reductions to ordinary differential equations. They studied the effects of a moving surface with vertical suction or injection through the porous surface and also analyzed the exact solution of boundary layer equations of a special non-Newtonian fluid over a stretching sheet by the method of Lie group transformation. Yurusoy et al. [42] investigated the Lie group analysis of creeping flow of a second grade fluid. They constructed an exponential type of exact solution using the translation symmetry and a series type of approximate solution using the scaling symmetry and also discussed some boundary value problems. But so far no attempt has been made to analyze the effect of temperature-dependent fluid viscosity with thermophoresis particle deposition on natural convection flow past a porous vertical stretching surface for various parameters using scaling group of transformations viz., Lie group transformations.

Section snippets

Mathematical analysis

We consider a free convective, laminar boundary layer flow and heat and mass transfer of viscous incompressible, Newtonian and electrically conducting fluid over a vertical stretching sheet emerging out of a slit at origin (x=0, y=0) and moving with non-uniform velocity U(x) in the presence of thermal radiation (Fig. 1). A uniform transverse magnetic field of strength B0 is applied parallel to the y-axis. The chemical reaction is taking place in the flow and the effects of thermophoresis are

Numerical solution

The set of non-linear differential Eqs. (24), (25), (26) with boundary conditions (27) was solved numerically using Runge Kutta Gill integration scheme [45] with a systematic guessing of f(0),θ(0)andϕ(0) by the shooting technique until the boundary conditions at infinity f′(0), θ(0) and ϕ(0) decay exponentially to zero with γc, ζ, τ and M2 as prescribed parameters. Effects of heat and mass transfer are studied for different values of thermophoresis particle deposition, strength of magnetic

Results and discussion

To analyze the results, numerical computation has been carried out using the method described in the previous section for various values of the chemical reaction parameter γc, temperature-dependent fluid viscosity parameter ζ, magnetic parameter M2, suction/injection parameter S, Prandtl number Pr, thermophoresis parameter τ and Schmidt number Sc. For illustrations of the results, numerical values are plotted in the Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11

Conclusions

Using scaling group transformation analysis, we first find the symmetries of the partial differential equations and then reduce the equations to ordinary differential equations by using scaling and translational symmetries. Exact solutions for translation symmetry and numerical solution for scaling symmetry are obtained. From the numerical results, it is predict that the effect of increasing temperature-dependent fluid viscosity parameter with uniform chemical reaction on a viscous

Acknowledgment

The authors wish to express their cordial thanks to our beloved Vice Chancellor and The Dean of Faculty of Science, Arts and Heritage, UTHM, Malaysia for their encouragements.

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