Publication on International Journal of Biology, Physics & Mathematics
Publication Date: March, 2020
Abiodun O. Ajibade & Bolaji A. Shehu
Department of Mathematic, Ahmadu Bello, University, Zaria
Department of Mathematics and Statistics, Kaduna Polytechnics, Kaduna
This study is about the natural convection flow of a fluid that conducts electricity in a vertical channel with induced magnetic field, viscous and Ohmic heating together with variable thermal conductivity and variable viscosity. A transverse magnetic field is employed on the fluid through the plates. One of the plates is subjected to an isothermal condition. While the other plate remains at a temperature equal to zero. The dimensional governing equations are altered (using suitable transforms) into dimensionless coupled system of ordinary differential equations. These equations are nonlinear and hence solved numerically using the differential transform method (DTM). The consequences of varying the Eckert number, magnetic parameter, variable viscosity, variable thermal conductivity and the other relevant dimensionless parameters involved in the study on the velocity, temperature, magnetic field and the current density are observed. It is obtained that increase in the viscous dissipation of the fluid results in the increase in its velocity, temperature, current density and the heat transfer. While it decreases the magnetic field profile. In addition, the skin friction on both plates grows and the heat transfer increases as the magnetic parameter decreases.
Keywords: Viscous dissipation; Ohmic heating; induced magnetic field; variable fluid properties; differential transform method (DTM).
Fluid flow through natural convection is caused entirely by buoyancy forces, which arise from density variations in a field of gravity. Unlike in forced convection where the velocity of flow of the fluid does not depend on temperature difference, in natural convection, the distribution of velocity and temperature are interconnected (coupled) and always considered together. If the fluid is incompressible then the density variation due to changes in pressure are negligible. However, density changes due to non-uniform heating of the fluid cannot be neglected. Since such changes are responsible for initiating free convection. An interesting feature of magneto-hydrodynamic heat transfer problems is that, the usual Reynolds analogy between skin friction and heat transfer, as in non-conducting fluids, does not, in general, hold good. This is because, in addition to viscous heating, there is a Joule heating due to the flowing electric current in the fluid.
Studies involving viscous dissipation and Ohmic heating on MHD natural convection flow of an electrically conducting fluid in a vertical channel with induced magnetic field is attracting a lot of interest because of its importance in the design of various industrial devices, which are subjected to large variation of gravitational force. Its application is found in heat exchangers designs, wire and glass fiber drawing and in nuclear engineering in connection with the cooling of reactors. Furthermore, Ohmic heating processes makes it possible to use high temperature at short time on suspended materials. This results in increase in the quality of products. A lot of researches considering different aspect of natural convection flow of fluids with viscous dissipation and Ohmic heating in different channels with the fluid being electrically conducting or not; and with or without magnetic field considerations were conducted recently.
Some of the works in this regard include the study by Jha and Ajibade (1) on the Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions. They solved the governing equations analytically and discovered that heat is being transferred from the fluid with viscous dissipation to the plate when the fluid has small Prandtl. Parveen and Alim (2) used implicit finite difference method of solution to the governing nonlinear differential equation as a tool to studied Joule Heating effect on Magneto-hydrodynamic Natural Convection Flow along a Vertical Wavy Surface. They observed that the effect of increasing Joule heating parameter results in the boundary layers becoming thicker; decrease in the local rate of heat transfer and increasing the local skin friction coefficient. The effects of hall current and viscous dissipation on MHD free convection fluid flow in a rotating system are studied by Abdul Quader and Alam (3). They deduced that the fluid temperature increases due to increase in the Eckert number. While it decreases due to increase in the prandtl number after solving the governing equation using the explicit finite difference scheme. Alam et al (4) used the finite difference method together with Newton’s linearization approximation to discuss the conjugate effects of viscous dissipation and variable viscosity on free convection flow over a sphere with joule heating and heat conduction. They argued that Increase in the values of viscous dissipation parameter enhances the velocity profile, the temperature profile and the local skin friction coefficient; while it decreases the heat transfer rate. That an enhanced Joule heating causes an increase in both the velocity and temperature profiles. Lakshmi et al (5), studied thermal energy increase and dissipation on MHD heat and mass concentration gradient flow over a plate. They employed the Crank-Nicolson’s implicit finite difference scheme to solve the governing differential equations and deduced that temperature of the fluid increases as the viscous dissipation parameter increases. Kabir et al (6) investigated the influence of viscous dissipation and thermal energy increase on free convection flow down a wavy surface. The equation of the problem was solved numerically by the keller-box method and they concluded that the velocity, temperature of the fluid and its local skin friction coefficient increases with increasing value of the viscous dissipation; whereas the rate of heat transfer decreases. Jaber (7) investigated the viscous dissipation and joule heating effects on the flow of a magneto-hydrodynamics fluid having variable properties past a vertical plate. He solved the governing equations numerically and deduced that the velocity increases, the temperature increases, the shear stress and the heat transfer increases at the wall as the viscous dissipation and the thermal conductivity parameters increase. Haque et al (8), worked on the effects of viscous dissipation on MHD natural convection flow over a sphere with temperature dependent thermal conductivity with heat generation. From this work, they were able to discover that the velocity and the temperature of the fluid improved with increasing thermal conductivity, increasing heat generation and viscous dissipation parameters. That skin friction increases with increasing viscous dissipation, heat generation and the thermal conductivity variation parameter. In another study, Nath and Parveen (9) used Keller-Box method to work on viscous dissipation and heat absorption effect on natural convection flow with uniform surface temperature down a wavy surface. In this work they concluded that increase in the viscous dissipation has increasing effect on the skin friction and decreasing effect on the rate of heat transfer. Emad et al (10) analysed the influence of dissipation and resistive heating on MHD free convection flow over a flat plate with the combined effect of Hall currents for the case of power-law variation of the wall temperature. They were able to do this by solving the transformed governing equation using a fifth-order Runge–Kutta–Fehlberg scheme with Newton–Raphson shooting method and concluded that the presence of dissipation and resistive heating decreases the local Nusselt number. Kishore et al (11) discussed the result of heat exchange and viscous dissipation on MHD temperature dependent natural convection flow over a vertical plate. They solved the governing equations numerically by the implicit finite difference method of Crank – Nicolson’s type, and discovered that velocity increases with increase in the thermal Grashoff number, acceleration parameter, Eckert number and time
Induced magnetic field plays a great role in the flow of fluids, which conducts electricity because it also produces additional magnetic field in the flowing fluid. This magnetic field generates a resistive force on the fluid flow and as a result, it modifies the original magnetic field. Therefore, in many concrete situations it is important to consider the influence of magnetic induction in the magneto-hydrodynamic equations. In the studies above, the influence of magnetic induction has been neglected so as to simplify the solution method of the problem. On account of this, Sarveshanand and Singh (12) discussed Magneto-hydrodynamic natural convection along vertical porous plates in which the induction of magnetic field is considered. They observed that the magnetic field induced in the fluid is significantly increased when the suction parameter, Prandtl number and the Hartmann number are increased. Prakash et al (13) in their study of MHD mixed convective flow over vertical plate where heat is generated due to viscosity and magnetization considered the influence of magnetic induction. And it is found that intensity of the induced magnetic field reduces when the quantity of the magnetic parameter, magnetic prandtl number and the viscous dissipation increases. Magnetic induction was also considered by Raju et al (14) in their work on the effect of heat transfer on a flowing fluid which dissipates heat due to viscous actions of its molecules past a vertical plate. They used a perturbative technique to solve the governing equations and deduced that the induced magnetic field decreases as the magnetic prandtl number is increased.
This study considers the influence of induced magnetic field and the heat generated due to the passage of the conducting fluid, through the magnetically induced vertical channels and that generated due to the interaction of the molecules of the viscous fluid on the MHD free convective flow with variable viscosity and thermal conductivity. The solution of the transformed governing boundary value equations involving the velocity, temperature and the induced magnetic fields are obtained by the differential transform method. Also, the expression for the electric current induced, the skin friction and the thermal exchange rate in terms of the Nusselt number are obtained. The corresponding results displayed in graphs and tables show that the Joule heating, viscous dissipation and the magnetic parameter have significant effects on the fluid flow considered in this study.
2. A GENERAL DESCRIPTION OF THE DIFFERENTIAL TRANSFORM METHOD (DTM)
Given a function y(x) its differential transform is defined by Zhou (15) as follows:
The inverse of which is defined as equation (2) below
In Equations (1) and (2), y(x) is the given function and Y (k) is the transformed function.
From the two Equations, we obtain equation (3) below.
From equation (3) it can be deduced that the notion of DTM is obtained from Taylor series expansion, but the differential transform method does not evaluate the derivatives symbolically as does in Taylor’s series. Rather, the relative derivatives are calculated by an iterative method. Following Zhou (15) and Ahmad et al (16) the basic operations of the DTM is defined as shown on table (1) below
Table 1. Basic operations of the one-dimensional differential transform method
In this study the given function is represented by the lower letter case and upper letter case represents the transformed function. The basic operations on table 1 are proved using equations (1) and (2). In practice, the function y(x) is written as a finite series using equation (2) in the form of equation 4 below
The value of n is decided by the convergence of the natural frequency in this work. Also, the sum of further terms written as the equation (5) below is taken to be negligibly small.
3. DISCRIPTION OF THE PROBLEM AND THE GOVERNING EQUATIONS
Consider a steady natural convection boundary layer flow of a viscous incompressible and electrically conducting fluid with induced magnetic field between two vertically parallel and infinitely long plates as illustrated in figure1 below.
Figure 1. Geometry of the problem
Taking and as the components of the velocities (along the plates) and (normal to the plates), respectively and h the distance between the two permeable vertical plates (figure 1). Then, keeping one of the plates at constant heat flux ( ) and the other maintained at constant temperature , so that for the steady incompressible boundary layer flow, the governing equations for the continuity, momentum and energy equations can be written as in Sarveshanand and Singh (12) adding the viscous dissipation and the Ohmic heating terms to the Temperature equation shown below.
With the boundary conditions
and k* represents the variable viscosity and variable thermal conductivity respectively. The fluid density is represented by ρ; Cp is the specific heat of the fluid at constant pressure. Because of the fluid’s electrical conductivity (σ) a magnetic field is induced as the fluid moves along the Equation (5) above is the expression for the velocity field with the temperature dependent viscosity taken as an inverse function of temperature following Singh and Agarwal (17) as
This is equivalent to equation (12) below
Equation (6) is the expression for the induced magnetic field and equation (7) is the expression for the Temperature field with the viscous dissipation and Joule heating terms. The variable thermal conductivity taken as an inverse function of temperature following Hazarika and Konch
Where a, b and T0 are constants and their values depend on the fluid’s reference state and its heat conductivity properties. That is the kinematic viscosity (ʋ) and thermal conductivity (k). Also, for liquids a > 0; b > 0 and for gases a < 0; b < 0.Where the dynamic viscosity and thermal conductivity of the baseline fluid (ambient fluid) are µ0 and k0 respectively.
From equation (11) let λ = aΔT which is the viscosity variation parameter. It is obvious that it increases with increasing temperature. So that equation (11) becomes equation (14) below
From equation (14), it is clear that the value of the dimensionless viscosity decreases with increasing temperature. That is when λ > 0. Since increasing temperature decreases the viscosity of liquids; while it increases that of gases, λ is represented with different signs for both. Likewise the same argument can be put forward for the dimensionless thermal conductivity presented in equation (15).
Where ε = bΔT is taking as the thermal conductivity variation parameter; which is taken to be negative in liquids and positive in gases.
The non-dimensional parameters given in equations (17a) and (17b) are employed to change the governing equations (5) to (9) to the non-dimensional form.
Hence, the nondimensional systems of nonlinear ordinary differential equations of the second order (18) to (22) are obtained.
Where Ha is the Hartmann number, Pm is the magnetic Prandtl number, Pr is the prandtl number, Ec is the Eckert number and M is the magnetic parameter.
4. SOLUTION BY DIFFERENTIAL TRANSFORM METHOD
In order to solve the non-linear governing continuity, momentum and energy equations, the equations (5) to (9) are reduced into the equations (18) to (22) using the set of non-dimensional parameters (17a) and (17b). The equations obtained constitute a system of nonlinear-coupled second order ordinary differential equations. These are solved numerically by using the Differential Transform Method (DTM). By which we obtain the iterative relations (23) to (26) below for the respective equations (18) to (22) in the system.
Also, applying DTM to the boundary conditions gives the expressions (26) below
U (0) = 0; U (1) = a; T (1) = -1; T (0) = c; B (0) = 0; B (1) = d. (26)
The expressions U (K), B (K) and T (K) are the differential transforms of the velocity equation U(y), the induced magnetic field equation B(y) and the temperature equation T(y). The constants ‘a, c and d’ are values from the boundary conditions and will be determined. Using the symbolic software Maple (21) we obtain the following solutions:
The solution of the induced current density equation is given by equation
The skin friction is given by the equation (31). The solution of which is given by equation (32); and those at y = 0 and y = 1 are given by the respective equations (33) and (34).
5. RESULTS AND DISCUSSIONS
A semi-analytic technique called the Differential Transform Method is used to solve the equations of free convection flow of an electrically conducting fluid in a vertical channel with induced magnetic field, viscous dissipation and Ohmic. For the purpose of discussing the results, the approximate solutions are obtained for the various flow parameters, which includes the Eckert number (Ec), the magnetic parameter (M), the magnetic Prandtl number (Pm), the thermal conductivity parameter(ε), the Prandtl number ( Pr ), the Hartmann number ( Ha ) and viscosity parameter( λ ). For these parameters, various computations have been carried out with values in the following range: , , , , . By so doing, relations for the velocity
profile, magnetic field profile, temperature profile, skin friction, the Nusselt number (which measures the heat flux) and the induced current density are obtained. As established in the work of Oke (19) the convergence of the iteration process is quite rapid. The effects of the flow quantities which affects the velocity field (U), the magnetic field (B), the temperature field (T) and the current density profile (J) are discussed with the help of the figures (2), (3), (4) and (5) respectively. Also, the effects of the relevant parameters on the skin friction (τ) is shown in table( 2) and on the Nuselt number (Nu) ( or the rate of heat transfer ) is shown in tables (3). Table (4), is used to compare the result of the present work with that f Sarveshnand and Singh (12).
Figure 2, Velocity profile varying the parameters (a) Ec, (b) M, (c) λ, (d) ε, (e) Pm, (f) Ha (g) Pr
In figure 2(a), it is seen that the velocity profile is increases as the Eckert number increase. Eckert number is the viscous dissipation parameter, so an increase in it means the dissipative force increases, which energize the fluid. Hence, the increase in its velocity profile. Figure 2(b) depicts the velocity variations for varying values of (M). It shows that as the magnetic parameter decreases the velocity increases. Magnetic parameter depicts the ratio of magnetic induction to the viscous force. Also, figure 2(c) shows an enhanced velocity profile as the viscosity parameter (λ) increase. This is expected since the viscosity parameter bears an inverse relationship with the viscosity of the fluid. The velocity is seen to be enhanced with decreasing thermal conductivity parameter as shown in figure 2(d). Increased velocity profile is also observed, as Pm is decreased shown in figure 2(e). When Pm is reduced, the strength of the magnetic field reduces which reduces the development of the flow resistive Lorentz force. This improves velocity of the flow. From figure 2(f) it is seen that decrease in Ha results in increase of the velocity of the flow. As the Hartmann number decreases, the viscous force becomes stronger than the magnetic force of the fluid thereby limiting the resistive influence of the Lorentz force. Hence, an enhanced motion of the fluid is achieved. Figure 2(g) shows that decreasing the prandtl number results in increase in the velocity of the fluid. This is because heat diffuses more readily than momentum.
Figure 3. Induced magnetic field profile varying the parameters (a) Ec (b) M (c) λ (d) ε (e) Pm (f) Ha (g) Pr.
In figure 3(a) it is seen that as the Ec increases, Pm is decrease. Increase in the Eckert number implies greater viscous dissipative heat, which is a consequence of work done against viscous stresses, which causes the reduction of the magnetic field. Figure 3(b) show that as the magnetic parameter increases the magnetic field also increases. This happens because increasing the magnetic parameter makes the magnetic induction in the fluid to be greater than the viscous force. Figure 3(c) shows that as the viscosity parameter decreases, the magnetic field increases. It is observed in figure 3(d) that increase in the thermal conductivity parameter causes the induction of magnetic field to increase. In figure 3(e) as Pm decreases, the magnetic induction is enhanced. Figure 3(f) depicts that as the Hartmann number increases, the magnetic induction decreases. From figure 3(g) it can be seen that as the prandtl number increases, the magnetic field decreases. All these observations agree with what is obtained in literature.
Figure 4, Temperature profile varying the parameters (a) Ec (b) ε (c) M (d) Pr.
The rise in temprature depicted in figure 4(a) is as a result of increasing the Eckert number (Ec). Increasing the Eckert number results in greater viscous dissipative heat which helps in raising the temperature profile. Figure 4(b) depicts that a rise in the magnetic parameter (M) results in the decrease of temperature. The Lorentz force developed due to the increase of M reduces the fluid velocity leading to the decrease of the temperature. Figure 4(c) shows that as the thermal conductivity parameter increases the temprature of the fluid decreases. This is borne from the fact that thermal conductivity affects the molecular diffusion of the fluid. Similarly, as Pr increases shown by figure 5, the temperature profile decreases. When the prandtl number is increased, the molecular movement of the fluid is reduced causing a drop of temperature in the flow field.
Figure 5, Induced current profile varying Magnetic Prandtl number (Pm)
Figure 20 shows that as the Eckert number increases, the induced current density also increases. When the dissipative force is increase as the Eckert number is being increased the fluid flows faster. This generates greater induced current due to the presence of the transverse magnetic field in the flowing fluid. In figure 21, we can see that as the magnetic parameter is increased, the induced current profile reduces. A growing magnetic parameter retards the velocity of the flow due to the magnetic pull caused by the Lorentz force which develops in the flow field. A consequence of the reduced velocity of a conducting fluid within transverse magnetic field is the reduced generation of induced current density. It is observed in figure 22 that as the viscosity decreases, the induced current (J) profile increases. This happens because the electrically conducting fluid is moving faster in the flow field, which contains magnetic fields. Figure 23 depicts that as the thermal conductivity parameter increases, J decreases. In figure 24 it is observed that as the prandtl number increases, the J also increases. As the Hartman number increases shown in figure 25, the J increases too. The induced current density increases when the Hartmann number (Ha) increases. Increase in Ha occurs when the electromagnetic force increases against the viscous force. This causes a faster motion of the fluid within the channel and resulting into greater induced current. Finally, in figure 26 it is seen that as the Pm decreases, the current also decreases.
Table 2 below shows the influence of the physical parameters namely Eckert number, magnetic parameter, viscosity variation parameter, thermal conductivity variation parameter, Hartmann number, magnetic prandtl number and the prandtl number on the coefficient of skin-friction which represent the shearing stress at the two plates. Table 3 represents the values of the heat transfer coefficient called the Nusselt number (Nu) for the various relevant parameters. Table 4 shows a comparison of the present work with that of Sarveshanand and Singh (12) with constant viscosity and thermal conductivity in the absence of viscous dissipation and Ohmic heating.
Table 2, The effect of (a) Ec and M (b) λ and ε (c)Ha and Pm (d) Pr, on heat transfer coefficient Nu
It is observed in table 2(a), that as the Eckert number (Ec) increases the skin friction (τ) also increases in the first plate; but in the second plate it decreases between the range 0.001 to 4.5 and increases between the ranges 4.5 to 8.0. In the same table as the magnetic parameter (M) increases, τ decreases on both plates. Table 2(b) shows how the viscosity parameter (λ) and the thermal conductivity parameter (ε) affect the skin friction around the two plates. The result is that decreasing (λ) decreases the skin friction on both plates and as ε increases, the fluid does not get heated easily which affect the velocity negatively. Hence, the result is a decrease in the skin friction as is shown in the table. Table 2(c) shows the effects of Hartmann number (Ha) and the Magnetic Prandtl number (Pm) on the skin friction. By increasing the (Ha) we observe the decreasing of the skin friction on both plates. Decrease in Pm number, results in increase in the skin friction on both plates. This is the case because increasing Ha and Pm reduces the velocity of the fluid due to Lorentz force pulling effect. Table 2(d) shows that increasing the prandtl number the skin friction on the first plate increases. Whereas it decreases from 0.7 to 4.0 and it increases from 4.0 to 6.0 on the second plate. This is because heat spreads away from the wall faster for higher values of Pr. Whereas, for smaller Pr the thermal boundary layer thickens increases hence; the rate of heat transfer is reduced. This is in agreement with the result obtained by Dada and Oladesusi (20).
Table 3, The effect of (a) Ec and M (b) λ and ε (c)Ha and Pm (d) Pr, on heat transfer coefficient Nu.
The effects of the Eckert number and the magnetic parameter on the heat transfer, taking place in the fluid are displayed in table 3(a). It is observed that as the Eckert number increases the Nuselt number increases but it behaves differently with the increase of the magnetic parameter. In table 3(b), as the viscous variation parameter (λ) increases negatively between – 0.005 to – 0.5 the Nuselt number increases but it decreases between the values – 0.5 to -1.0. Increase in the values of thermal conductivity parameter (ε) results in the decrease of the Nusselt number. The effect of Ha and the magnetic prandtl number on the Nusselt number shown on table 3(c). It shows that as Ha increase the Nu increases whereas the Nusselt number decreases as the magnetic prandtl number increases. Prandtl number affects the Nu in the same direction as shown on table 3(d).
Table 4, Comparison of the present work with Sarveshanand and Singh (12).
Table 4 shows the results of the comparison of the present work with that of Sarveshanand and Singh (12). We let Ec = 0, M = 0 which means that there are no effects of viscous dissipation and Joule heating. λ = ε = 0 means that the viscosity and thermal conductivity are constants. The table shows that the results agree to a very appreciable degree.
The presence of Joule heating, viscous dissipation as well as the magnetic parameter has significant effects on the free convection flow of a conducting fluid in a vertical channel with induced magnetic field just as it is presented in this study. The main findings together with the effects of various other paramount parameters on the velocity field, induced magnetic field, temperature field, induced current density, skin friction profiles and Nusselt number has been depicted in graphs and tables as seen above. It is found that:
(1) Increased dissipation in the fluid, results in the increase in its velocity, temperature, current density and the thermal exchange rate. While it decreases the magnetic induction. Furthermore, increasing the viscous dissipation of the fluid leads to increase of the skin friction in the first plate while it fluctuates in the second plate between some ranges.
(2) Decreasing the ratio of magnetic induction to the viscous force (magnetic parameter) in the fluid leads to increase in the velocity, the temperature of the fluid and the quantity of current induced in the fluid. Also, the drag caused by friction on both plates grows and the heat exchange rate increases as the magnetic parameter decreases. The magnetic induction of the fluid is directly related to the magnetic parameter.
(3) Enhanced velocity and electric current induction results from the reduction of the fluid viscosity. While increased induced magnetic field profile and reduced skin friction in both plates occurs as the viscosity of the fluid increases. In addition it is observed that the heat exchange rate increases at lower viscosity and decreases at higher viscosity.
(4) The temperature of the fluid and its induced current density decreases as the thermal conductivity parameter increases. So also the thermal exchange rate and the skin friction on both plates decrease as the thermal conductivity parameter increases. Whereas the velocity of the fluid increases as the thermal conductivity parameter decreases. While the magnetic induction increases with increasing thermal conductivity parameter.
(5) When the momentum diffusivity is lower than the magnetic diffusivity, the skin friction on both plates, the velocity of the fluid and the magnetic induction are enhanced; the induced current density is reduced. While the thermal exchange rate also decreases with decreasing magnetic prandtl number
(6) The velocity and the skin friction enhances with decreasing Hartmann number. The magnetic induction decreases as the Hartmann number increases. While the current produced by induction and the heat exchange rate increases as the Hartmann number increases.
(7) Also, the fluid velocity is enhanced when the prandtl number decreases. For increasing prandtl number the magnetic induction and the fluid temperature decreases. Induced current and the rate at which heat is transferred in the fluid have a direct variation with the prandtl number. Skin friction grows with increasing prandtl number on the first plate but it decreases for smaller values and increases for bigger values on the second plate.
The numerical results of this work, are compared with previously published ones, for some special cases and found to be in excellent agreement. It is hoped that the present results be used for understanding more complex problems dealing with natural convection flow of an electrically conducting fluid in a vertical channel with induced magnetic field, viscous dissipation and Ohmic heating.
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
8. Declaration of Conflicting Interests.
The Authors declare that there is no conflict of interest
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