ADESANYA Samuel O., MAKINDE Oluwole D.

1. Department of Mathematical Sciences, Redeemer’s University, Redemption, Nigeria,E-mail: adesanyaolumide@yahoo.com 2. Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

Introduction

Studies related to conservation of thermal energy in a moving non-Newtonian fluid through vertical channel has several applications in Carnot engines,geology, metallurgical and petro-chemical engineering.As a result of the hyper-viscosity of some non-Newtonian fluids, technological processes occur at a very high temperature by either direct heating of the fluid or by simple exothermic chemical reaction.

In this paper, emphasis is laid on the flow of reactive non-Newtonian fluid that is induced by chemical reaction. It is well known that internal heat generation is connected with moving fluid undergoing exothermic chemical reaction or combustion. Furthermore,the inability of the classical linear stress-strain relation for Newtonian fluids to explain the complex rheological properties of some non-Newtonian fluids like heavy hydrocarbon oil, bitumen, soaps, shampoos,certain oils, etc. has led to the development of many constitutive relations in literature. For example, the Eyring-Powel model[1-3], the couple stress model[4-6]and many more. In recent years, a lot of quality research work has been done on third grade fluid model by Hayat and his collaborators[7-12](and references therein). This is because third grade fluid model has the ability to describe the shear thinning/thickening even in steady flow situations past rigid boundaries but lack the ability to capture the polar and particle size effects.

However, all the work in Refs.[7-12] neglects the thermodynamic analysis of the fluid flow. From application point of view, the performance of engines that utilize heat as the working medium can be measured by analysing the entropy-exergy of the thermal system.Similarly, thermodynamic irreversibility arises due to heat transfer and fluid friction. These indices help to determine the efficiency of a thermal system. Recently,a number of studies have been done on the entropy generation in a moving fluid. For example, Adesanya and Makinde[13]investigated the entropy generation in couple stress fluid flowing steadily through a porous channel with slip at the isothermal walls by using the Navier model. Similarly, Adesanya and Makinde[14]examined the effect of couple stresses on the entropy generation rate of an incompressible viscous fluid through a porous channel with convective heating at the walls. Makinde and Osalusi[15]presented the secondlaw analysis of a laminar falling viscous incompressible liquid film along an inclined porous heated plate in which the upper surface of the liquid film is considered free and adiabatic. Makinde[16]studied the inherent irreversibility in the flow of a variable-viscosity fluid through a channel with a non-uniform wall temperature with the assumption that the fluid viscosity varies linearly with temperature. While thermal stability and entropy generation in a temperature dependent viscous with Newtonian heating was investigated in the work of Makinde[17].

Motivated by the above discussions, the objective of the present study is to investigate the thermodynamic properties of a reactive third grade fluid flow through a vertical channel taking the effect of internal heat generation into consideration. The out-coming result is very useful in many geological and petro-chemical engineering systems, in which exergy is maximized. One case in mind is the enhanced oil recovery duringin-situcombustion of heavy oil. That is, air is introduced into bitumen or heavy oil bed to induced chemical reaction of the hydrocarbon. The heat of reaction has melting effect on the hyper-viscous fluid and flow is induced by the combination of natural convection resulting from density changes and pump action for further processing in the refinery.

The dimensionless problem is coupled and nonlinear. Hence, analytical solution will be obtained by using rapidly convergent Adomian decomposition method. The method has been shown to be convergent and successfully applied to several linear and nonlinear mathematical models in Refs.[18-20].

Fig.1 Flow geometry

1. Mathematical analysis

Consider the steady mixed convective flow of third grade fluid through the space between two infinite parallel isothermal plates of distance 2hapart as shown in Fig.1. All fluid properties are assumed constant except for fluid density and internal heat generation that varies linearly with temperature. The fluid is assumed chemically active and chemical reactions occur in the middle of the channel. Hence, heat flows symmetrically from the centreline of the channel toward the fluid layer close to the cold plates.

Therefore, the entropy lost in the hot region is gained in the cold region in accordance to the second law of thermodynamics. The velocity field is given by

Under this configuration, the conservation of mass,conservation of momentum and balanced energy equation for an incompressible fluid are given by

where D/Dtis the material derivative,cpis the specific heat,T′ is the fluid temperature,T0is the referenced fluid temperature,gis the gravitational force,βthe coefficient of volume expansion due to temperature,Lis the velocity gradient,Qis the reactant concentration that measures internal heat generation andSis the Cauchy stress tensor,kthe thermal conductivity,ρis the fluid density.

The Cauchy stress tensorSis given by[7,8]

whereIis the identity tensor,αi(i=1,2) andβi(i=1,2,3)are the material constants,μis the dynamic viscosity,p1is the pressure,Ai(i=1,2,3) are the kinematic tensors in which the first three kinematic tensorsAi(i=1,2,3) are defined by

where ▽ is the gradient operator. As was shown in Refs.[7]-[12] (and references therein), if the motion of the fluid are compatible with the thermodynamics,then the Clausius-Duhem inequality together with the assumption that the Helmholtz free energy is minimum at equilibrium implies that

For thermodynamically compatible third grade fluid,Eq.(5) becomes

where the effective shear-dependent viscosity is

then the thermodynamically developed flow are governed by the momentum equation

and the energy equation

and the entropy generation is given as

together with boundary conditions

hereu′ is the fluid velocity,T′ the fluid temperature,p′ the pressure,EGis the entropy generation parameter. Introducing the following dimensionless parameters and variables

we obtain the dimensionless problem together with the appropriate boundary conditions

whereAis the dimensionless pressure gradient taken to be unity,γis the dimensionless third grade material parameter,δrepresents internal heat generation parameter,Brviscous heating Brinkman number,θis the dimensionless temperature,uis the dimensionless velocity andGris the Buoyancy parameter andNSis the dimensionless entropy generation parameter,Uis the characteristic velocity andΩis the temperature difference parameter.

2. Method of solution

The differential Eqs.(16)-(17) can be written in the integral form as follows:

where the constantsa0,b0are to be determined later by using the other boundary conditions aty=1. The nonlinear terms in Eqs.(19)-(20) can be represented as

such that by introducing an infinite series solutions in the form

the Adomian polynomials can be computed as follows:

Similarly, using forms (22) in Eqs.(19)-(20) leads to the following recursive relation:

Equations (23)-(26) are then coded on Mathematica software to obtain the partial sums

as the approximate solutions and the graphical results are presented as Figs.2-17.

Fig.2 Effect of Grashof numbers on velocity profile

Fig.3 Effect of Grashof numbers on temperature distribution

Fig.4 Effect of internal heat generation parameters on velocity profile

Fig.5 Effect of internal heat generation parameters on temperature distribution

Fig.6 Effect of Brinkman numbers on velocity profile

Fig.7 Effect of Brinkman numbers on temperature profile

Fig.8 Effect of non-Newtonian material parameters on velocity profile

Fig.9 Effect of non-Newtonian material parameters on temperature profile

whenevern＞n0andεis sufficiently small.

Proof: It is required to show that the sequences are convergent (Cauchy sequence) in the Hilbert spaceH. Let us define

Then for everyε＞0 there existn0∈Nsuch that for alln,m＞n0and for everyy∈[-1 ,1], we havewhenever

Then we have,

From Cauchy’s inequality, we get

Now suppose the continuity condition holds, then let us chooseε=εα＞ 0 whereαrepresent a positive constant. For everyn≥n0we get

Hence, the sequenceun(y) →um(y) for sufficiently smallε＞ 0, thenun(y) converges tou(y). Using the same procedure the convergence of theθ(y) can be established. Table 1 confirms the rapid convergence of the series solution for certain values of the flow parameters given.

Table 1 Convergence result for Br =0.02, γ=0.01, Gr=1, δ=0.1

Error analysis: letunandun+1as any two consecutive sequences together with a positive constantκ.Then

then by subtraction, Eqs.(33) becomes

an estimate of the integral over boundaries can be obtained in the form

by Cauchy estimates, then

that is

ifκis sufficiently small, then inequality (37) becomes

Evidently, [κn+2/(1 -κ)] →0 asn→ ∞ whenever 0＜κ≤1 showing that the error is negligibly small.Substituting forms (27) in Eq.(18), we get the entropy generation rate. Due to large size of the solution, only the graphical results are presented as Figs.10-13.While the irreversibility ratio can be computed from the entropy generation rate due to the contribution from heat transfer and that from fluid friction in the form

Such that the irreversibility ratio denoted by Bejan number (Be) can be written as

Fig.10 Effect of Brinkman numbers on entropy generation rate

Fig.11 Effect of internal heat generation on entropy generation rate

Fig.12 Effect of Grashof numbers on entropy generation rate

Fig.13 Effect of non-Newtonian material parameters on entropy generation rate

3. Discussion of results

To understand the coupling between the fluid velocity and temperature, analytical results in the form of ADM solution are presented for various flow parameters.

Figure 2 represents the effect of variations in the Grashof number on the velocity profile. An increase in the Grashof number is observed to increase the flow velocity maximum. This is true since an increase in the Grashof number implies a decrease in the dynamic viscosity of the fluid. This enhances the volumetric expansion within the channel. Moreover, Fig.3 shows that an increase in the Grashof number has an increasing effect on the temperature distribution within the channel. This is because as the Grashof number increases, there is a rise in the volumetric expansion within the channel. As can be observed in Fig.4, as the internal heat generation parameter increases, there is a corresponding rise in the fluid velocity. This is simply due to the fact that the fluid absorbs its own emissions.This lowers the viscosity of the hyper-viscous fluid.Similarly, a rise in the internal heat generation parameter is connected with the decrease in the thermal conductivity of the fluid. This shows that the heat dissipated will continue to rise within the flow channel as the heat generated due to chemical reactions increases as shown in Fig.5. Moreover, Fig.6 shows the effect of the Brinkman number on the velocity profile. From the figure it is observed that an increase in the Brinkman number increases the fluid flow velocity.This is due to the fact that the Brinkman number is an additional heat source and the kinetic energy of the moving fluid is converted into heat energy. On the other hand, as the Brinkman number increase, the thermal conductivity of the fluid decreases. As a result of collision of the fluid particles, heat is generated which eventually leads to an increase in the fluid temperature distribution as shown in Fig.7. Also, Fig.8 represents the effect of the variation of the non-Newtonian material parameter on the fluid flow velocity. From the figure it is observed that an increase in the non-Newtonian parameter leads to a decrease in the fluid velocity due to fluid thickening. As a result, the interparticle frictional force increases and the temperature of the fluid decreases within the channel as observed in Fig.9.

In Fig.10, it is observed that entropy generation is higher at the heated walls due to the rise in the viscous dissipation parameter. This is because heat is transferred from the central line of the channel to the fluid layers close to the walls. Based on this, the result is well behaved and as such, the entropy generation decreases symmetrically towards the central line of the channel where the exergy is at the peak. Similarly,the heat released due to chemical interactions is observed to enhance the entropy generation rate at the walls. This is because heat is transferred from the central line of the channel to the fluid particles close to the walls. Hence the contribution of additional heat from combustion reaction will lead to excess heat near the cold walls while entropy is minimum at the central line of the channel as shown in Fig.11. Meanwhile,Fig.12 depicts the effect of the Grashof number on the entropy generation rate, and as seen from the figure,an increase in the Grashof number leads to a rise in the entropy generation at the walls. This is a result of heat transfer from the volumetric expansion of the fluid to the walls. While an increase in the non-Newtonian material parameter is observed to discourage entropy generation at the walls as observed in Fig.13. This is due to the fact that as the fluid viscosity rises, fluid friction also rises, thus minimizing entropy generation towards the cold walls.

Fig.14 Effect of Grashof numbers on irreversibility ratio

Fig.15 Effect of internal heat generation on irreversibility ratio

Fig.16 Effect of Brinkman numbers on irreversibility ratio

Fig.17 Effect of internal heat generation on Bejan number irreversibility ratio

Figure 14 shows the effect of the Grashof number on the irreversibility ratio. As can be observed from the plot, an increase in the Grashof number shows that heat transfer dominates over the fluid friction across the channel but this dominance is more pronounced at the in the fluid layer close to the cold plate. This is due to the fact that decrease in fluid viscosity encourages the rise in the Grashof number.Hence, heat transfer is enhanced. Figure 15 shows the effect of internal heat generation parameter on the irreversibility ratio. As is observed, heat transfer dominates over the fluid friction throughout the channel.This is due to the fact that as the internal heat generation parameter increases there is a decrease in the fluid thermal conductivity. This encourages heat transfer from the channel centreline to the walls. Similar behavior is seen in Fig.16 where the effect of the Brinkman number on irreversibility of heat is presented. Finally,fluid friction dominates over heat transfer as the non-Newtonian material parameter increases as shown in Fig.17 due to the hyper-viscosity of the fluid.

4. Conclusions

In the present paper, the thermodynamics analysis of a reactive non-Newtonian fluid flow through a channel with uniform wall temperature is presented.Analytical solutions of the coupled nonlinear boundary valued problem are obtained using the rapidly convergent ADM. These solutions are used to compute the irreversibility ratio and entropy generation within the channel. Major contributions in this paper are as follows:

(1) The exergy of the system increases with the rise in the non-Newtonian parameter while buoyancy force, internal heat generation and viscous heating of the fluid deplete the exergy level of the thermal system.

(2) Fluid viscosity dominates the channel with the increase in the non-Newtonian material parameter while heat transfer dominates the channel with increase in the Grashof number, internal heat generation and Brinkman number.

Acknowledgement

The authors gratefully thank the anonymous reviewers of this manuscript for their useful comments and suggestions.

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