RUNDORA Lazarus， MAKINDE Oluwole Daniel. Department of Mathematics and Applied Mathematics， University of Limpopo， Turfloop Campus， South Africa，E-mail： lazarus.rundora@ul.ac.za. Faculty of Military Science， Stellenbosch University， Saldanha， South Africa

Effects of Navier slip on unsteady flow of a reactive variable viscosity non-Newtonian fluid through a porous saturated medium with asymmetric convective boundary conditions*

RUNDORA Lazarus1， MAKINDE Oluwole Daniel2
1. Department of Mathematics and Applied Mathematics， University of Limpopo， Turfloop Campus， South Africa，E-mail： lazarus.rundora@ul.ac.za
2. Faculty of Military Science， Stellenbosch University， Saldanha， South Africa

（Received December 22， 2013， Revised October 12， 2015）

A study on the effects of Navier slip， in conjunction with other flow parameters， on unsteady flow of reactive variable viscosity third-grade fluid through a porous saturated medium with asymmetric convective boundary conditions is presented. The channel walls are assumed to be subjected to asymmetric convective heat exchange with the ambient， and exothermic chemical reactions take place within the flow system. The heat exchange with the ambient obeys Newton's law of cooling. The coupled equations， arising from the law of conservation of momentum and the first law of thermodynamics， then the derived system are nondimensionalised and solved using a semi-implicit finite difference scheme. The lower wall slip parameter is observed to increase the fluid velocity profiles， whereas the upper wall slip parameter retards them because of backflow at the upper channel wall. Heat production in the fluid is seen to increase with the slip parameters. The wall shear stress increases with the slip parameters while the wall heat transfer rate is largely unaltered by the lower wall slip parameter but marginally increased by the upper wall slip parameter.

Navier slip， saturated porous medium， third-grade fluid， temperature dependent viscosity， convective boundary conditions

Introduction0F

The majority of fluids encountered in industrial applications are non-Newtonian in character. Non-Newtonian is a generic term that incorporates a variety of phenomena which are highly complex and require sophisticated mathematical modelling techniques for proper description［1］. Examples of such fluids are geological materials， liquid foams， polymeric fluids，slurries， drilling mud， clay coatings， elastomers， emulsions， hydrocarbon oils and a variety of food products［2］. Theoretical consideration of the flow of such fluids in porous media has received considerable attention in recent years， and it is reasonable to pin down the interest on the wide range of scientific， technological and engineering applications as well as the theoretical complexity of the mathematical problems that arise［3］. Fluids of the differential type of third grade are an example amongst the fluids referred to［4，5］.

Although several studies involving heat and mass transfer in non-Newtonian third grade fluids have been conducted［6-8］a systematic and rational treatment of the thermodynamics of the problem with respect to the combined effects of porous media， unsteadiness，temperature dependent viscosity and asymmetric convective boundary conditions on the flow system was only carried out recently in Ref.［2］. As it has become almost the norm and tradition， the so-called no-slip boundary condition， namely the fluid velocity relative to the solid is zero on the fluid-solid interface［9］， has been assumed in many such researches including in Ref.［2］. Indeed this has become a common practice despite the fact that the no-slip condition is a hypothesis rather than a condition deduced from any principle. It is due to this fact that its validity has been continuously debated in scientific literature［10］. Evidences of slip of a fluid on a solid surface were reported by many authors［11，12］. The importance of studies involving flow and heat transfer in channels with wall slipin improving the design and operation of many industrial and engineering devices cannot be overlooked.

The present work seeks to extend the problem in Ref.［2］ by investigating the effects of Navier slip on unsteady flow of a reactive temperature-dependent viscosity third grade fluid through a porous saturated medium w ith asymmetric convective boundary conditions. The mathematical formulation of the problem is given in the second section. We implement， in the third section， a semi-implicit finite difference scheme in pursuit of the solution to the problem. In the fourth section， graphical simulations are presented and analyzed quantitatively and qualitatively w ith respect to the many parameters deriving the system.

Fig.1 Schematic diagram of the problem

1. Mathematical formulation

Consider an unsteady flow of an incompressible，third-grade， variable viscosity， reactive fluid through a channel filled w ith a homogeneous and isotropic porous medium as depicted in Fig.1. The plate surfaces are assumed to be subjected to asymmetric convective heat exchange w ith the surrounding medium as a result of unequal heat transfer coefficients. The fluid motion is induced by an applied axial non-constant pressure. Follow ing［4，5，13-17］， and neglecting the reacting viscous fluid consumption， the governing equations for the momentum and heat balance can be w ritten as：

The additional viscous dissipation term in Eq.（2） is due to Ref.［17］ and is valid in the limit of very small and very large porous medium permeability. The appropriate initial and boundary conditions are

here T is the absolute temperature，ρis the density， β0，β1are the lower and upper walls slip parameters， cpis the specific heat at constant pressure，t is the time，h1is the heat transfer coefficient at the lower plate，h2is the heat transfer coefficient at the upper plate，T0is the fluid initial temperature，Tais the ambient temperature，k is the thermal conductivity of the material，Q is the heat of reaction，A is the rate constant，E is the activation energy，R is the universal gas constant，C0is the initial concentration of the reactant species，a is the channel w idth，l is Planck's number，h is Boltzmann's constant，n is the vibration frequency，K is the porous medium permeability， α1and β3are the material coefficients，P is the modified pressure， and m is the numerical exponent such that m∈｛-2，0，0.5｝， where the three values represent numerical exponents for sensitised， Arrhenius and bimolecular kinetics respectively （see Refs.［16，18，19］）. The temperature dependent viscosity（μ）can be expressed as

where b is a viscosity variation parameter and μ0is the initial fluid dynam ic viscosity at temperature T0. We introduce the follow ing dimensionless variables into Eqs.（1）-（6），

and obtain the following dimensionless governing equations：

where λ represents the Frank-Kamenetskii parameter， n1，n2are the lower wall slip parameter and the upper wall slip parameter respectively，Pr is the Prandtl number，εis the activation energy parameter，δis the material parameter，γis the non-Newtonian parameter，G is the pressure gradient parameter，Dais the Darcy number，αis the variable viscosity parameter，?is the viscous heating parameter，θais the ambient temperature parameter，Sis the porous medium shape parameter，Bi1and Bi2are the Biot numbers at the lower and upper channel walls respectively. The skin friction （Cf）at the channel walls is given as

where τw=μ（T）?u/?yis the shear stress evaluated at the wall y=0，a. The other dimensionless quantity of interest is the wall heat transfer rate（Nu）given by

In the following section， Eqs.（8）-（14） are solved numerically using a semi-implicit finite difference scheme.

2. Numerical solution

Our numerical algorithm is based on the semiimplicit finite difference scheme［20-24］. Implicit terms are taken at the intermediate time level（N+ξ）where 0≤ξ≤1. The discretization of the governing equations is based on a linear Cartesian mesh and uniform grid on which finite differences are taken. We approximate both the second and first spatial derivatives with second-order central differences. The equations corresponding to the first and last grid points are modified to incorporate the boundary conditions. The semi-implicit scheme for the velocity component is

In Eq.（15）， it is understood that ?#/?t：=［#（N+1）-#（N）］/Δt. The equation for w（n+1）then becomes

where

with μ=exp（-αθ）and γ˙=wy. The solution procedure for w（n+1）thus reduces to inversion of tri-diagonal matrices， which is an advantage over a full implicit scheme. The semi-implicit integration scheme for the temperature equation is similar to that for the velocity component. Unmixed second partial derivatives of the temperature are treated implicitly：

The equation for θ（N+1）thus becomes

where r=ξΔt/Δy2. The solution procedure again reduces to inversion of tri-diagonal matrices. The schemes （16） and （18） were checked for consistency. For ξ=1， these are first order accurate in time but second-order accurate in space. The schemes in Ref.［22］ have ξ=0.5which improves the accuracy in time to second order. Following the work in Refs.［23，24］ we use ξ=1so that the choice of larger time steps is possible and still obtain convergence to the steady solutions.

Fig.2 Transient and steady state velocity profiles

Fig.3 Transient and steady state temperature profiles

3. Results and discussion

3.1 Transient and steady state solutions

In Fig.2 and Fig.3 transient and steady state velocity and temperature profiles are displayed respectively. In both cases， there is a transient increase in profiles until steady state is reached. In Fig.2 negative velocity profiles signify backflow as a result of slip at the channel walls.

Fig.4 Blow-up of fluid temperature for large λ

Fig.5 Effects of the porous medium parameter，S， on velocity profiles

3.2 Blow-up of solutions

Figure 4 shows that if fluid temperature is not carefully monitored and controlled accordingly， it maynot be possible to obtain steady state profiles as depicted in Figs.2， 3. Figure 4 shows that failing to control particularly the reaction parameter λ， in conjunction with other parameter values， there will be blow-up of temperature soon after the value of λ surpasses a value of about 0.9. The consequences of this could be detrimental to life and property.

Fig.6 Effects of the porous medium parameter，S ， on temperature profiles

3.3 Parameter dependence of solutions

The present study is validated with the earlier result of Makinde et al.［2］in the absence of Navier slip at the channel walls i.e.，（n1=n2=0）and a perfect agreement is achieved for the velocity and temperature profiles （see Figs.13-16）. Fig.5 and Fig.6 show the behaviour of the velocity and temperature profiles respectively as the porous medium parameter，S ， is varied. The velocity profiles are observed to diminish rapidly with increasing values of porous medium parameter. An increase in the porous medium parameter values means that the pore spaces in the porous matrix are reduced and this has a dampening effect on the flow of the fluid particles. As the velocity is drastically reduced， the viscous heating source terms in the fluid temperature equation are reduced and fluid temperature also decreases as a result as shown in Fig.6.

Fig.7 Effects of the temperature-dependent viscosity，α， on velocity profiles

The effects of the temperature dependent viscosity parameter on velocity and temperature profiles is displayed in Fig.7 and Fig.8. As the temperature dependent viscosity parameter is increased， it reduces the fluid viscosity and this increases the rate of flow of the fluid particles as shown in Fig.7. Figure 8 shows that the temperature dependent viscosity parameter has no significant effect on the temperature profiles.

Fig.8 Effects of the temperature-dependent viscosity，α， on temperature profiles

Fig.9 Effects of the non-Newtonian parameter，γ， on velocity profiles

Fig.10 Effects of the non-Newtonian parameter，γ， on temperature profiles

The effect of increasing the non-Newtonian parameter，γ， is to increase the non-Newtonian properties of the fluid， and these characteristics （such as viscoelasticity） bring resistance to the rate of flow of the fluid. Velocity profiles， as in Fig.9， are seen to retard as a result. The effects on temperature profiles， as seen in Fig.10 are less noticeable.

Fig.11 Effects of the reaction parameter，λ， on velocity profiles

Fig.12 Effects of the reaction parameter，λ， on temperature profiles

The Frank-Kamenetskii parameter，λ， measures the reaction rate of the chemical reaction in the process. Since， as pointed out earlier， the reaction is exothermic， increasing λ will inevitably result in an increase in the temperature of the fluid. This is seen in Fig.12. As pointed out earlier （Fig.2）， if this parameter is not carefully controlled chemical explosions will be difficult to combat. As Fig.11 shows， the coupling effect means that the velocity profiles are also increased，albeit at a lower scale， by increased reaction rate.

Fig.13 Effects of the lower wall slip parameter，n1， on velocity profiles

Figures 13-16 show the effects of the channel wall slip parameters on the fluid velocity profiles as well as the fluid temperature profiles. The effect of increasing the lower wall slip parameter，n1， on the velocity field is depicted by Fig.13. Fluid velocity is seen to increase with an increase in the lower wall slip parameter. On the contrary， the upper wall slip parameter n2is observed to retard fluid velocity as shown in Fig.15. Significant backflow at the upper wall which is as a result of slip at the wall is the cause of this drop in velocity. The backflow also renders rapid mixing of the fluid particles， and this results in significant friction that induces raise in the fluid temperature as observed in Fig.14 and Fig.16.

Fig.14 Effects of the lower wall slip parameter，n1， on temperature profiles

Fig.15 Effects of the upper wall slip parameter，n2， on velocity profiles

Fig.16 Effects of the upper wall slip parameter，n2， on temperature profiles

Fig.17 Effects of the parameterm on fluid temperature profiles

Fig.18 Effects of the activation energy parameter，ε， on fluid temperature profiles：m=0.5

Fig.19 Effects of the activation energy parameter，ε， on fluid temperature：m=0

The numerical exponent m=0.5means that the type of exothermic chemical reaction is bimolecular， m=0means that the type of reaction is Arrhenius and m=-2means that the reaction is sensitised. Figure 17 shows that most heat is generated under a bimolecular type of exothermic chemical reaction and the least heat is generated when the reaction is one of sensitised type. Figure 18 shows that the effect of an increase in the activation energy parameter， when m=0.5， is also to increase the heat of the reaction. However when m =0and m=-2（Fig.19， Fig.20 respectively） the activation energy parameter，ε， induces effects that are exactly opposite that depicted when m=0.5. As explained in Ref.［2］， this is due to the fact that in the temperature equation， when m≤0， the function （1+εθ）mexp［θ/（1+εθ）］ decreases with increasingε.

Fig.20 Effects of the activation energy parameter，ε， on fluid temperature：m=-2

Fig.21 Effects of the Biot number Bi2on fluid temperature profiles

Fig.22 Effects of the Prandtl number，Pr ， on fluid temperature profiles

Fluid temperature is observed to decrease significantly with an increase in the Biot number as well as an increase in the Prandtl number. It is important to note that in the limit of Bi1→0and Bi2→0， the channel walls are insulated with no heat loss while thecase of Bi1→∞and Bi2→∞correspond to a scenario where both the ambient temperature and that of the fluid at the wall are the same. Moreover， as the parameter values of Bi1and Bi2increase， the convective heat loss to the ambient from both walls increase，leading to a decrease in the fluid temperature. Hence，an increase in the Biot number signifies higher degrees of convective cooling at the channel walls and this induces a significant temperature drop in the bulk of the fluid. Figure 21 illustrates this phenomenon. A similar trend is observed in Fig.22 where fluid temperature drops with increasing Prandtl numbers. Higher Prandtl numbers generally signify a decrease in fluid thermal conductivity. Figure 23 shows the fluid temperature growing with the viscous heating parameter?.

Fig.23 Effects of the viscous heating parameter，?， on fluid temperature profiles

Of engineering importance are the two quantities namely， the wall shear stress （skin friction） and the wall heat transfer rate （Nusselt number）. Figure 24. displays graphs showing variation of the wall shear stress with the reaction parameter λ and the variable viscosity parameter， the non-Newtonian parameter，the lower wall slip parameter and the upper wall slip parameter. The figures are plotted up to the solution blow-up values of the Frank-Kamenestkii parameter λ. Skin friction is observed to diminish with increasing values of the temperature dependent viscosity parameter and the non-Newtonian parameter. On the other hand skin friction is noted to increase with increasing wall slip parameters.

Figure 25 illustrates variation of the wall heat transfer rate， up to blow-up values of λ， withλ and the variable viscosity parameter， the non-Newtonian parameter， the wall slip parameters and the porous medium parameter. While the wall heat transfer rate is largely unaltered by the variable viscosity parameter，the non-Newtonian parameter， the lower wall slip parameter and the porous medium parameter， the upper wall slip parameter marginally increases it. It is reasonable to tie this observation with the observed backflow at the upper wall that tend to rigorously mix fluid particles resulting in increased heat production at the wall.

Fig.24 Variation with λ and α，γ，n1，n2of the wall shear stress

Fig.25 Variation with λ andα，γ，n1，n2，Sof the wall heat transfer rate

Table 1 displays thermal critical values of the reaction parameter λ for various parameter variations. It is important to do this exercise as， depending on other flow parameters， values of the reaction parameter above a certain threshhold leads to blow up of solutions. It is therefore a safety precaution to know in advance the blow up values of λ in relation to other parameter values. We particularly note from the table that increasing the Biot number increases the blow up value of the reaction parameter. However the general trend that is noted is that the blow up values of the reaction parameter are not affected by slight changes in most parameter values.

4. Conclusion

We computationally investigate the effects of Navier slip on unsteady flow of a reactive variable viscosity third-grade fluid through a porous saturated medium with asymmetric convective boundary conditions. It has been concluded that the lower wall slip parameter increases the fluid velocity profiles， whereas because of backflow at the upper channel wall， the upper wall slip parameter retards them. Heat production in the fluid increases with the magnitude of the slip parameters. This is also the case with the wall shear stress. The wall heat transfer rate is largely unaltered by the lower wall slip parameter but marginally increased by the upper wall slip parameter. Our results will no doubt be of significant interest in the area of petroleum exploration and refinery. Petroleum is very reactive and non-Newtonian in nature. Its reservoir can be found within porous rocks， such as sandstonewhile the refinery is made up of several pipes with porous matrix. Moreover， both exploration and refining process of petroleum involve slip flow of a reactive variable viscosity non-Newtonian fluid through a porous saturated medium.

Table 1 Thermal criticality values of λ for different parameter values

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* Biography： RUNDORA Lazarus （1972-）， Male，Ph. D.， Senior Lecturer