DYNAMICAL ANALYSIS OF UNSTEADY POISEUILLE FLOW OF TWO-STEP EXOTHERMIC NON-NEWTONIAN CHEMICAL REACTIVE FLUID WITH VARIABLE VISCOSITY

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 12, December 2018, pp , Article ID: IJMET_09_12_062 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed DYNAMICAL ANALYSIS OF UNSTEADY POISEUILLE FLOW OF TWO-STEP EXOTHERMIC NON-NEWTONIAN CHEMICAL REACTIVE FLUID WITH VARIABLE VISCOSITY S.O. Salawu Department of Mathematics, Landmark University, Omu-aran, Nigeria. H.A. Ogunseye School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Scottsville, Pietermaritzburg 3209, South Africa. A.M. Olanrewaju Department of Mathematics, Covenant University, Ota, Nigeria ABSTRACT Investigation into the effects of two-step exothermic third-grade chemical reactive poiseuille fluid flow through fixed walls with exponentially heat dependent viscosity on the flow and heat transfer rate is carried out. The reactive non-newtonian fluid is propelled by constant pressure gradient under Bimolecular kinetic rate law and heat absorption. The reactive fluid is stimulated by periodic variations in time. The upper wall of the channel is exposed to coolant convection heat exchange with the environment and it satisfied the Newton s cooling law. The dimensionless velocity and energy balance flow equations are computationally solved by applying a stable and convergent finite semi-implicit method of an even finer mesh. The response of the flow fluid and temperature to variational rise in some values of the embedded fluid parameters are graphically presented. The bifurcation and thermal criticality results of the reactive fluid under different chemical kinetics are also obtained and conferred quantitatively. Keywords: Thermal criticality; Viscous combustible; Reactive fluid; Non-Newtonian; Convective cooling. IJMET/index.asp 596 editor@iaeme.com

2 Dynamical Analysis of Unsteady Poiseuille Flow of Two-Step Exothermic Non-Newtonian Chemical Reactive Fluid with Variable Viscosity Cite this Article: S.O. Salawu, H.A. Ogunseye and A.M. Olanrewaju, Dynamical Analysis of Unsteady Poiseuille Flow of Two-Step Exothermic Non-Newtonian Chemical Reactive Fluid with Variable Viscosity, International Journal of Mechanical Engineering and Technology, 9(12), 2018, pp INTRODUCTION Theoretical inquiry on the liquids that change their viscosity or conduct when subjected to stress has in modern times attracted substantial and elevated status due to their uses in the process industry. As a result of their involvedness, no constitutive single model can explain non-newtonian liquids and therefore, several non-newtonian constitutive models have been developed for various classes of fluids. Among the several recommended models is the third ordernon-newtonian fluids that are difficult to obtain analytically even for a simple flow except the uses of computational techniques of solution. Broad investigations into such fluids have been done by many researchers [1-5]. In the work [6], investigation of the uniqueness and stability of the thermodynamics models of different kinds with the special case of third grade liquids was done. Also, third grade heat transfer problems with comprehensive study of the constitutive function in thermodynamics were carried out by [7]. The third grade viscoelastic liquid has significance in industry, chemical technology and dynamic fluid geophysical. [8] examined complex viscoelastic Rivlin-Ericksen flow fluid with bounds rate growth in thermosolutal through permeable media. [9] presented non- Newtonian hydromagnetic flow with variable porousness past a penetrable medium. [10] studied third grade compressible fluid with thermal instability of suspended dust particles in permeable media. While, [11] discussed unsteady Rivlin-Ericksen MHD convective flow with variable temperature and heat sink effect in a permeable semi-infinite motioning plate. It was obtained from their findings that the temperature profile decreases as the values of heat absorption term increases and also, for transverse boundary layer, there a reduction in the velocity distribution as the material coefficient parameter values is enhanced. In nature, and principally in industry, swift exothermic non-newtonian reaction processes occur with the release of great quantities of heat are substantially essential. The processes are called the combustion processes. Combustion is widespread and significant in the flow systems involving chemical reaction with applications in the prevention of fire, propulsion of rocket and jet, pollution control, material industrial processing of materials and many more [12]. Several researchers on combustion of chemical reactive fluid have examined many idealized problems in one-step chemical reaction of flow system model [5,13-18]. Though this conjecture can be correct for some problems, nevertheless, in many reaction processes of combustion, one-step exothermic reaction is not enough to define ignition and flame diffusion in a system see [19]. For example conversion of an automobile catalytic used for exhaust system offers a base for exothermic two-step chemical reaction where complete combustion of unburned hydrocarbons takes place. This assists in lessening the emissions of carbon monoxide (CO) that causes toxic environmental hazard by [20]. The purpose of the study is to examine the dynamical analysis of unsteady poiseuille gravity driven flow of exothermic non-newtonian chemical reactive fluid with variable viscosity. When the effect of heat dependent variable viscosity is considered, the flow properties change meaningfully when compared to the constant physical properties. Therefore, the viscosity is assumedto be an exponential function of temperature. The reactive flow formulation is described inthesection two while in section three, the finite difference semi-implicit scheme in space and time ispresented and implemented. In section four, the IJMET/index.asp 597 editor@iaeme.com

3 S.O. Salawu, H.A. Ogunseye and A.M. Olanrewaju numerical solutions with the graphical resultsare presented and discussed with respect to existing fluid parameters entrenched in the flow. 2. MATHEMATICAL FORMULATION Consider an isothermal, transient incompressible fluid and heat dependent variable viscosity of two step chemical combustible reactive poiseuille fluid flow along horizontally fixed channel of width. Since the modeled is by Bimolecular kinetic law, the viscosity function reduces exponentially with temperature and the chemical reactive fluid is encouraged by the fully developed axial uniform pressure gradient. The viscoelastic effect is induced by the employment of non-newtonian third grade fluid model. The upper surface of the channel is exposed to coolant heat convection with the ambient temperature, thus providing an asymmetric cooling effect. Follow [18,20-22], and the above assumptions, The flow coordinates geometry is illustrated as figure 1, the mechanisms of two step chemical reaction schemes in an auto-catalytic converter, and the transport equations of the reactive liquid are presented as follows: Figure 1. The formulation geometry Mechanisms of two-step chemical reaction of ethanol oxidation is given as: The equations governing the flow * + ( ) (1) ( ) ( ) ( ) ( ( ) ) (2) Subject to relevant boundary conditions where,,,,,,,,,,,, and, are respectively the dimensional flow direction, fluid temperature, wall temperature, fluid velocity, fluid viscosity, thermal conductivity, fluid density, planck s number, constant specific heat, channel width, frequency of vibration, constant universal gas and coefficient of heat transfer and coefficients of material,. While the parameters,,,,, and, are respectively first and second reaction heat, reacting species, rate constant and activation energy. is the numerical index that is { } respectively denotes the sensitized, arrhenius and bimolecular (3) IJMET/index.asp 598 editor@iaeme.com

4 Dynamical Analysis of Unsteady Poiseuille Flow of Two-Step Exothermic Non-Newtonian Chemical Reactive Fluid with Variable Viscosity kinetics that present the reactants of energy along with their orientation. The exponentially temperature dependent viscosity is taken as The following non-dimensional quantities are used (4) ( ) (5) Introducing equations (4) and (5) in equations (1)-(3) to obtain ( ) ( ) (6) ( ) ( ) [ ( ) ] (7) with the corresponding boundary conditions (8) where,,,,,,,,,,,,, and are respectively the dimensionless initial temperature parameter, temperature, velocity, variable viscosity parameter, non-newtonian term, material parameter, pressure gradient term, Brinkman number, second step chemical reaction term, Frank-Kamenetskii term, heat sink term, Biot number, Prandtl number, activation energy term and ratio of activation energy. The nondimensional quantities of engineering concerned are the skin friction and thermal gradient rate defined as give (9) Equations (6)-(9) are solved numerically by applying finite difference technique of semiimplicit. 3. METHOD OF SOLUTION The implemented numerical technique for the heat and flow rate equations is finite semiimplicit method as in Chinyoka, (2008), the method used implicit terms in the range for a time level. In order to have higher steps of time, is expected to be 1. Definitely, being entirely implicit, the applied numerical technique offered in this study is speculated to be suitable for any time steps estimation. The main equations governing the problem are discretized on a Cartesian uniform grid with linear mesh on which the finite differences are taken. Approximating the derivatives of first and second spatial with order two of central differences, the consequential equations of the grid points are adjusted to integrate the boundary conditions. The semi-implicit expression for the non-newtonian reactive flow rate module is written as follows: ( ) ( ( ) ) ( ) (10) the equation for takes the form: IJMET/index.asp 599 editor@iaeme.com

5 S.O. Salawu, H.A. Ogunseye and A.M. Olanrewaju [ ] (11) where [ ] with. The forward difference techniques are used for all time dependent derivatives. The solution for decreases to inversion tri-diagonal matrices. The semi-implicit method for the heat module resembles that of flow rate equation. The unvarying derivatives of the heat equation takes the form: the equation for is written as: [ ( )] ( ) (12) [ ( )] ( ) (13) where.the solution for changes to inversion tri-diagonal matrices. The schemes (11) and (13) were checked for uniformity. When allow a large time steps of first and second order in time and space respectively. As it was initially assumed, the scheme allows any time step values! Maple code is written to carried out the transient analysis. 4. RESULTS AND DISCUSSIONS The initial temperature of the reactive fluid is taken to be equal to wall temperature, therefore parameter. The parameters values adopted for this study based on existing theoretical results are,,,,,,,,,,,. Table 1 Computations results for thermal criticality under various chemical kinetics Table 1 portrays the changes in the conditions of thermal criticality and temperature for diverse arrangement of entrenched fluid terms under various chemical kinetics. The thermal criticality magnitude reduces under sensitized kinetic with rising values of variable viscosity and second-step reaction rate terms. This shows that successive explosion and thermal instability is encouraged. While there is respective decrease and increase in the magnitude of thermal criticality under arrhenius kinetic as the values of Brinkman number IJMET/index.asp 600 editor@iaeme.com

6 Dynamical Analysis of Unsteady Poiseuille Flow of Two-Step Exothermic Non-Newtonian Chemical Reactive Fluid with Variable Viscosity and activation energy ratio rises. Also, enhancing in the values of pressure gradient and activation energy terms respectively diminishes and enhances the thermal criticality under bimolecular kinetic. Thermal criticality explosion does not dependents on the reaction type but explosion occur faster under bimolecular reaction than sensitized and Arrhenius reactions due to lesser value of thermal criticality in bimolecular kinetic reaction. Figures 2 and 3 represent the transient results which are defined on an even finer mesh. The figures depict a transient rise in both the fluid flow rate and heat transfer until a steady state is obtained. The time of reaching the steady state for fluid flow rate and temperature for various parameter values are not the same. That is, the time of achieving no variation in the fluid velocity and heat distributions depend highly on the parameter values. Figure 2. Transient velocity profile Figure 3. Transient heat profile The response of fluid momentum to an increase in the pressure gradient parameter values is presented in Figure 4. The flow rate is at it maximum as the values of rises. The higher the pressure applied the faster the breaking down in the fluid bonding force which resulted into great collision in the fluid particles. Hence, the velocity field increases. The influence of variable viscosity term on the fluid velocity is shown in Figure 5. Enhancing the parameter values affect significantly the flow characteristics by discouraging the viscosity of the fluid and weaken the fluid s opposition forces to the flow. This correspondingly leads to a rise in the fluid velocity as portrayed in the Figure. The viscous heating source terms in the heat balance equation rises which in turn enhances the velocity profile. Figure 4. Effects of on velocitye Figure 5. Effects of on velocity IJMET/index.asp 601 editor@iaeme.com

7 S.O. Salawu, H.A. Ogunseye and A.M. Olanrewaju The reaction fluid flow rate to an increase in the non-newtonian term is depicted in Figure 6. Increasing the parameter values has a substantial effect on the flow properties by boosting the viscosity of the fluid and strengthens the fluid s opposition forces to the flow. This consequentially causes a reduction in the fluid momentum as shown in the Figure. The heat source terms in the velocity equation reduce which in turn declined the velocity distribution. While in Figure 7, the influence of the material parameter on the fluid momentum is presented. A variational rise in the parameter encourages viscoelastic effect and fluid s opposition forces which then reduces the reactive Poiseuille flow rate in the channel. Therefore, the third grade parameter values needs to be kept low because it dampens shear rate terms in the equations. Hence, boosting the parameter values diminishes the velocity distribution. Figure 6. Effects of on velocity Figure 7. Effects of on velocity Figure 8 and 9 depict the impact of activation energy and activation energy ratio on the exothermic chemical reaction temperature profiles. The effect of parameters and on the temperature corresponding to the exothermic chemical Bimolecular reaction. The temperature of the non-newtonian liquid increases as the values of and rises. Activation energy is an energy needed for a chemical reaction process to occur, hence an increase in the values inspires Poiseuille fluid flow reaction process and as well the temperature field of the system as shown in the graphs. Figure 8. Effects of on temperature Figure 9. Effects of on temperature IJMET/index.asp 602 editor@iaeme.com

8 Dynamical Analysis of Unsteady Poiseuille Flow of Two-Step Exothermic Non-Newtonian Chemical Reactive Fluid with Variable Viscosity The reaction of the fluid temperature distributions to an increase in the Frank- Kamenettski term and Brinkman number is demonstrated in Figures 10 and 11. The flow heat field of the system is motivated with varying in the parameters and as seen in the plots. A rise in the dissipating viscous heating is observed which is attributed to stimulation in the internal heat source term due to a rise in the Brinkman number and Frank- Kamenettski terms, this is significantly manifested to strengthen the fluid temperature. Therefore, the temperature profile is prompted. Figure 10. Effects of on temperature Figure 11. Effects of on temperature Figure 12 shows the variants in the wall shear stress to increasing in the pressure gradient. An initial rise in the shear stress near the wall is noticed in the plot as the values of increases which later decreases as it moves farther away from the channel wall at towards the free flow under the influence of second step exothermic chemical reaction parameter. In Figure 13, the variation of wall heat transfer rate to an increase in the Brinkman number parameters is presented. There is an increase in the thermal gradient heat transfer rate at the wall as the parameter values rises. The parameter enhance energy transfer coefficient at the wall by causing the reactive fluid particles to collide faster which leads to generation and distribution of more heat along the channel walls. Figure 12. Skin friction with and Figure 13. Nusselt number with and IJMET/index.asp 603 editor@iaeme.com

9 S.O. Salawu, H.A. Ogunseye and A.M. Olanrewaju 5. CONCLUSION The computational dynamical of a transient poiseuille pressure driven flow of two step exothermic third-grade chemical reactive fluid with variable viscosity and upper wall convective cooling was carried out using a developed unconditionally convergent and consistence finite difference of semi-implicit scheme for any time step size. An enhancement in the heat dependent viscosity strengthens the velocity gradient, and the fluid velocity is at the peak, it also inspired the viscous dissipation which encourages huge temperature within the main channel of the system. A transient stimulation in the temperature is observed within the channel as the second step exothermic chemical reaction parameter increases due to an increase in the temperature source terms. This shows that second step reaction parameter indeed supported complete combustion in an engine. Also, a transient decline in the fluid momentum is obtained with a rise in the non-newtonian behavior at small values of the thirdgrade material term, while high values of the parameter can cause solutions blow up and it is found that the thermal criticality conditions with the right combination of thermo-fluid parameters governing the system, the likely solutions finite time blow up can be prevented. Therefore, thermal stability can be sustained. REFERENCES [1] Abu-Hijleh, B. Natural convection and entropy generation from a cylinder with high conductivity fins. Numerical Heat Transfer Part A. 39, 2001,pp [2] Asghar, S., Hanif, K., Hayat, T. Flow of a third grade fluid due to an accelerated disk, International Journal for Numerical Methods in Fluids,63, 2010, pp [3] Ellahi, R., Afzal, S. Effects of variable viscosity in a third grade fluid with porous medium: an analytic solution, Communications in Nonlinear Science and Numerical Simulation,14, 2009, pp [4] Hayat, T., Momoniat, E., Mahomed, F.M. Peristaltic MHD flow of third grade fluid withan endoscope and variable viscosity, Journal of Nonlinear Mathematical Physics 15, 2008, pp [5] Salawu, S.O., Oke, S.I. Inherent irreversibility of exothermic chemical reactive thirdgrade poiseuille flow of a variable viscosity with convective Cooling, J. Appl. Comput. Mech., 4, 2018, pp [6] Rajagopal, K.R. On Boundary Conditions for Fluids of the Differential Type: NavierStokes Equations and Related Non-Linear Problems, Plenum Press, New York, 1995, p [7] Fosdick, R.L., Rajagopal, K.R. Thermodynamics and stability of fluids of third grade, Proceedings of the Royal Society of London, Series A 339, 1980p [8] Daleep, K., Sharma, A., Banyal, S. Bounds for complex growth rate in thermosolutal convection in RivlinEricksen viscoelastic fluid in a porous medium. Int J EngSciAdvan Technol2, 2012,pp [9] Noushima, H., Ramana Murthy, M.V., Reddy, C.K., Rafiuddin, M., Ramu, A., Rajender, S. Hydromagnetics free convective RivlinEricksen flow through a porous medium with variablepermeability. Int J ComputAppl Math 5, 2010,pp [10] Rana, G.C. Thermal instability of compressible RivlinEfficksen rotating fluid permeated with suspended dust particles in porous medium. Int J Appl Math Mech, 8, 2012,pp [11] Ravikumar, V., Raju, M.C., Raju, G.S.S. Combined effects of heat absorption and MHD on convective Rivlin-Ericksen flow past a semi-infinite vertical porous plate with variable temperature and suction, Ain Shams Engineering Journal, 5, 2014, pp IJMET/index.asp 604 editor@iaeme.com

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