ADOMIAN DECOMPOSITION METHOD FOR STEADY FREE CONVECTIVE COUETTE FLOW IN A VERTICAL CHANNEL WITH NON-LINEAR THERMAL RADIATION, DYNAMIC VISCOSITY AND DYNAMIC THERMAL CONDUCTIVITY EFFECTS

In this paper, we investigate steady free convective Couette flow in a vertical channel with nonlinear thermal radiation, dynamic viscosity and dynamic thermal conductivity effects. The investigation is motivated by the studies of some researchers which assumed linear thermal radiation and constant fluid properties. However, this is uncalled for; as these assumptions do not reflect true behavior of the flow. For instance; increase in temperature affects fluid viscosity, thermal conductivity thereby changing the transport phenomenon. Here; the investigation considers both the fluid viscosity and thermal conductivity to be dependent on temperature with the thermal radiation adopting nonlinear form. Due to this reasons, the associated flow equations are highly nonlinear and exhibit no analytical solution and therefore require the use of Adomian decomposition method (ADM) of solution. The attained ADM solution is then coded into computer algebra package of mathematica where results under the parameters of interest are presented and discussed. Results of the investigation show that raising the thermal radiation leads to corresponding rise in both the velocity and temperature of the fluid in the channel. Furthermore; lessening the viscosity and thermal conduction of the fluid were identified to escalate both velocity and temperature of the fluid.


INTRODUCTION
Flow of fluid induced by density difference occurring between the fluid particles due to temperature gradients is referred to as free convection flow. This type of flow has fundamental importance in many technological and industrial applications such as nuclear reactor, radiators, furnaces, rapid cooling process, sewage disposal and many more. Natural convection flows due to the movement of bounding surface surrounding the fluid is termed as "Couette flow". This type of flow occurs in fluid machineries involving moving parts; especially in hydrodynamics lubrications. Couette flow has been used as a fundamental method for measurement of viscosity and as a means of estimating drag force in many wall driven applications (Yasutomi (1984)). A situation in flow formation which is not time dependent is called steady flow. This type of flow has applications in many engineering devices like boiler, turbine, condenser and water pump that run nonstop for many months before they are shut down for maintenance. Several scholars considered steady natural convection flow through channels due to its significance in engineering technology; especially; in cooling/heating applications. For example; it is used in computer engineering where electronic cabinets containing circuits are design in channel forms so as to enhance cooling of the computer system; in civil engineering, channels are used for irrigation purposes, measuring discharge of water in a river, studying the spread of pollutants and so on. In relation to this, Ostrach (1952) investigated steady laminar natural convection flow of viscous incompressible fluid between two vertical walls while Ostrach (1954) and Sparrow et al. (1952) studied combined effects of steady free and forced convective flow and heat transfer between vertical walls. The study of Miyatake and Fuzii (1972) presented results for steady natural convection between vertical walls on considering different physical situations of the flow process. Transient flow between two vertical walls heated/cooled asymmetrically was investigated by Singh and Paul (2006) and revealed that formation of upward flow occurs near the heated wall with down ward flow achieved near the cooled wall. Couette flow of heat generating/absorbing fluid was investigated by Jha and Ajibade (2010) and their result shows that reverse flow of the fluid is achieved with external heating of the moving plate. The study of Miyatake et al. (1973) pointed that the rate of heat transfer near the hotter wall is enhanced by the buoyancy force with the reversal flow attained near the cooler wall. Other connected studies can be witnessed in Mandal et al. (2014), Jha and Ajibade (2011). Nelson and Wood (1989a,b) and Jha et al. (2012).
Studies related to viscous fluid with temperature-dependent viscosity are of paramount importance; especially in petroleum industries for purification and filtration processes; it is also used in food processing and coating of metals. The ancient expression of temperature-dependent viscosity was first given by Reynold (1984). With the advent of this; several scholars have modeled the expression for temperature-dependent fluid viscosity in different forms; all of which revolved around the ancient Reynold's expression. These can be viewed in Elbashbashy et al. (2000), Mukhapdyay et al. (2009) andVanden Berg et al. (2005); just a few to mention among others. In a related article, Carey and Mollendorf (1978) affirmed that when the viscosity of water is raised from its viscosity is decreased by 240% while that of Grey et al. (1982) conveyed that; when the viscosity of a fluid is temperature-dependent, the flow mechanism of the fluid changes significantly compared to the assumption of constant viscosity. Mehta and Sood (1992) disclosed that; the usual assumption of constant viscosity of fluids evaluated at some reference temperature is not sufficient to describe a correct situation in the transport characteristics of viscous fluids. Temperature-dependent viscosity on free convective laminar boundary layer flow past a vertical isothermal flat plate was studied by Kafousius and Williams (1995) while the effect of temperature-dependent viscosity on mixed convection flow past a vertical flat plate in the region near a leading edge was investigated by Kafousius and Rees (1998). In the afore mentioned studies the latter researcher disclosed that when viscosity of fluid is sensitive to temperature change, the effect of temperature-dependent viscosity has to be taken into cognizance or else significant errors may occur in the flow mechanisms. Furthermore, Makinde and Ogulu (2011) concluded that a reduction in fluid viscosity amounts to the rise in its velocity. Interrelated scholarly articles can be seen in Costa and Macedonio (2003), Seddeek and Salem (2006), and Hossain et al. (2001).
Temperature-dependent thermal conductivity in the study of flow of viscous fluids has been considered by scholars due to its solicitations in technological innovations like in the extrusion of plastic sheets, polymer processing, spinning of fibers, cooling of elastic sheets etc. For instance, in heat sink/source applications; materials of high thermal conductivity are used while those of low conductivity are usedin designing insulators. Similarly, metals in liquid form with small Prandtl number in the interval of 0.01 -0.1 are commonly used for cooling purposes because of their high thermal conductivity. Numerous scholars investigated flow of viscous fluids on the assumption of constant thermal conductivity. However this is uncalled for; as variation in temperature affects the thermal conduction of the fluid. This is evidently observed in the study of Adrian et al. (1997) Van den Berg et al. (2001) divulged that the use of variable thermal conductivity to study flow of molten magma can delay secular cooling of the mantle with constant viscosity model. Sharma and Aisha (2014) submitted that thermal conduction of fluid increases with decrease in Prandl number. Other associated studies can be referred to the articles of Rihab et al. (2017), Dubuffet et al.(1999, Starlin (2000), Blas (2019) and Hofmeister (1999).
Release of energy in the form of electromagnetic waves by hot objects is termed thermal radiation. This has fundamental importance in cooling/heating processes; especially in the aspect of engineering applications for human survival on the earth. For instance thermal radiation is used in sterilization of medical instruments, toasting of bread, treatment of cancer and tumor, air conditioners and heaters. Due to this, Rosseland (1931), first gave the expression for thermal radiation and this expression was further simplified by Sparrow and Cess (1962). The simplified form is being used by scholars to study flow of fluids with thermal radiation; refer to Makinde et al. (2007), Makinde and Ibrahim (2017), Ganji et al. (2015), Sheikholesmi (2015), Makinde (2008) and Abel and Mashesha (2007). In view of this novelty, some researchers mentioned above have discussed the effect of thermal in their flow formation using linearized temperature in in their flow formation. The usage of this was queried by Magyari and Patokratoras (2011) arguing that the flow behavior is not accurately predicted via this procedure. They therefore proposed alternative method which adopted the use of nonlinear temperature in the expression for thermal radiation. In recognition to this, connected studies can be witnessed in Ajibade (2018a, 2020), Ajibade and Yusuf (2019), Yabo et al. (2016) and Jha et al. (2017).
There exists different method of solving differential equations arising from fluid flows. These include the method of undetermined coefficients, Runge-Kuta method, finite difference method, Laplace transform, Adomian decomposition method (ADM) and many others.
The present article investigates steady free convective Couette flow in a vertical channel on adopting nonlinear thermal radiation, dynamic viscosity, dynamic thermal conductivity and ADM method of solution (Adomian (1994)). This investigation is motivated by the works of some authors which failed to adopt the above parameters upon which the flow behaviors are either under-determined or over-determined. The choice of ADM is due to the following reasons: the technique avoids perturbation, it gives efficient, accurate and approximate solution, it does not require discretization of the solution, does not results to large equations. Furthermore; the method is not affected by computational round off errors, consumes less time and less amount of computer memory (Makinde et al. (2007). Figure 1 consists of an infinite vertical channel formed by two parallel plates kept h distance apart. The channel is filled with an optically thick viscous incompressible fluid at the expense of radiative heat flux of intensity r q ; which is absorbed by the plates and transferred to the fluid. Neglecting the effect of viscous dissipation and assuming all the fluid's physical properties are constant except for its viscosity and thermal conduction which are assumed to be temperature-dependent. The ′ -axis coordinate is taken along the channel in the vertically upward direction, being the direction of the flow while the ′ -axis is taken normal to it. Also; assuming effect of radiative heat flux in the ′ -direction to be negligible compared to that in the ′ -direction with the temperature of

FORMULATION OF THE PROBLEM
with the radiative heat flux of Sparrow and Cess.(1962) given as: Carey and Mollendorf (1978); the dynamic viscosity and dynamic thermal conductivity of the fluid are respectively expressed in the form: with the boundary conditions for the velocity and temperature fields as: 0 ,

Non-dimensional of the Problem
The problem under consideration involves quantities in different dimensions and so equations (1-6) are therefore required to be transformed into non-dimensional form using the quantities: Using equation (4) and (7); the momentum equation (1) is transformed into dimensionless form and the following equation is obtained: The radiative heat flux in equation (3) is expanded nonlinearly on adopting Magyari and Pantokratoras (2011) Now substituting equation (4), (7) and (9) into equation (2) gives the equation: Again, using equation (7) in equation (5) and (6), the boundary conditions are: Meaning of the parameters involved in equations (1-13) see the table of nomenclature. . Mathematical Description of ADM Consider the differential equation in Adomian form: where u is unknown function which is to be determined by a recursive relation, L is the highest order derivative which is also invertible, S is the remainder of the linear operator whose order is less than L , Nu represents the nonlinear terms and g is the system input.
Operating 1  L to both sides of equation (14) and using the given initial boundary conditions, the following differential equation is obtained: where w represents the term arising from integrating and the auxiliary conditions. According to ADM the solution is defined by the series: and Nu comprises the series of the Adomian polynomials: where are Adomian polynomials generated from the equation: The solution components 0 , 1 , 2 , … … are determined recursively as: where w is referred to as the zeroth-order component.

ADM Solution of the Problem
The differential equations (8) and (10) subject to equations (11) and (12)

Convergence/Termination Criteria of the ADM Solution
It has been proven in Adomian (1994) and Cherruault (1990) that convergence of ADM solution always exists and is rapidly. Based on this the convergence of the solution is not tested here.
For the termination criteria; the ADM solutions for u and  are all paused after the 3 rd terms as subsequent terms after these contribute insignificantly to the final solution. The final solutions are not presented here due to their cumbersomeness but are used for discussing the results.

Nusselt Number and Skin Friction on the Channel Plates
The Nusselt number on the channel plates are evaluated on adopting Kay (2017) figure 3 where the fluid velocity is also seen to rise with increase in , these ehaviors are the attributes of the decrease in thermal conductiom of the fluid.   figure 7 where the fluid velocity also increases with increase in .  The effect of varying Gr on the fluid velocity is depicted in figure 9 where the figure shows that the fluid velocity increases with increase in Gr. This is the consequence of the increase in the buoyancy force of the fluid molecules within the channel.

VALIDATION OF THE RESULTS
This section validates the accuracy of the results realized in this investigation. In order to do this, the parameters    , 32: 1965-1983. Andrian , N, Stefan , N,. Martins, N. and Holger, V. (1997.