MHD natural convection flow of Casson fluid in an annular microchannel containing porous medium with heat generation/absorption

Abstract This study has been conducted to focus on natural convection flow of Casson fluid through an annular microchannel formed by two cylinders in the presence of magnetic field. The process of heat generation/absorption is taken into consideration. Combined effects of various parameters such as porous medium, velocity slip and temperature jump are considered. Solution of the present mathematical model is obtained numerically using fourth-fifth order Runge-Kutta-Fehlberg method. The flow velocity, thermal field, skin friction and Nusselt number are scrutinized with respect to the involved parameters of interest such as fluid wall interaction parameter, rarefaction parameter, Casson parameter and Darcy number with the aid of graphs. It is established that higher values of Casson parameter increases the skin friction coefficient. Further it is obtained that rate of heat transfer diminishes as fluid wall interaction parameter increases.


Introduction
Increase in trend towards micro scale system leads to analyze the uid ow through micro passage such as micro heat exchangers, micro nozzles and microchannel etc. Micro scale system has tremendous application in the eld of biomedicine, pharmaceutical chemistry, micro processing, engineering applications and electronics. In uence of Soret and Dufour e ect on MHD ow of Casson uid over stretching sheet has been studied by Hayat et al. [1] using Homotopic method. They conclude that rise in values of Casson parameter and Hartmann number decreases the velocity. Heat and mass transfer analysis of Casson uid over a stretching surface was conducted by Raju et al. [2]. A comparative study of Casson and Newtonian uids is also done in this work. They obtained that enhancement in thermal boundary layer is achieved by increasing viscous dissipation. Mixed convection ow over a stretching surface of Casson uid has been investigated numerically by Rana et al. [3] with homogeneous and heterogeneous heating. It is found that increase in Casson and slip parameter decreases the heat ux. Kumar et al. [4] examined the heat transfer of a Carreau uid over a stretching sheet. They established that thickness of thermal boundary layer declines for large Prandtl number. Scrutinization of MHD Casson uid ow past a wedge has been analyzed by Zaib et al. [5] considering activation energy and binary chemical reaction. It is found that velocity for both Newtonian and non-Newtonian uid increases for increasing magnetic parameter. In uence of homogeneous and heterogeneous reactions on Casson uid has been studied by Khan et al. [6]. It has been shown that temperature pro le diminishes for higher values of Prandtl number. MHD ow of Casson uid over a moving cylinder of linear velocity has been investigated by Tamoor et al. [7]. It is evident that skin friction coe cient diminishes for higher values of Casson parameter. Casson liquid ow induced over a sheet with nonlinear velocity has been studied by Khan et al. [8]. They obtained that concentration pro le decrease for increasing values of homogeneous and heterogeneous parameter. Rheological Casson uid model was employed in this study. Scrutinization of heat and mass characteristics of Casson and Williamson uid has been discussed numerically by Kumaran and Sandeep [9] by considering Brownian and thermophoresis e ect. It is concluded that impact of Lorentz force is higher on Williamson uid than Casson uid. E ect of viscous dissipation on MHD ow of Casson liquid over a vertical stretching plate has been studied by Gireesha et al. [10]. They found that for higher values of magnetic parameter, momentum boundary layer thick-ness diminishes. Kumar et al. [11] deliberated the mixed convection ow of Casson uid over a vertical plate considering nonlinear radiation. It is shown that, temperature increases as Dufour e ect and Soret parameter increases.
Analytical study of natural convection in a vertical microchannel has been carried out by Chen and Weng [12]. They conclude that volume ow rate at micro scale is higher compared to macro scale for natural convection in most situation. Impact of magnetic eld on Casson liquid has been studied by Akbar et al. [13]. It is obtained that rise in values of magnetic parameter leads to increase of velocity pro le. Impact of suction/injection on porous microchannel has been studied by Jha and Aina [14]. The result depicts that rate of heat transfer increases as mixed convection parameter increases. Jha et al. [15] investigated the natural convection ow in an annular microchannel analytically in the presence of velocity slip and temperature jump. It is found that increasing Knudsen number enhances the volume ow rate. Reddy et al. [16] investigated the entropy generation analysis of hydromagnetic Casson uid over a cylinder. It is shown that increasing Casson parameter leads to enhancement in Bejan number. Convection ow of Casson uid through a vertical channel has been studied by Khan et al. [17]. They employed Laplace method to describe velocity and temperature pro les. Pop et al. [18] numerically studied Casson uid in a square cavity considering the combined e ect of thermal radiation and viscous dissipation. The result shows that heat transfer rate enhances for increasing values of Casson parameter. Jha and Aina [19] investigated the in uence of induced magnetic eld on fully developed convection ow in an annular microchannel. It is obtained that for induced magnetic eld, skin friction at the annular microchannel surface is higher.
Soret induced natural convection ow in an annular channel has been studied by Jha et al. [20]. They deliberated the heat and mass transfer analysis through channel. It is obtained that volume ow rate enhances with increasing mass transfer area. Sheremet and Pop [21] discussed the ow behavior of free convection in a triangular cavity lled with porous medium employing Bongiorno's model. It is evident that Sherwood number is an increasing function of Lewis number. Makinde et al. [22] explored the characteristics of heat and mass transfer through porous plate. MHD boundary ow is considered for this investigation. They studied the e ect of temperature dependent viscosity on Nusselt number. Rate of heat transfer rises with temperature dependent viscosity. Mabood et al. [23] explored the MHD ow of nano uid. E ect of chemical reaction, viscous dissipation and radiation are taken into account. It is obtained that skin friction coe cient increases as perme-ability parameter increases. Fully developed ow between two vertical cylinders were analysed by Jha and Yusuf [24] considering porous medium. The outcome of this study reveals that velocity is the function of Darcy number and annular gap. Dar and Elangovan [25], predicted heat and mass transfer of couple stress uid ow through channel. It is obtained that velocity pro le increases as couple stress parameter increases. MHD ow of Carreau uid over a stretching sheet in the presence of homogeneous and heterogeneous reaction was investigated by Hayat et al. [26]. They concluded that thermal boundary layer enhances for increasing values of Biot number. Ree-Eyring uid over a parallel plate has been studied by Ramesh and Eytoo [27] considering heat transfer and slip boundary conditions. They analysed the e ect of porous medium and Ree-Eyring uid parameter. It is clear that velocity is an increasing function of Ree-Eyring uid parameter.
Two dimensional laminar ow of a incompressible uid over a at surface has been carried out by Chamkha and Issa [28]. They obtained that local Nusselt number is the function of Hartmann number and wall suction velocity. Natural convection ow of viscous incompressible uid over a wavy surface has been studied by Molla et al. [29] using Finite Di erence Method. Impact of heat generation/absorption on velocity boundary ow is observed. Finite Volume Method has been employed by Chamkha [30] to study the laminar convection ow of an electrically conducting uid considering heat generation/absorption. It is concluded that average Nusselt number decreases in the presence of internal heat generation. Impact of heat generation/absorption on nano uid ow over a permeable stretched surface has been examined by Alsaedi et al. [31]. In uence of heat generation and absorption on Casson uid ow over a vertical cone and at plate are studied by Mythili et al. [32]. They found that heat generation/absorption plays signi cant role in de ning heat transfer through moving uid. Heat generation/absorption uid in an annular microchannel has been studied by Jha and Aina [33]. It is evident that heat generation parameter enhances the volume ow rate. Inuence of sinusoidal heating in a porous cavity has been examined by Cheong et al. [34] using nite di erence method. It is concluded that rate of heat transfer is higher for wavy cavity than square cavity. Gireesha et al. [35] numerically studied the in uence of exponential heat source on Casson nano liquid ow over a stretching plate considering activation energy and chemical reaction. It is obtained that value of Nusselt number is higher for nonlinear radiative heat process compared to linear radiation e ect.
Micro uidics has tremendous application in science and engineering technology. More speci cally, electronic cooling systems micromixers and micro heat exchanger. Therefore, the present work is intended to explore the impact of Casson uid ow in an annular microchannel lled with porous material. In this study heat generation/absorption characteristic of Casson uid through microchannel is investigated. The variation in velocity and temperature pro le for di erent parameters is examined and is displayed through graphs. The important parameters such as skin friction and Nusselt number are discussed.

Mathematical analysis
Natural convection ow of an incompressible Casson uid in a vertical annular microchannel formed by two concentric cylinders with porous medium is considered. Let Xaxis is parallel to gravitational eld g in opposite direction while r axis is in the radial direction. The radius of inner cylinder is r and that of outer cylinder is r . The outer surface of the inner cylinder is heated to a temperature T in such a way that the surrounding uid having temperature T < T . Here the inner surface of the outer cylinder is maintained at temperature T . Natural convection occurs due to this temperature di erence. Due to the ow is fully developed and cylinders are of in nite length, the transport phenomenon depends only on r. A uniform magnetic eld of strength B is applied normal to direction of ow. Heat generation/absorption process is considered in this analysis and is given by: Physical model of the present problem under Boussinesq approximation is given as follows; Momentum equation: The boundary conditions for both velocity and temperature eld are: On introducing following dimensionless quantities From equations (2)- (7), one can get is uid wall interaction parameter and β is Casson parameter.
The corresponding boundary conditions are as follows: Solution for energy equation is scrutinized considering two cases such as heat generation and absorption process. For heat generating case second term of equation (10) has positive sign. It has negative sign for heat absorption case.

Case-I: Heat generation
For this energy equation is as follows;

Case-II: Heat absorption
The two parameters of physical interest are rate of heat transfer/Nusselt number and skin friction at the cylinder surface are given by: Here, Nu , Nu , τ and τ are the Nusselt number and skin friction at the inner and outer cylinder surfaces respectively.

Solution procedure
The systems of ordinary di erential equations (9)-(10) representing the de ned ow problem along with the boundary conditions (11)- (14) are highly nonlinear. The boundary value problems are converted into initial value problem. The initial conditions are accessed with the help of shooting technique. Initially the equations are reduced to a system of rst order simultaneous equations considering unknowns as follows: X for heat absorption. (24) The corresponding boundary conditions (11)-(14) are reduced as below; These equations are solved by adopting Maple symbolic software which is capable of solving nonlinear ordinary di erential equations using Runge Kutta Fehlberg fourth fth order method.

Results and discussion
In this section we explored the nature of velocity prole, temperature pro le, skin friction and Nusselt number for di erent ow parameters such as rarefaction parameter β v Kn, uid wall interaction parameter ln, magnetic parameter M, heat generation/absorption parameter S, Darcy number Da and Casson parameter β with the help of graphical representation. The investigation is carried out for di erent values of β v Kn = . , ln = .
, M = . , S = β = . , n = . and Da = . . Figure 2 displays the e ect of rarefaction parameter β v Kn on velocity pro le. As β v Kn increases, we observe an increase in velocity slip at the surfaces, diminishes the retarding e ect on the boundaries. This leads to enhancement in uid velocity. It is obtained that velocity pro le increases for rise in values of rarefaction parameter for both heat generation and absorption cases. Figure 3 exhibits the in uence of uid wall interaction parameter ln on velocity pro le. It is shown that velocity pro le decreases at inner cylinder surface whereas it increases at outer cylinder wall for both heat generation and absorption. Figure 4a  and 4b predict the in uence of heat generation/absorption parameter S on velocity pro le for heat generation and absorption cases respectively. From Figure 4a it is obtained that velocity pro le enhances as the value of S increases. Also, velocity retards with increasing values of heat absorption parameter. The impact of larger values of magnetic parameter M on velocity pro le is displayed in Figure  5. For larger magnetic parameter M, uid velocity decelerates. This is due to the fact that the strong magnetic eld develops Lorentz force (a resistive kind of force) which slows down the ow velocity. Figures 6-9 represents the in uence di erent parameters on enhancing heat transfer characteristics. The impact of increasing values of rarefaction parameter β v Kn on   Figure 6 for heat generation and absorption. Temperature pro le shows decrease in trend as rarefaction parameter β v Kn increases at inner cylinder. This is due to less interaction of uid molecules and the hot wall of the cylinder. It is also observed that temperature enhances at outer cylinder. Figure 7 portrays the in uence of uid wall interaction parameter ln on temperature pro le for both heat generation and absorption. It is observed that as uid wall interaction parameter ln increases temperature decreases at the inner cylinder whereas it shows increase in trend towards outer cylinder surface. Figures 8a and 8b interpret the e ect of heat generation/absorption parameter S on temperature pro le for both heat generation and absorption respectively. Figure 8a shows that as heat generation/absorption parameter S increases temperature also increases. This is because heat generation/absorption parameter S warms the uid.
Hence an increment in the uid temperature is observed.
On rising values of heat generation/absorption parameter S the uid temperature decreases for heat absorption shown in Figure 8b.  In uence of Casson parameter β and magnetic parameter M on skin friction at the inner and outer cylinder is depicted in Figure 9 and 10 respectively. Higher value of Casson parameter enhances the skin friction coe cient. Variation of skin friction coe cient for increasing values of uid wall interaction parameter and rarefaction parameter at inner cylinder R = and outer cylinder R = are shown in Figure 11 and Figure 12 respectively for both heat generation and absorption. For increasing values of uid wall interaction parameter ln from − , skin friction decreases at the outer surface of the inner cylinder is shown in Figure 11. It is observed from Figure 12 that there is an enhancement in skin friction coe cient for increasing values of uid wall interaction parameter ln at the outer cylinder. Figure 13a exhibit the e ects of heat generation parameter S and rarefaction parameter β v Kn on the skin friction at the inner cylinder. It is obtained that skin friction coecient shows increase in trend for increasing values of heat generation parameter S. Impact of heat absorption parameter S and rarefaction parameter β v Kn at the inner cylinder is represented in Figure 13b, which shows decreasing trend for rising values of S. Figure 14a displays the e ects of heat generation parameter S and rarefaction parameter β v Kn on skin friction coe cient at the inner surface of the outer cylinder R = . It is obtained that skin friction coe cient  Figure 15 and Figure 16 respectively. It is clear that rate of heat transfer diminishes for both heat generation and absorption at inner (R=0) and outer cylinder (R=1) surface for increasing values of ln.

Conclusions
Fully developed steady state natural convection ow of Casson liquid through an annular microchannel lled with porous material is considered. The present investigation explores the impact of heat generation/absorption and applied magnetic eld on Casson uid ow in an annulus. Consequences of various ow parameters on ow velocity, temperature, Nusselt number and skin friction are discussed by means of graphs. The key outcome of the present analysis is as follows: • Higher rarefaction parameter enhances the ow velocity of the Casson uid. Nomenclature: w = channel width; Pr = Prandtl number; T = Temperature; u = Velocity; Cp = speci c heat at constant pressure; k = thermal conductivity; β t , β v = dimensionless variables; Nu = Nusselt number at the inner cylinder; g = acceleration due to gravity; Nu = Nusselt number at outer cylinder; ln = uid wall interaction parameter; Kn = Knudsen number; T = free stream temperature; M = Hartmann number; τ = Skin friction at inner cylinder surface; τ = Skin friction at outer cylinder surface; and Q , r = radial coordinate.