Conjugate Effects of Heat and Mass Transfer on Natural Convection Flow Across an Isothermal Horizontal Circular Cylinder with Chemical Reaction

Abstract. Natural convection flow across an isothermal cylinder immersed in a visc ous incompressible fluid in the presence of species concentration and chemic al rea tion has been investigated. The governing boundary layer equations are transf o med into a system of non-dimensional equations and the resulting nonlinear system of partia l differential equations is reduced to a system of local non-similarity boundary layer eq uations, which is solved numerically by a very efficient implicit finite difference method tog ether with the Keller-box scheme. Numerical results are presented by the velocity, temperature and species concentration profiles of the fluid as well as the local skin-friction coefficient, local heat transfer rate and local species concentration transfer rate fo a wide range of chemical reaction parameter γ (γ = 0.0, 0.5, 1.0, 2.0, 4.0), buoyancy ratio parameter N (N = −1.0,−0.5, 0.0, 0.5, 1.0), Schmidt number Sc (Sc = 0.7, 10.0, 50.0, 100.0) and Prandtl number Pr (Pr = 0.7, 7.0).

Nomenclature a radius of the circular cylinder g acceleration due to gravity C species concentration in the fluid Gr Grashof number C ∞ species concentration with fluid J w concentration flux away from the cylinder K thermal conductivity C p specific heat at constant pressure K 1 chemical reaction parameter C w species concentration at the surface N buoyancy ratio parameter of the cylinder Nu x local Nusselt number C f x local skin-friction P r Prandtl number D chemical molecular diffusivity

Introduction
The application of boundary layer techniques to mass transfer has been of considerable assistance in developing the theory of separation processes and chemical kinetics. Some of the interesting problems that have been studied are mass transfer from droplets, free convection on electrolysis in non-isothermal boundary layer. Heat, mass and momentum transfer on a continuously moving or a stretching sheet has several applications in electrochemistry and polymer processing [1][2][3][4]. Gebhart and Pera [5] investigated the nature of vertical natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion. Diffusion and chemical reaction in an isothermal laminar flow along a soluble flat plate was studied and an appropriate mass-transfer analogue to the flow along a flat plate that contains a species say, A slightly soluble in the fluid say, B has been discussed by Fairbanks and Wick [6]. Hossain and Rees [7] have investigated the combined effect of thermal and mass diffusion in natural convection flow along a vertical wavy surface. The effects of chemical reaction; heat and mass transfer on laminar flow along a semi-infinite horizontal plate have been studied by Anjalidevi and Kandasamy [8].
By taking advantage of the mathematical equivalence of the thermal boundary layer problem with the concentration analogue, results obtained for heat transfer characteristics can be carried directly over to the case of mass transfer by replacing the Prandtl number P r by the Schimdt number Sc. However, the presence of a chemical reaction term in the mass diffusion equation generally destroys the formal equivalence with the thermal energy problem and moreover, generally prohibits the construction of the otherwise attractive similarity solutions. Takhar et al. [9] for example, considered diffusion of chemically reactive species from a stretching sheet.
The application of the boundary layer theory with chemical reaction has been applied to some problems of free and mixed convection flow from the surface of simple geometry by the above authors. Chiang et al. [10] investigated the laminar free convection from a horizontal cylinder. Sparrow and Lee [11] looked at the problem of vertical stream over a heated horizontal circular cylinder. They have obtained a solution by expanding velocity and temperature profiles in powers of x, the co-ordinate measuring distance from the lowest point on the cylinder. The exact solution is still out of reach due to the non-linearity in the Navier-Stokes equations. It appears that Merkin [12,13] was the first who presented a complete solution of this problem using Blasius and Görtler series expansion method along with an integral method and a finite-difference scheme. Ingham [14] investigated the free convection boundary layer flow on an isothermal horizontal cylinder. Recently, Nazar et al. [15] have considered the problem of natural convection flow from lower stagnation point to upper stagnation point of a horizontal circular cylinder immersed in a micropolar fluid.
In the present study the focus is given on the effects of heat and mass transfer on the natural convection boundary layer flow across a horizontal circular cylinder with chemical reaction when the surface is at a uniform temperature and a uniform mass diffusion. Here it has been assumed that the level of species concentration is very low and that the heat generated during chemical reaction can be neglected. The basic equations are transformed to local non-similarity boundary layer equations, which are solved numerically using a very efficient finite-difference scheme together with the Keller-box method [16]. To the best of our knowledge, this problem has not been considered before. Consideration is given to the situation where the buoyancy forces assist the natural convection flow for various combinations of the chemical reaction parameter γ, buoyancy ratio parameter N , Prandtl number P r and Schimdt number Sc. The results allow us to predict the different kinds of behaviour that can be observed when the relevant parameters are varied.

Formulation of problem
Let us consider a steady two-dimensional laminar free convective flow that flows across a uniformly heated horizontal circular cylinder of radius a, which is immersed in a viscous and incompressible fluid. It is assumed that the surface temperature of the cylinder is T w , where T w > T ∞ . Here T ∞ is the ambient temperature of the fluid and T is the temperature of the fluid. The configuration considered is as shown in Fig. 1. An appropriate mass transfer analogue to the problem shown in Fig. 1 would be the flow across the surface of a horizontal circular cylinder that contains a species A slightly soluble in the fluid B. The concentration of the reactant is maintained at a constant value C w at the surface of the cylinder and the solubility of A in B is D and the concentration of A far away from the surface of the cylinder is assumed to be C ∞ . Let the reaction of a species A with B be the first order homogeneous chemical reaction with rate constant, K 1 . It is desired to analyse the system by a boundary layer method. It is assumed that the concentration of dissolved A is small enough and the physical properties ρ, µ and D are virtually constant throughout the fluid. Under the usual Boussinesq approximation, the governing equations of the flow are The boundary conditions of equation (1) to (4) are where ( u, v) are velocity components along the ( x, y) axes, g is the acceleration due to gravity, ρ is the density, k is the thermal conductivity, β T is the coefficient of thermal expansion, β c is the coefficient of concentration expansion, µ is the viscosity of the fluid, C p is the specific heat due to constant pressure and D is the molecular diffusivity of the species concentration. We now introduce the following non-dimensional variables: where ν = (µ/ρ) is the reference kinematic viscosity and Gr is the Grashof number, θ is the non-dimensional temperature and φ is the non-dimensional species concentration function. Substituting the variables (6) into equation (1)-(4) leads to the following non-dimensional equations The boundary conditions (5) become where N is the ratio of the buoyancy forces due to the temperature and concentration, γ is the chemical reaction parameter are defined respectively as To solve equations (7)- (10), subject to the boundary conditions (11), we assume the following variables where φ is the non-dimensional stream function defined in the usual way as Substituting (13) into equations (8)-(10) we get, after some algebra the following transformed equations 1 Sc along with boundary conditions It can be seen that near the lower stagnation point of the cylinder i.e. x ≈ 0, equations (15)- (17) reduce to the following ordinary differential equations: 1 subject to the boundary conditions at In the above equations primes denote differentiation with respect to y.
In practical applications, the physical quantities of main interest are the shearing stress, the rate heat transfer and the rate of species concentration transfer in terms of the skin-friction coefficients C f , Nusselt number Nu and Sherwood number Sh respectively, which can be written as Using the variables (6), (13) and the boundary condition (18a) into (23)-(24),we get We also discuss the effect of the chemical reaction parameter γ and buoyancy ratio parameter N on the velocity, temperature and concentration distribution. The values of the velocity, temperature and concentration distribution are calculated respectively from the following relations: u = ∂f ∂y , θ = θ(x, y), φ = φ(x, y).

Results and discussion
Equations (15)- (17) subject to the boundary conditions were solved numerically using a very efficient implicit finite difference method together with the Keller-box scheme, which is described by Cebeci and Bradshow [17].  [12] and Nazar et al. [15] has been made in Table 1. It is clearly seen that there is an excellent agreement among the respective results. The effect of the chemical reaction parameter γ, on the reduced local skin-friction coefficient C f x , local Nusselt number Nu x and the local Sherwood number Sh x is shown in Figs. 2(a) Here it is found that both the velocity and concentration profiles decrease significantly and the non-dimensional temperature profile increases slightly with the increase of chemical reaction parameter.
In Table 2 we have entered the numerical values C f x , Nu x and Sh x for different values of P r (P r = 0.7, 7.0) while N = γ = 0.5 and Sc = 0.7. It is observed that for increasing axial distance parameter x, the values of skin-friction increase and the values of rate of heat transfer and that of the rate of species concentration decrease for both the values of Prandtl number P r. On the other hand both the C f x and Sh x decrease and Nu x increases accordingly as Prandtl number P r increases. And these changes in C f x , Nu x and Sh x due to an increase in P r are consistent with a free convection boundary layer.

Conclusions
The effect of chemical reaction, heat and mass diffusion in natural convection flow from an isothermal horizontal cylinder has been investigated numerically. New variables to transform the complex geometry into a simple shape have been introduced and the resulting non-similarity boundary layer equations are solved by a very efficient implicit finite difference method together with the Keller-box scheme [16]. From the present investigation the following conclusions may be drawn: • Both the local skin-friction coefficient C f x and the local Nusselt number Nu x decrease and the local Sherwood number Sh x increases when the value of the chemical reaction parameter γ increases.
• As γ increases both the velocity and concentration distribution decrease significantly and the temperature profile increases slightly at x = π/2 of the surface.
• An increase in the values of N leads to an increase in the values of the local skinfriction coefficients C f x , local Nusselt number Nu x and the local Sherwood number Sh x .
• The skin-friction coefficient C f x and local Nusselt number Nu x decrease and the rate of species concentration Sh x increases within the boundary layer as the value of Sc increases.
• The skin-friction coefficient C f x and the rate of species concentration Sh x decrease and local Nusselt number Nu x increases for increasing values of P r.