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An analysis is performed to study thermo-diffusion and diffusion-thermo effects on mixed convection heat and mass transfer boundary layer flow along an inclined (solar collector) plate. The resulting governing equations are transformed and then solved numerically using the local nonsimilarity method and Runge-Kutta shooting quadrature. A parametric study illustrating the influence of thermal buoyancy parameter (<i>ζ</i>), Prandtl number (P<sub>r</sub>), Schmidt number (S<sub>c</sub>), Soret number (S<sub>r</sub>), Dufour number (D<sub>u</sub>) and concentration-to- thermal-buoyancy ratio parameter, <i>N</i>, on the fluid velocity, temperature and concentration profiles as well as on local skin-friction, Nusselt and Sherwood numbers is conducted. For positive inclination angle of the plate (<i>γ</i> = 70 degrees), flow velocity (<i>f'</i>) is strongly increased i.e. accelerated, with thermal buoyancy force parameter (<i>ζ</i>), in particular closer to the plate surface; further into the boundary layer, <i>ζ</i> has a much reduced effect. Conversely temperature (<i>θ</i>) and concentration (<i>ψ</i>) is decreased with increasing thermal buoyancy parameter, <i>ζ</i>. For negative plate inclination, the flow is accelerated whereas for positive inclination it is decelerated i.e. velocity is reduced. Conversely with negative plate inclination both the temperature and concentration in the boundary layer is reduced with the opposite apparent for positive inclination. Increasing Prandtl number strongly reduces temperature in the regime whereas an increase in Schmidt number boosts temperatures with temperature overshoots near the plate surface for S<sub>c</sub> = 3 and 5 (<i>i.e</i>. for S<sub>c</sub> > 1). Concentration is reduced continuously throughout the boundary layer, however, with increasing Schmidt number. A positive increase in concentration-to-thermal-buoyancy ratio parameter, <i>N</i>, significantly accelerates the flow in the domain, whereas negative <i>N</i> causes a deceleration. A velocity overshoot is also identified for <i>N</i> = 20, at intermediate distance from the plate surface. Negative <i>N</i> (thermal and concentration buoyancy forces oppose each other) induces a slight increase in both fluid temperature and concentration, with the reverse observed for positive <i>N</i> (thermal and concentration buoyancy forces assisting each other). Increasing Dufour number respectively causes a rise in temperature and a decrease in concentration, whereas an increase in Soret number cools the fluid i.e. reduces temperature and enhances concentration values. In the absence of Soret and Dufour effects, positive N causes a monotonic increase in local Nusselt number, Nu<sub>x</sub>Re<sub>x</sub><sup>-1/2</sup> with <i>ζ</i> Cos <i>γ</i>, for <i>N</i> = -1 the local Nusselt number remains constant for all values of parameter, <i>ζ</i> Cos <i>γ</i>. Local Sherwood number, Sh<sub>x</sub>Re<sub>x</sub><sup>-1/2</sup> is boosted considerably with higher Schmidt numbers and also with positive <i>N</i> values. The computations in the absence of Soret and Dufour effects correlate accurately with the earlier study by Chen <i>et al</i>. (1980).

Combined heat and mass transfer from inclined surfaces finds numerous applications in solar energy systems, geophysics, materials processing etc. Many studies have appeared concerning natural and also mixed convection flows. Kierkus [

Consider the combined thermal convection and diffusion mass transfer in laminar boundary layer flow parallel to a flat plate which is inclined to the vertical with angle, , with free stream velocity, , as depicted in figure 1. The temperature of the ambient medium is and wall temperature is. The flow along the plate contains a species, A, slightly soluble in the fluid, B with the concentration at the plate surface being and the solubility of A in B far away from the plate is. The streamwise coordinate, x, is measured from the leading edge of the plate, parallel to the plate and the transverse coordinate, y, normal to the plate in the outward direction, for flow regimes both above and beneath the plate. Following Sparrow et al. [

The forced flow, following Chen et al. [

(2)

The first two terms on the RHS of equation (2) correspond to the streamwise pressure gradients induced by the combined buoyancy forces, with the plus and the minus signs representing, respectively, flows above and below the plate. The third and fourth terms correspond to the buoyancy forces generated by thermal and mass diffusion, with the plus and minus signs referring, respectively, to upward and downward forced flows. The final term in (2) on the right hand side is the viscous diffusion term. The initial and boundary conditions at the plate and in the free stream are:

, , at x = 0

, , , at y = 0

, , as (5)

where u and v denote the velocity components in the xand y-directions respectively, v is the kinematic viscosity, and are the coefficients of thermal expansion and concentration expansion, respectively, T and C are the temperature and concentration, respectively, is the density, D_{m} is the mass diffusivity, c_{p} is the specific heat capacity, c_{s} is the concentration susceptibility, is the thermal diffusivity, T_{m} is the mean fluid temperature, K_{T} is the thermal diffusion ratio. Also, as indicated by Chen et al. [

Chen et al. [

where denotes the local Reynolds number. Effectively the condition (6) or (7) is valid for Tan 3 ~ 30 i.e. angles of inclination, 72 ~ 88 degrees. In this situation, the buoyancy-induced streamwise pressure gradient terms are omitted in equation (2) which reduces to the much simpler form:

The governing boundary layer equations then comprise equations (1), (3), (4) and (8), with boundary conditions (6) subject to the condition given by (6) and (7). The special case of a vertical plate is retrieved from (8) for i.e.. Proceeding with the analysis we introduce the following dimensionless variables:

where h is the pseudo-similarity variable, x is transformed x-coordinate which represents the thermal buoyancy effect, f is a reduced stream function, q is dimensionless temperature and f is the dimensionless concentration function. The stream function satisfies the mass conservation equation (2) with

Implementing equations (9) and (10) into equations (8) and (3), (4), (5), we obtain:

The transformed dimensionless boundary conditions are:

wheredenotes differentiation with respect to, Pr is the Prandtl number, Sc is the Schmidt number, the thermal buoyancy force parameter, (which is a measure of thermal buoyancy force effect on forced convection), is a parameter quantifying the relative effects between buoyancy forces that arise from mass diffusion and thermal diffusion, denotes local Grashof number for thermal diffusion,

denotes the local Grashof number for mass diffusion respectively,

is the Dufour number,

is the Soret number. The engineering design quantities of physical interest include the skin-friction coefficient (a function of local Reynolds number), Nusselt number function and Sherwood number function, which in dimensionless form are given by:

We now obtain approximate solutions to Equations (11) - (13) based on the local similarity and local nonsimilarity methods developed by Sparrow et al. [

subject to the boundary conditions:

For the second level of truncation, we introduce,

and restore all of neglected terms in the first level of truncation. Thus, the governing equations are:

subject to the boundary conditions:

The introduction of the three new dependent variables in the problem requires three additional equations with appropriate boundary conditions. This can be obtained by differentiating equations (21) with respect to and neglecting the terms and, which leads to:

(23)

The coupled non-linear eqautions (18), (21) and (23) with the boundary conditions (19), (22) and (24) are solved using fourth order Rung-Kutta method with a shooting technique. This method has also been applied very recently to simulate viscoelastic flow in porous media by Bég and Makinde [

The values of Sr and Du have been selected to ensure that the product Sr Du is constant, assuming that the mean temperature is constant. The default values for the control parameters are selected as: Pr = 0.7 (air), Sr = 1, Du = 0.05 [i.e. Sr Du = 0.05], N = 1, Sc = 0.2 (hydrogen gas as the species diffusing in air). In all computations we present the variation of f', q and f versus h for the velocity, temperature and species diffusion boundary layers, and also

and versus x as a simulation of skin friction function, Nusselt number function and Sherwood number, respectively. In figures 2 to 4 we have plotted the variation of dimensionless velocity, temperature and concentration function with the thermal buoyancy parameter,.

For positive inclination angle of the plate (g = 70 degrees), we observe flow velocity (f') is strongly increased i.e. accelerated, with a rise in thermal buoyancy force parameter (x), in particular closer to the plate surface; further into the boundary layer, the profiles begin to converge for all values of x i.e. they tend towards unity as specified in the boundary conditions. A distinct velocity overshoot is identified for x values greater than or equal to unity; for x = 0 and 0.1, this overshoot vanishes. Strong thermal buoyancy therefore has a significant acceleration effect on the boundary layer flow. Temperature function (q) however, as illustrated in figure 2 is adversely affected with increasing thermal buoyancy parameter, x. Profiles all descend smoothly from the maximum value at the wall to zero in the free stream. The value N = 1 implies that the thermal and concentration (species diffusion) buoyancy forces are of the same order of magnitude. Dimensionless concentration function (f) is decreased also with increasing thermal buoyancy parameter, x, as shown in figure 4. The differences in values are however smaller over the same range of x values, compared with the temperature distribution (figure 3). In all cases the velocity, temperature and concentration are minimized with x = 0.

In figures 5 to 7 the effects of plate inclination on the dimensionless velocity, temperature and concentration function with coordinate transverse to the plate (h) are illustrated, again with both Soret and Dufour effects present, for N = 1. When g < 0 i.e. negative plate inclination, in figure 5, the velocity (f') is reduced at first i.e. flow is initially decelerated nearer the plate surface; however further away it is accelerated and infact overshoots. For the case of the vertical plate (g = 0 degrees) and for positive inclination, velocities are always monotonic distributions and no overshoot is present.

With g > 0 i.e. 30 degrees and 80 degrees, velocity is reduced i.e. flow is decelerated, largely owing to gravitational effects. Conversely in figure 6, with negative plate inclination (g < 0) the temperature (q) decreases slightly;

for a vertical plate temperatures are greater than for the negatively inclined plate; temperatures are further increased marginally with positive inclination of the plate. In figure 7, a similar response is observed for the concentration distributions (f) which as with temperature are reduced with negative plate inclination, but enhanced with positive inclination and also for a vertical plate scenario (g = 0 degrees).

_{ }values imply a thinner thermal boundary layer thickness and more uniform temperature distributions across the boundary layer. Smaller Pr_{ }fluids have higher thermal conductivities so that heat can diffuse away from the vertical surface faster than for higher Pr fluids (thicker boundary layers). Physically Pr = 0.7 is representative for air or hydrogen and Pr = 1 for water. Pr = 10, 20 may correspond to oils and lubricants. Pr encapsulates the ratio of momentum diffusivity to thermal diffusivity. With a decrease in Pr temperatures as seen in figure 8 are substantially increased. The lowest temperatures correspond to the highest value of Prandtl number (Pr = 20). No temperature overshoot is observed. In figure 9 the effect of Schmidt number, Sc on temperature profiles (q) is plotted, for low value of thermal buoyancy parameter (x = 0.1). Larger values of Sc correspond to Methanol diffusing in air (Sc = 1.0) and Ethylbenzene in air (Sc = 2.0), as indicated by Gebhart et al. [