**1. Introduction**

The impact of temperature and species concentration distribution on heat and mass transfer of fluid flow has received renewed interest to researchers and the academic community due to its multiple application areas notably in physical and chemical processes, food and manufacturing industries, geophysics, oceanography, and photosynthesis. These occurrences of thermo-solutal convection not only involve temperature variation but also concentration variation. Free convection problems driven only by temperature difference have been studied extensively by many investigators notably Akter et al. [1], Schlichting [2], Venkateswara [3], Gebhart and Pera [4], Khair and Bejan [5], and Mongruel et al. [6]. Species concentration variation sometimes plays a major role in creating the buoyancy needed in driving flow and influencing rate of heat transfer. Double

diffusion has been studied by few investigators over the years. The pioneer of this area of research is the work of Gebhart and Pera in 1971 [4] where they investigated the combined buoyancy effects of thermal and mass diffusion on natural convection flow. Also, Bejan and Khair [7] carried out some analysis on heat and mass transfer by natural convection in porous media. Furthermore, the Schmidt number is the appropriate number in the concentration equation for *Pr* < 1 regime, while in the *Pr* > 1, regime Lewis number is the appropriate dimensionless number for vertical walls and this extends to inclined walls. This important criterion is sometimes omitted from heat and mass transfer studies. Allain et al. [8] also considered the problem of combined heat and mass transfer convection flows over a vertical isothermal plate. These contributors used a combination of integral and scaling laws of Bejan for their investigations. Their work was restricted to cases where two buoyancy forces aid each other; however, it was observed that heat diffusion is always more efficient than mass diffusion meaning that Lewis number is always greater than unity in most cases. It has been recommended in some previous works that more numerical or experimental results covering a wide range of Prandtl and Schmidt numbers are needed to be obtained by further investigations.

**2. Problem formulation and scale analysis**

*DOI: http://dx.doi.org/10.5772/intechopen.90896*

the following:

*u ∂v ∂x* þ *v ∂v <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>ϑ</sup> <sup>∂</sup>*<sup>2</sup>

**Figure 1.**

**27**

**Energy equation**

**Continuity equation**

**Momentum equation**

*v*

*Physical model of double diffusive free convection over vertical wall.*

**Species concentration equation**

*u ∂T ∂x* þ *v ∂T <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>α</sup><sup>T</sup>*

*u ∂C ∂x* þ *v ∂C <sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>D</sup> <sup>∂</sup>*<sup>2</sup>

The problem of combined heat and mass transfer over a heated semi-infinite inclined solid wall is considered. The fluid is assumed to be steady, Newtonian, viscous, and incompressible. It is assumed that the wall is maintained at uniform surface temperature *Tw* and concentration *Cw* and it is immersed in fluid reservoir at rest which is kept at uniform ambient temperature *T*<sup>∞</sup> and concentration *C*<sup>∞</sup> such that *Tw*>*T*<sup>∞</sup> and *Cw*>*C*∞. Boundary layer flow over an inclined wall driven by both thermal gradient and concentration gradient, respectively, are thereby set up due to the difference between wall values and quiescent fluid values. Hence, it is called combined heat and mass transfer phenomenon over an inclined wall (**Figure 1**).

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass…*

This problem is governed by the non-linear and coupled conservation equations. Using the Boussinesq approximation and boundary layer simplifications, we have

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0, (1)

*<sup>∂</sup>x*<sup>2</sup> (3)

*<sup>∂</sup>x*<sup>2</sup> (4)

*<sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>gβT*ð Þ *<sup>T</sup>* � *<sup>T</sup>*<sup>∞</sup> cos *<sup>α</sup>* <sup>þ</sup> *<sup>ρ</sup>gβc*ð Þ *<sup>C</sup>* � *<sup>C</sup>*<sup>∞</sup> cos *<sup>α</sup>* (2)

*∂*2 *T*

*C*

*∂u ∂x* þ *∂v*

Some other research studies carried out were by Angirasa and Peterson [9], who considered free convection due to combined buoyancy forces for *N* = 2 in a thermally stratified medium, and, recently, other contributors have considered flow of power law fluids in saturated porous medium due to double diffusive free convection [10]. Other effects such as Soret and Dufour forces in a Darcy porous medium were considered by Krishna et al. [11]. The problem of mass transfer flow through an inclined plate has generated much interest from astrophysical, renewable energy system, and also hypersonic aerodynamics researchers for a number of decades [1]. It is important to note that combined heat and mass flow over an isothermal inclined wall has received little contributions from scholars [12, 13]. The key notable ones in the literature include the general model formulation of natural convection boundary layer flow over a flat plate with arbitrary inclination by Umemura and Law [14]. Their results showed that flow properties depend on both the degree of inclination and distance from the leading edge. Other investigations considered the problem of combined heat and mass transfer by MHD free convection from an inclined plate in the presence of internal heat generation of absorption [15], natural convection flow over a permeable inclined surface with variable temperature, momentum, and concentration [16], investigations on combined heat and mass transfer in hydro-magnetic dynamic boundary layer flow past an inclined plate with viscous dissipation in porous medium [17], a study on micro-polar fluid behavior in MHD-free convection with constant heat and mass flux [18] and investigations on mass transfer flow through an inclined plate with porous medium [19].

However, research conducted to critically analyze fluid behavior with the effect of species concentration and thermal diffusion on heat and mass transfer particularly for low Prandtl flows past an inclined wall is very rare. This gap has been captured in this study. The objective of this research is to investigate the effect of combined heat and species concentration involving a low Prandtl number fluid flow over an inclined wall using the method of scale analysis in formulation of the model along with the similarity transformation technique to convert partial differential equations to ordinary differential equation. The resulting dimensionless coupled and non-linear equations are solved using differential transform method. The numerics of the computation are discussed for different values of dimensionless parameters and are graphically presented.

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass… DOI: http://dx.doi.org/10.5772/intechopen.90896*
