Effect of Fluctuating Surface Heat and Mass Flux on Natural Convection Flow along a Vertical Flat Plate

An investigation has been carried on double diffusive effect on boundary layer flow due to small amplitude oscillation in surface heat and mass flux. Extensive parametric simulations were performed in order to elucidate the effects of some important parameters, that is, Prandtl number, Schmidt number, and Buoyancy ratio parameter on flow field in conjunction with heat and mass transfer. Asymptotic solutions for low and high frequencies are obtained for the conveniently transformed governing coupled equations. Solutions are also obtained for wide ranged value of the frequency parameters. Comparisons between the asymptotic and wide ranged values are made in terms of the amplitudes and phases of the shear stress, surface heat transfer, and surface mass transfer. It has been found that the amplitudes and phase angles obtained from asymptotic solutions are found in good agreement with the finite difference solutions obtained for wide ranged value of the frequency parameter.


Introduction
Heat and mass transfer are kinetic processes that may occur in nature, studied separately or jointly. Studying them apart is simpler; however, it is more efficient to consider them jointly. Double diffusive convection creates a buoyancy force so that fluid flow occurs and can be seen in many natural and technological processes. Besides, heat and mass transfer must be jointly considered in some cases like evaporative cooling and ablation. Because of the coupling between the fluid velocity field and the diffusive fields, flow becomes more complicated than the convective flow. Therefore, different behavior may be expected and thus many investigators are still interested in double diffusive flow.
The transport processes due to double diffusion occur in both nature and many engineering applications. Some very important examples of engineering applications include chemical reactions in reactor chamber, chemical vapor deposition of solid layers, combustion of atomized liquid fuels, and dehydration operations in chemical and foundry plants.
An extensive literature survey on this topic has been carried out by Ostrach [1], Huppert and Turner [2], Bejan [3], Gebhart and Pera [4], and Mongruel et al. [5]. Important information and a framework of this type of flow can also be found in the works of these investigators. In their studies, simultaneous heat and mass transfer in buoyancy induced laminar boundary layer flow along a vertical plate have been investigated substantially. Mongruel et al. [5] have proposed a novel method to solve double diffusive boundary layer flow over a vertical flat plate. They considered a vertical flat plate which is immersed in a viscous fluid or in a fluid saturated porous medium. They proposed the integral boundary layer equations and scaling analysis approach. The study of laminar boundary layer flow in presence of an oscillatory potential flow with a steady mean component was first undertaken by Lighthill [6]. He considered the effects of small fluctuations in the free stream velocity on the skin friction and the heat transfer for plates and cylinders by employing the Karman-Pohlhausen approximate integral method. Later, Eshghy et al. [7] and Nanda and Sharma [8] extended Lighthill's theory for free convection flows. Muhuri and Maiti [9] investigated the free convection flow and heat transfer along a semi-infinite horizontal plate with small amplitude surface temperature oscillation about a nonzero mean, with the same method that has been mentioned. The problem of natural convection flow with an oscillating surface heat flux has been studied by Hossain et al. [10]. The effect of transverse magnetic field on the same type of problem was imposed by Kelleher and Yang [11] and more recently this was also premeditated by Siddiqa et al. [12]. Combined heat and mass transfer above a near-horizontal surface in a fluid saturated media were also studied by Hossain et al. [13]. Less attention has been given to the study of unsteady flow due to double diffusion. Moreover, only the search of similarity solutions has attracted much attention. This is because similarity formulation transforms easily the transport equations into a set of ordinary differential equations which can be solved numerically for different values of the parameters involved. However, some researchers, for example, Khair and Bejan [14] and Trevisan and Bejan [15], set out a framework to solve nonsimilarity solutions for depicting heat and mass transfer with great success. Hossain and Mondal [16] investigated the effect of mass transfer and free convection on the unsteady MHD flow past a vertical plate with constant suction. Later on, the same problem considering variable suction was also investigated by the same authors. Hussain et al. [17] investigated the steady natural convection flow due to combined effects of thermal and mass diffusion from a permeable vertical flat plate. This study focused on the boundary layer regime promoted by the combined events in the permeable surface when the surface is at a nonuniform temperature and a nonuniform mass diffusion but with a uniform rate of suction. Hossain et al. [18] presented the results of unsteady natural convection flow along vertical flat plate subjected to the oscillatory boundary conditions on both surface temperature and species concentration. It has been assumed that both the surface temperature and species concentration have small amplitude temporal oscillations with nonzero means. The mean temperature and mean species concentration are assumed to vary as a power of of the distance measured from the leading edge. Roy and Hossain [19] studied the effects of conduction-radiation on natural convection flow. In their work, the fluid has been considered as incompressible and unsteady boundary conditions were taken for both surface temperature and mass concentration. Jaman and Hossain [20] presented the results for the flow along a vertical cylinder with elliptic cross section. Hussain [21,22] focuses on the numerical simulations and analysis of the flow pattern in a spacer-filled flat channel. To find an optimal spacer design, it is essential to determine the flow pattern and turbulence distribution in a spacer-filled channel or spiral-wound module. With the development of more powerful computing techniques, such as computational fluid dynamics (CFD), it has become possible to simulate flow in spacer-filled channels. The basic spacer geometry is modeled as a series of cylindrical filaments partially obstructing the channel. Roy et al. [23] studied the effects of oscillating free stream and surface temperature on natural convection flow along a vertical wedge. In this investigation, the effects of Richardson's number and the Prandtl number have been illustrated. Different numerical techniques are employed to simulate the governing equations and calculated results are compared. Adequate agreement amongst all these calculated values can be noted from these comparisons.
In this present problem, double diffusive flow through a vertical flat plate has been studied extensively. The most important parameter that determines the relative strength of the two buoyancy forces is the ratio parameter , and, for a positive , the buoyancy forces are cooperating and drive the flow in the same direction which is considered in this present investigation. This parameter measures the relative importance of solutal and thermal diffusion in causing the density changes which drive the flow. It can be observed that = 0 corresponds to no species diffusion and ∞ to no thermal diffusion. Similar to any double diffusive study, the governing equations of the flow field are simulated for two different diffusive parameters, Pr and Sc. The values of these two parameters depend on the nature of the fluid and on the physical mechanisms governing the diffusion of the heat and chemical species. As the most important fluids are atmospheric air and water, the results are presented here for Pr = 0.7 that represent the air at 20 ∘ C at 1 atmosphere against the transpiration parameter and Sc ranged from 0.1 to 1.6. Another important parameter, , has also been taken into account widely. In this present study, numerical simulations are carried out in detail and results are illustrated in both figures and tabular forms.

Formulation of the Problem
A two-dimensional unsteady free convection flow of a viscous incompressible fluid flow along a vertical flat plate in the presence of a soluble species is considered in this present study. It is assumed that both the surface heat flux and surface species concentration flux exhibit small amplitude oscillations in time about a steady nonzero mean temperature and concentration. A semi-infinite vertical flat plate is placed at = 0 in Cartesian coordinate system. And ≥ 0, so that the distance from the leading edge along the plate measuring and is measured in outward normal direction from the plate. The ambient fluid temperature and species concentration are taken as ∞ and ∞ . In the case where the surface heat flux and mass flux are considered time dependent, the governing equations of the flow are given by the following sets of Navier-Stokes equations: Mathematical Problems in Engineering   3 where and V are the and components of velocity field, respectively, is the gravitational acceleration, and are the volumetric expansion coefficients for temperature and concentration, respectively, is the thermal diffusivity, and is the molecular diffusivity of the species concentration. Moreover = − ∞ and = − ∞ are the differences of temperature and species concentration between fluid and ambient flow. Considered boundary conditions, under which (1) are solved, are as follows: This set of boundary conditions suggests the form of the solutions of (1) as where represents the frequency of oscillation and , which is very very small positive number, measures the amplitude. Considering these forms of solutions, the steady mean flow is governed by the following set of equations: subject to the boundary conditions: The unsteady flow field is governed by the set of following differential equations and corresponding boundary conditions (9): To get the similarity equations the following transformations are commenced: where ( ) = 0 , Mathematical Problems in Engineering where 0 is the stream function which satisfies the continuity equation (4) and 0 and 0 are the constants related to mean surface heat flux and mass flux, respectively. By introducing the above-mentioned set of transformations, the following sets of equations along with the boundary conditions are found for the steady flow: and the transformations for the nonsimilarity equations are as follows: Equations for the unsteady flow field are as follows: 1 1 Corresponding boundary conditions are Unsteady shear stress, surface temperature, and surface concentration are the most important quantities which should be taken into account to understand the flow field clearly and to elucidate the effects of corresponding important parameters on the flow pattern. These quantities can be calculated from the solutions of (11), (12), and (13)- (17). In this present study, these quantities are calculated and presented in terms of amplitude and phase angels. The following expressions are used to calculate the amplitude and phase of regarding quantities: where ( , ), ( , ), and ( , ) represent the real and imaginary part of ( , ), ( , ), and ( , ), respectively. The solution methodologies for different parts of the flow field are discussed in brief in the following sections.

Solutions Methodologies
Three different techniques are used to solve the governing equations of the flow field. The implicit finite difference method of Keller [24] is put into operation for the entire regime, extended series solution (ESS) for small which corresponds to the region near the leading edge, and asymptotic solution (ASS) for large , corresponding to the region far from the leading edge. Comparisons amongst the results simulated by these three different techniques are elucidated in tabular form as well as by graphs. Excellent agreement amongst the simulated results by different numerical techniques ensured the validity of the model assumptions and efficiency of the numerical techniques that are applied here.

Extended Series Solutions (ESS).
The results considering finite number of terms are valid only for very small range of frequencies.
Since small values of correspond to small Mathematical Problems in Engineering 5 frequencies also, it can be predicted that the flow would be adjusted quasi-statically to the fluctuating rate of both heat and mass transfer in the boundary layer. For small values of which corresponds to near the leading edge, the functions , , and ℎ are expanded in power of as given below: Introducing the above-mentioned series in (13)-(16) and equating the terms of similar powers of to zero, the following sets of equations can be obtained: where = 1, 2, 3, . . . and the respective boundary conditions are where primes denote the derivatives with respect to as convention. In the above, 0 , 0 , and ℎ 0 are the well-known free convection similarity solutions for steady flow field and the functions , , and ℎ are the higher order corrections to the flow due to the effect of the transpiration of fluid through the surface of the plate. Moreover it can be observed that the equations are linear but coupled. Thus it can be assumed that the solutions can be calculated by pairwise sequential solution. Here, pair of equations are integrated using implicit Runge-Kutta-Butcher [25] initial value solver together with Nachtsheim and Swigert [26] iteration scheme.
In this investigation, 8 pairs of equations are considered and solved numerically. Simulated results are compared with the results that are obtained by finite difference method and nice agreement was found amongst these three types of results.

Asymptotic Solution for Large .
To study the flow pattern far from the leading edge, that is, when the values of are large, the asymptotic solutions are carried out. From the results obtained by Keller-Box method it is shown that for larger values of the unsteady response is confined to a thin layer adjacent to the surface. Thus as frequency approaches towards infinity, the solutions tend to be independent of the distance measured downstream from the leading edge, similar to the shear wave solution in the corresponding forced flow problem. This suggested once again another set of series expansion utilizing the limiting solutions as the zeroth-order approximation: Introducing these transformations, (14)-(16) take the form 1 Sc respectively. We can express the functions , Θ, and Φ with fine accuracy as power series. This is because the above equations represent the region which is confined to a thin layer adjacent to the surface. Here, the following series representations are used: Implementing those above expansions, solutions of (14)- (16) can be found in the form of After substituting (33) into (14)- (16) and collecting similar powers of , the following equations can be obtained: 1 1 1 1 1 1 1 1 In these equations, primes denote the differentiation with respect to , and the associated boundary conditions are The solutions of (31)-(42) subject to the boundary conditions (43) give the following expressions for shear stress, surface temperature, and species concentration, respectively: ) −1/2 The above expressions are valid only for Pr ̸ = 1 and Sc ̸ = 1. If it is necessary to calculate the values for Pr = 1 and Sc = 1 then the limiting values as Pr → ∞ and Sc → ∞ should be calculated.

Results and Discussions
Since natural convection flow due to combined effects of thermal and mass diffusion is very important in practical point of view, an extensive investigation for this type of model flow field has been carried out through numerical simulations. In view of the fact that Hossain et al. (1998) also examined this type of flow field only for thermal diffusion, in this present investigation, similar types of results, discussed by Hossain et al. (1998), are produced first. Then the model has been extended to study the flow field with both thermal and mass diffusion. All the model assumptions are kept similar to Hossain et al. (1998). Both the steady and fluctuating parts of the problem are analyzed by the Keller-Box method for the entire frequency regime. The fluctuating part of the problem is investigated by three different methodologies. Results are presented in amplitude and phase angles forms for variation of different parameters in both tabular and graphical forms. The forgoing formulations may be analyzed to indicate the nature of the interaction of the various contributions to buoyancy. These may aid or oppose one another and be of different magnitudes characterized by the value of . When the thermal and solutal effects are opposed, the value of is negative in order to ensure that the flow is in positive   Table 1. During the simulations, the value of Prandtl number, Pr, is chosen as constant value 0.7, representing air, and all other parameters are varied. It can be observed from Table 1 that the values of shear stress, surface temperature, and surface concentration decrease as the values of , , and Sc become higher. Tables 2 and 3 show the comparison of the results obtained by Keller-Box method and perturbation method for local surface temperature and local mass concentration, respectively. For these simulations, the values of the parameters , , Pr, and Sc are taken as 0.5, 0.5, 0.7, and 0.22, respectively, and the quantities represented against ranging from 0.00 to 70.00. The required quantities for the small value of (0.0-0.9) are obtained from extended series solution method and for the higher values of (1.0-70.0) the respective quantities are taken from the results by asymptotic series solution method. For both amplitude and phase angels of the respective quantities, nice agreement is found amongst the results calculated by different methodologies. Here also plodding decrement of amplitude of shear stress, surface temperature and surface concentration, and increment of phase angels of the respective quantities along with the increment of the values of can be observed. The effects of Prandtl number Pr and the exponent parameter on the shear stress are presented in Figures 1  and 2. All the graphs for depicting the amplitudes and phase angels of shear stress are drawn for the results obtained from numerical simulations by Keller-Box method, extended series solution method, and asymptotic series solution method. All the figures clearly show that results obtained for the entire regime are significantly close to the results that are calculated for the region near and far from the leading edge.
For different values of Pr, while all other parameters are kept constant ( = 0.5, = 0.5, and Sc = 0.6), remarkable increase can be seen in amplitudes of the shear stress due to the increment of the Prandtl number, Pr. The phase angels are zero under quasi-steady conditions and decrease monotonically towards the asymptotic values −90 ∘ as → ∞.    In Figure 4, the effect of exponent parameter on the surface temperature is presented. Similar to shear stress, here also the values of amplitude of surface temperature become little smaller as the values of become higher and far from the leading edge; that is, for the large values of , these changes become ignorable for the variation of and tend toward the value of zero. For the phase angels, as expected, the opposite behavior is observed; that is, very small increment of the quantities is achieved because of small decrement of values of .
where is the steady mean shear stress, is the steady surface temperature, and is the steady surface concentration, respectively. The values of , , and are calculated first and

Conclusion
The purpose of this study is to investigate the velocity flow field in terms of local shear stress. Local heat and mass transfer resulting from buoyancy forces which arise from a combination of temperature and species concentration effects of comparable magnitude are also studied rigorously. A linearized theory has been utilized and detailed numerical calculations are carried out for wide ranges of parameters. The important findings of this study can be summarised as follows.
(i) Exceptive agreement amongst all the results calculated by different numerical methods established the validity of the simulations as well as the assumptions of the mathematical model that are made for the respective flow field. (ii) From the analysis of hydrodynamics heat and mass transfer, a complex correlation can be observed among the flow pattern, wall shear stress, and mass transfer enhancement along the flow channel being dependent on the different parameters.
(iii) From the observations of the simulated results it can be concluded that the amplitude of the shear stress, local heat transfer, and local mass transfer decreased as the frequency increases despite the consequences of the Prandtl number, Schmidt number, and the surface heat and mass flux exponent.
(iv) The phase angles for both heat and mass transfer decrease towards the asymptotic value −45 ∘ , while the respective quantity for shear stress reaches the value of −90 ∘ in a decreasing manner.
(v) The heat and mass flux exponent parameter has no significant effects on both amplitude and phase angels as the values of become very large.