Double-Diffusive Natural Convective Flow of a Nanofluid past an Inclined Wavy Plate in a Non-Darcy Porous Medium

In this paper, the double-diffusive convective flow along an inclined semi-infinite wavy plate in a nanofluid saturated non-Darcy porous medium is investigated numerically. Following Prandtl’s transposition theorem, a coordinate transformation is used to transform the irregular wavy surface into a smooth surface. The convective type thermal boundary condition is taken into account and also the Brownian motion and thermophoresis effects are considered into the present nanofluid model. The governing transport equations are initially reshaped into a system of coupled ordinary differential equations by choosing suitable similarity transformations and then solved numerically by using the Spectral Local Linearization Method (SLLM). The effects of various flow influenced parameters on the fluid flow, heat and mass transfer characteristics are explored and exhibited graphically.


Introduction
The double-diffusion is one of the interesting topics in fluid dynamics and it arises in oceans where the convection process is generated by two different density gradients. The doublediffusive natural convection in a Darcy and non-Darcy porous medium has been investigated by Nithiarasu et al. (1997). Khanafer and Vafai (2002) discussed the double-diffusive mixed convection flow in a non-Darcy porous medium. An exhaustive report on the heat and mass transfer due to various fluid flows in a porous medium (both Darcy and non-Darcy models) has been presented by Nield and Bejan (2013).
Most of the authors have analyzed the convective flows with heat transfer over various geometries by assuming the fluid as Newtonian. But, nowadays, the problem of convective flows with heat transfer of non-Newtonian fluids has been one of the active areas in the computational fluid dynamics and one of such fluids is Nanofluid. A suspension of nano-sized fibres or solid particles in conventional fluids (ethylene glycol, water, oil, etc.) is called nanofluid and it is recommended by Choi and Eastman (1995). Both the experimental and theoretical studies have been carried out by several researchers to get enhanced heat transfer rate and higher energy efficiency in various thermal exchange systems for numerous industrial applications. Buongiorno (2006) explored seven slip mechanisms such as fluid drainage, thermophoresis, inertia, Magnus effect, Brownian diffusion, diffusiophoresis, and gravity settling. With this experimentation, he found that the Brownian diffusion and thermophoresis are more significant effects to investigate the nanofluid flows. A detailed review of nanofluids and their relevant applications have been presented by Das et al. (2007), Stephen (2009), Kakac andPramuanjaroenkij (2009).
The study of heat and mass transfer over irregular bodies (wavy or non-uniform or rough surfaces) is essential for various heat transfer applications such as flat plate condensers and collectors in refrigerators. The effect of cross diffusion in natural convection flow of a Newtonian fluid along a vertical wavy plate embedded in a Darcy porous medium has been studied by Lakshmi Narayana and Sibanda (2010) whereas Mahdy and Ahmed (2012) considered the problem of laminar natural convective flow along a wavy plate in a porous medium saturated with nanofluid. Further, the natural convection heat and mass transfer over an inclined wavy plate is generally experienced in various devices like electroplating, solar water heaters, processing of heavy metals, etc. The double diffusive natural convection over an inclined wavy plate embedded in a porous medium has been studied by Cheng (2010) and also he investigated the free convection and heat transfer along a wavy inclined surface in a porous medium (see Cheng (2013)). The numerical solutions to natural and mixed convective flows of a nanofluid over an inclined wavy surface in a Darcy porous medium have been presented by Vijay kumar (2015, 2016).
The present numerical study aims to investigate the double-diffusive convection in a nanofluid flow over an inclined wavy plate in a non-Darcian porous medium. Pseudo-spectral collocation method along with local linearization technique is employed to solve the present problem numerically [for more details, see Canuto et al. (2006), Motsa (2013), Motsa and Animasaun (2015)]. The effects of relevant physical parameter on the nanofluid fluid flow, heat and mass transfer characteristics have been discussed and shown graphically. Consider the laminar, incompressible and 2-D flow over a semi-infinite wavy inclined surface in a nanofluid. The 2-D co-ordinate system is exhibited in Figure 1. Let A 00 (0 90 ) A  be the inclined angle of the wavy surface and the surface of the wavy inclined plate is given by

Mathematical Analysis
in which 2L and a % are the characteristic length and amplitude of the wavy surface. The wavy surface kept at uniform wall concentration w C % and it assumed to be greater than the ambient concentration C  % at any arbitrary reference point in the medium. Further, the ambient temperature and solid volume fraction are considered as T  % and C  % , respectively.
By assuming Oberbeck-Boussinesq approximation and using standard boundary layer approximations, the governing equations can be written as 0 uv xy uv %% are the Darcy velocities in ( , ) xy %%-directions, ,, TC  are the temperature, solid volume fraction, regular concentration, g is the acceleration due to gravity, K is the permeability, , as (1) automatically and also we introduce the following dimensionless variables Using Eq. (7), the governing equations (1)-(5)    (1 ) Furthermore, we recommend the following similarity variables to reduce the coupled system of PDEs (13)-(16) into a coupled system of ODEs Using Eq. (17), we obtain the following equations and the reduced B.C. are

Results and Discussion
The non-dimensional momentum, energy, solid volume fraction and regular concentration equations (18)-(21) related to the boundary conditions (22) are non-linear and coupled ordinary differential equations for which the analytical solution is out of the scope and thus we solved numerically. In this present work, the governing equations (18)

Conclusions
In this paper, the double-diffusive type nanofluid flow over an inclined wavy plate in a non-Darcy porous medium is investigated. The spectral local linearization method is employed to obtain the numerical solutions for various values of non-Darcy parameter, Biot number, inclined angle and amplitude. The main outcomes are:  With an increase in the angle of inclination, the velocity, solid volume fraction, local Nusselt, and regular Sherwood numbers increase and an opposite trend is detected for temperature, regular concentration and local nanoparticle Sherwood number.
 An increase in the wave amplitude enhances the velocity, solid volume fraction, whereas the temperature and regular concentration reduce. Also, the sinusoidal variations are noticed in the Nusselt and Sherwood numbers.
 An increase in the values of Gr * significantly reduces the velocity, local heat, and regular mass transfer rate, while the temperature and regular concentration increase.
 As Biot number enhances, the velocity, temperature, volume fraction, local Nusselt and regular Sherwood numbers enhance whereas the local nanoparticle Sherwood number diminishes.