Effects of thermal slip and chemical reaction on free convective nanofluid from a horizontal plate embedded in a porous media

1.   Introduction

Numerous studies were made on convective flow in porous medium owing to its far-reaching applications. Furthermore, in recent years some of the convective flow in porous media applications have been intensively studied due to its wide applications in engineering. For example, they are applied in the energy sectors, in which they are used in post-accidental heat removal in nuclear reactors, underground disposal of radioactive waste, solar collectors, and oil recovery ^[1,2,3,4]. Other applications include but are not limited to food processing, fibrous insulation, building construction, and nano-based thermal insulation for energy-efficient buildings ^[5,6,7,8]. The fluids containing the solid nanometer-sized (with length scales of 1-10 nm) particles dispersed in some basic fluid that conventionally transfer heat are defined as "nano-fluid". Due to these nanoparticles' thermal conductivity along with the coefficient of heat transfer of the base fluid remarkably increases.

Heat transfer fluids such as synthetic oil, ethylene glycol, and water have a low thermal conductivity ^[9]. Furthermore, thermal conductivity affects the heat transfer factor across the space spreading the heat transfer medium. In ^[9], Choi first presented the novel idea of jumbling metallic nanoparticles with non-metallic nanoparticles to a base fluid. The author reported several potential advantages to his idea, such as the rise of heat conveyance and having a smaller heat conveyance size. Choi et al. ^[10] were from the first to use the term "nano-fluid", and they proved that thermal conductivity augments with the inclusion of small nanoparticles less than 1% by volume. Many researchers ^[11,12,13] demonstrated that nano-fluids contain effective thermal conductivity to a great extent compared to base fluids. Consequently, such fluids have an excellent prospective for heat transfer enhancement. In ^[14], Ghasemi and Aminossadati analyzed natural convection heat transfer in a compound, placed at an inclined position, filled with a CuO/water nano-fluid. Their analysis points out that the inclusion of nanoparticles into pure water shows better performance for heat transfer. Recently, Godson et al. ^[15] presented an overview of the increase of heat transfer by using nano-fluids.

Various science and engineering studies show the occurrence of heat transfer in porous media of having a saturation of a nano-fluid along with a chemical reaction. On account of these occurrences, nano-fluids are extremely important to both scientists and engineers from the point of practical applications. Several industries make use of this type of flow. Moreover, various engineering applications such as safety measures of a radioactive reactor, combustion systems, solar thermal collectors, metallurgical processes, and chemical engineering contain many transport processes. Thermal and mass diffusion exerts a buoyancy force used by the transport processes of such engineering applications in conducting their working principle. All these processes are run under the effects of a chemical reaction. The convection in a horizontally placed layer with a porous medium was analyzed by Nield and Kuznetsov ^[16,17], and also in ^[18] by Cheng and Minkowycz. The aforementioned authors considered a porous medium saturated by nano-fluid. Gorla and Chamkha ^[19] investigated a porous medium with a nano-fluid saturation with natural convection along an isothermal horizontal plate in the medium.

Hamad et al. ^[20] have endeavored to deduce the similarity solutions of the flow of a 2-D laminar forced convection past a stretching permeable sheet within a porous medium, assuming a saturation of a nano-fluid in the medium. Recently, Ahmad and Pop ^[21] have used the nano-fluid model to research the steady mixed convection boundary layer flow along a firmly fixed plane surface placed vertically and surrounded in a porous medium containing a nano-fluid. Lately, an investigation was made by Arifin et al. ^[22] on the steady free as well as mixed convection boundary layer flow along a plane plate placed horizontally. They assumed the plate was inserted in a porous medium containing a saturation of a nano-fluid. The similarity solutions reduce the free variables of their problem, due to which such solutions are quite widely used. Derivation of all group-invariant similarity solutions of 2-D laminar boundary-layer equations is accomplished by applying Lie group transformations. However, at present, it is widely familiar that the classical Lie symmetry technique is applicable for obtaining similarity solutions, see Ovsiannikov ^[23], Ibragimov ^[24], and Bluman ^[25] et al.

In ^[26], Krishna et al. investigated the magneto-hydrodynamic convection flow of an electrically conducting viscous incompressible and heat-absorbing fluid through a porous medium over a vertical flat plate under the influence of a magnetic field. In addition, other heat transfers and flow problems on magneto-hydrodynamic nano-fluid with the effect of Hall and ion slip with infinite vertical plate embedded in a porous medium have been studied and analyzed mathematically, among others, by Krishna and his collaborators ^{[27,28,29,30,31]}. In ^[32], Patil et al. investigated the Prandtl magneto-hydrodynamic nanofluid aspects over a stretched sheet with convective boundary conditions.

Different works, such as those in ^{[33,34,35,36]}, use the group method to solve various transport problems. For example, Patil et al. ^[37] studied the unsteady magneto-hydrodynamic flow of a nano Powell-Eyring fluid near a stagnation point past a convectively heated stretching sheet with thermal radiation and a chemical reaction. The authors used theoretical group analysis to transfer the system of nonlinear PDEs into a system of nonlinear ODE's, which was then solved numerically. Affify and Elgazery ^[38] used scaling group transformations to study a steady two-dimensional stagnation point flow of heat and mass transfer over a heated porous stretching sheet embedded in a porous medium in the presence of a chemical reaction, with heat generation and absorption effects. Ferdows et al. ^[39] investigated such merged impacts on uniform MHD free convection flow that transfers mass as well as heat along a vertically stretched flat surface in motion. They considered the moving stretching surface to be permeable.

In ^[40], Uddin et al. studied the problem of a free convective boundary layer flow of a nano-fluid past a horizontal plate that is placed in a porous media. They considered convective boundary conditions along with suctions and injection at the wall. In ^[41], the authors investigated the amalgamated effects of convective surface boundary conditions along with thermal radiation. The effects were studied on a uniform magneto-hydrodynamic free convection flow that transfers heat and mass along a vertically stretched flat surface. In their investigation, the authors assumed that the permeable stretching surface was in motion. Thermal radiation and a convective surface boundary condition have a plethora of effects on fluid flow. Rashidi et al. ^[42] used optimal homotopy analysis to study the laminar incompressible free convection flow of a nano-fluid past a chemically reacted plate that is placed in a porous medium. The boundary conditions were considered to be convective along with suction and injection at the wall.

The influence of thermal slip boundary condition with chemical reaction and heat emission/absorption on free convection nano-fluid past a horizontal plate in porous media with free stream conditions has not been mentioned in the literature. Our present study aims to investigate the consequences of thermophoresis, Brownian motion, thermal slip, chemical reaction, and heat emission/absorption on the boundary layer flow of a nano-fluid. The flow is considered along a horizontal plate at an upward-facing position. The entire investigation is conducted to obtain the results numerically. The governing equations are translated into a group of nonlinear ODEs by using Lie symmetry group transformation. The corresponding boundary conditions are also analyzed. Finally, graphical analysis is presented on the resulting consequences of governing parameters on the dimensionless flow profiles.

This paper's outline is as follows: In Section 2, we formulate our model. In Section 3, we perform the Lie group analysis. After that, in Section 4, we discuss our physical quantities and the non-dimensional quantities. Then, in Sections 5 and 6, we present the results of our model simulations as well as show our analysis. Finally, in section 7, we conclude with a discussion of our results.

2.   Problem formulation

In this problem, we consider a laminar, 2-D free convective boundary layer flow of a nano-fluid. The nano-fluid is assumed to flow over a horizontally placed plate upwardly faced and held in a Darcy porous medium. We select a coordinate frame in which the $x$-axis is in the horizontal direction, and the $y$-axis is normal to it (Figure 1). Across the space separating the constitute of the plate and the fluid, a chemical reaction of order $n$ occurs. This chemical reaction is assumed to be homogeneous, isothermal, and reversible. On the surface, the fraction volume of the nanoparticle $\left(C\right)$ and the temperature $\left(T\right)$ are assumed to be constants and take the values ${C}_{W}$ and ${T}_{W}$, respectively. We denote the corresponding values of $T$ and $C$ as obtained at free stream by ${T}_{\infty }$ and ${C}_{\infty }$, respectively. As in ^[43], we also assume that ${T}_{W}>{T}_{\infty }$, and that the Boussinesq-approximation is used.

The following assumptions are considered:

● At the boundaries, it is assumed that $T = {T}_{w}$ and $C = {C}_{w}$ are both constants.

● The corresponding values of $T$ and $C$ as obtained at free stream are denoted by ${T}_{\infty }$ and ${C}_{\infty }$, respectively.

● It is assumed that ${T}_{w}>{T}_{\infty }$.

● It is assumed that the Boussinesq approximation is used.

As in references ^{[41,42,43,44]}, the conservation of mass, momentum, energy, and properties of the nanoparticles are respectively described by the following four field equations:

In the equations above, $\overrightarrow{V} = \left(\overline{u}\text{, }\overline{v}\right)$, and $\tau = \frac{{\left(\rho {c}_{P}\right)}_{P}}{{\left(\rho {c}_{P}\right)}_{f}}$. Also, the thermophoretic diffusion coefficient is denoted by ${D}_{T}$ is, the Brownian motion diffusion coefficient is denoted by ${D}_{B}$, the porous media porosity is denoted by $ϵ$, and finally, ${\left(\rho C\right)}_{f}$ and ${\left(\rho C\right)}_{P}$ denote the heat capacity of our fluid and nanoparticles, respectively. We assumed that we have a slow flow ignoring an advective term quadratic drag by Buongiorno ^[45] and Nield and Kuznetsov ^[46]. The momentum equation is linearized by assuming nanoparticle concentration is diluted with a befitting selection for the reference pressure. Thus, Eq (2) corresponds to the following form:

where ${\rho }_{P}$ and ${\rho }_{f}$ are the density of particles and base fluid, respectively.

The effect of volumetric heat generation as a function of temperature in the flow region is given by:

where ${Q}_{0}$ is the heat constant, and $L$ is assumed to be the length of our plate. The reaction rate also varies as the following function:

where ${k}_{0}$ denotes the constant reaction rate.

Considering the previous suppositions, and following references ^{[41,42,43,44]}, the governing equations are rewritten as:

where ${\alpha }_{m} = \frac{{k}_{m}}{{\left(\rho {c}_{P}\right)}_{f}}$ is the thermal diffusivity, ${k}_{m}$ is the thermal conductivity, the parameter $\text{τ}\text{ = }\frac{ϵ{\left(\rho C\right)}_{p}}{{\left(\rho C\right)}_{f}}$, and the boundary conditions take the form:

In Eq (13) above, ${D}_{1}$ is the thermal slip factor, and the studied boundary conditions are different from those in references ^{[41,42,43,44]}.

Now, to transform Eqs (6) to (11), we set:

Assuming $\psi$ to be a stream function such that

As in references ^[40] and ^[42], we substitute the stream function into Eqs (8)-(13) and further assume that the free stream velocity ${U}_{\mathrm{\infty }} = {x}^{m}$, which gives us:

The boundary conditions in Eq (13) are transformed to:

The parameters in Eqs (16)-(18) are$Nt$ (thermophoresis), $Nb$ (Brownian motion), $Nr$ (buoyancy ratio), $Q$ (heat generation), $K$ (chemical reaction), and $Le$ (Lewis number). Now, similar to Tapanidis et al. ^[47], we define our parameters as

3.   Lie group analysis

Scaling group transformations are a distinctive case of Lie group analysis ^[46,47]. We consider the following:

In the above equation, $ϵ$ is the parameter of group $\mathrm{\Gamma }$, and ${\alpha }_{i}\left(i = 1, 2, \dots, 6\right)$ are assumed to be real numbers. The transformations in Eq (21) can be considered as a point transformation that converts the coordinates from $\left(x, y, \psi, \theta, \varphi, {D}_{1}\right)$ into $\left({x}^{*}, {y}^{*}, {\psi }^{*}, {\theta }^{*}, {\varphi }^{*}, {D}_{1}^{*}\right)$. Now, we find the relation between ${\alpha }_{i}$ such that

where $j = 1, 2, 3$.

The differential forms ${\mathrm{\Delta }}_{1}, {\mathrm{\Delta }}_{2}, \;\mathrm{a}\mathrm{n}\mathrm{d}\;{\mathrm{\Delta }}_{3}$ are required to be conformally constant under group transformation in Eq (21). As done in references ^[48,49,50], we set the conversions in Eq (21) into Eqs (16)-(18) to give us the following:

The invariance under the group $\mathrm{\Gamma }$ is invoked, and from it, we obtain the following relationships:

The boundary conditions yield:

Now, solving Eqs (26) and (27) yields

The set of transformation under group $\mathrm{\Gamma }$ reduces to

Now, taking Taylor series expansion about $ϵ$ gives us:

As ${\alpha }_{2}\ne 0,$ we can rewrite the equations above as the following characteristic equations:

Thus, in terms of differentials, we have that:

3.1.   Similarity transformations

Now, integrating Eq (32) gives us:

where $\eta$ is a constant. Likewise, integrating Eq (33) yields:

where $f\left(\eta \right)$ is a constant, and integrating Eq (34) gives us:

where ${\left({D}_{1}\right)}_{0}$ is a constant. Finally, from Eq (35), we have that:

where $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ are constants. Therefore, from Eqs (36)-(39), we get that:

We note that the similarity transformations in Eq (40) are congruous with the conventional similarity transformations as described in Cheng and Chang ^[51] for $\text{λ}\text{ = 0}$. Thus, the expression of $u, v$ becomes:

3.2.   Similarity equations

Making substitution of the transformations in Eq (40) into the governing Eqs (16)-(18) gives us:

contingent on the boundary conditions

where, $b = {\left({D}_{1}\right)}_{0}/L$ is the thermal slip parameter.

4.   Physical quantities

In our present problem, the parameters of physical significance are the local Nusselt number $N{u}_{\overline{x}}$, local skin friction coefficient ${C}_{f\overline{x}}$, and the local Sherwood number $S{h}_{\overline{x}}$, respectively. Physically, ${C}_{f\overline{x}}$ is the shear stress of the wall, $N{u}_{\overline{x}}$ is the rate of transfer of heat in the wall, whilst $S{h}_{\overline{x}}$ symbolizes the rate of volume fraction of nanoparticles in the wall. The following relations are applied to obtain the aforementioned quantities:

Substituting Eqs (12) and (40) into Eq (46) shows that the physical quantities can be written in the following dimensionless form:

In the equation above, $R{a}_{\overline{x}} = \frac{gK\beta \left(1-{C}_{\infty }\right)\mathrm{\Delta }T\overline{x}}{{\alpha }_{m}v}$ is the local Rayleigh number, $pr = \frac{v}{{\alpha }_{m}}$is the Prandtl number, and ${U}_{r} = \frac{\left(1-{C}_{\infty }\right)gK\beta \mathrm{\Delta }T}{{\alpha }_{m}}$is the reference velocity.

5.   Numerical solutions

A two-point boundary value problem is formed by the set of coupled nonlinear ODEs in Eq (42) through Eq (44) with the boundary conditions in Eq (45). In ^[52], Aziz showed numerically using Runge-Kutta-Fehlberg's method in Maple that similarity solutions are possible for convective surface boundary conditions. Moreover, White and Subramanian ^[53] examined the above process's precision for different transport problems.

A finite value of 10 for the similar variable ${\eta }_{\text{max}}$ is used to replace the asymptotic boundary conditions given in Eq (43). The chosen value of ${\eta }_{\text{max}}$ assured the correct mannered approach of all numerical solutions to the far-field asymptotic values, which is often ignored. In ^[54], Pantokratoras studied convective heat transfer problems, and in his report, he found some erroneous results in the convective heat transfer problems. He identified the reason behind these errors as the graphs for the temperature, velocity, and the volume fraction (concentration) of nanoparticle distributions in the boundary layers do not tend to the accurate values in an asymptotic manner owing to the infinitesimal value of ${\eta }_{\text{max}}$.

Here we perform numerical computations for $0.1\le Nb\le 0.5, 0.1\le Nt\le 0.5, 0.1\le Nr\le 0.5, 1\le Le\le 10, -0.5\le Q\le 0.5, 0.5\le K\le 3, 0\le b\le 1, \mathrm{a}\mathrm{n}\mathrm{d}1\le n\le 3$. At the end of our discussion, we measured the wall shear stress, local heat transfer rate, and local mass transfer rate, respectively, with regard to the local skin friction coefficient, the reduced local Nusselt number $Nu$, and reduced local Sherwood number $Shr$. Finally, to evaluate our numerical computations' precision, we compared our results to that of Gorla and Chamkha ^[19]. Both sets of results are summarized in Table 1, and it is worth noting that we found our results to compare exceptionally well.

We note that when the case when the thermal slip parameter is ($b = 0$), buoyancy force ($Nr = 0$), thermophoresis ($Nt = 0$), heat generation ($Q = 0$), chemical reaction ($k = 0$), and Brownian motion ($Nb\to 0$), our undertaken problem transforms to the problem, which had been inspected by Cheng and Chang ^[50] for the case of $\text{λ}\text{ = 0}$ in their work.

6.   Results and discussion

Numerical solution to Eqs (42)-(44) with boundary conditions in Eq (45) is obtained by applying the Runge-Kutta-Fehlberg fourth-fifth order numerical method. We performed the computations using the software Maple 13, and our numerical results are displayed graphically for emphasizing the significant attributes of the flow characteristics.

In Figures 2 and 3, we have presented the velocity profiles within the boundary layer for both thermal slip parameter $b$, and no-thermal slip parameter ($b = 0)$ against several values of heat emission/absorption parameter Q, chemical reaction parameter $k$, and reaction order $n$, respectively.

Figure 2(a) shows that the velocity profiles inside the boundary layer become high; hence the boundary layer's thickness rises with the increase of heat generation/absorption parameter $Q$. As heat evolves, the buoyancy force increases, which in turn causes the flow rate to reduce, which triggers an increase in the velocity profile. However, with the intensification of heat absorption, the velocity is observed to decrease due to the buoyancy force's decrease. Figure 2(b) shows that the velocity profiles $f\text{'}\left(\eta \right)$ inside the boundary layer increase as the chemical reaction parameter $K$ increase, and the boundary layer's width also decreases. This is because physically, the fluid motion becomes thicker as $K$ increases. In both Figure 2(a), (b), we clearly see that an increase in the thermal slip parameter $b$ causes the velocity profiles $f\text{'}\left(\eta \right)$ to decrease. Therefore, the velocity profiles increase with the heat generation and order of chemical reaction, while they decrease when the thermal slip increases.

Figure 3 shows that the velocity profiles $f\text{'}\left(\eta \right)$ inside the boundary layer slightly expand, and consequently, the width of the boundary layer decrease as the order reaction parameter $n$ rises. As in Figure 2, we see that the velocity profiles decrease as the thermal slip thermal parameter $b$ increases.

In Figure 4(a), (b), we have presented the temperature profiles $\theta \left(\eta \right)$ within the boundary layer for both thermal slip parameter $b$ and no-thermal slip parameter ($b = 0)$ for several values of heat emission/absorption parameter $Q$, and order of reaction $n$, respectively. Figure 4(a) displays that the boundary layer's temperature profiles increase; hence, the boundary layer's thickness increases with the heat generation/absorption parameter. As heat generation ($Q > 0$), it is evident that there is a rise in the thermal state of the fluid. As a result, we notice that the temperature enhances as $Q$ attains larger values. However, when heat absorption ($Q < 0$), the exact converse happens, and this result agrees with previous work done by Alsaedi et al. ^[55]. Figure 4(b) shows that the boundary layer's temperature profiles increase; hence, the boundary layer's width extends as the order reaction parameter $n$ increases. As the thermal slip parameter becomes large, less heat is transferred from the surface to the fluid, and consequently, the temperature profiles decrease.

In Figure 5(a), (b), we have presented the nanoparticle volume fraction profiles $\varphi \left(\eta \right)$ within the boundary layer for various values of the order of reaction $n$, Lewis number $Le$, chemical reaction parameter $K,$ and the thermal slip parameter $b$, respectively. Figure 5(a) shows that the nanoparticle volume fraction profiles within the boundary layer increase; hence, the boundary layer's width becomes greater as the reaction order becomes large. This is because the Lewis number $Le$ is inversely proportional to the diffusion coefficient. Consequently, a rise in $Le$ yields a reduction in diffusion, which lastly results in a drop in nanoparticle concentration. Figure 5(b) shows that the nanoparticle volume fraction profiles within the boundary layer decrease; hence, the boundary layer's width enhances as the chemical reaction parameter rises. This result agrees with previous work by Abdul-Kahar et al. ^[35]. For both the "slip thermal" and "no-slip thermal" cases, the nanoparticle volume fraction profiles decrease monotonically as $K$ increases. These physical behaviors are demonstrated because of the coalesced effects of the Brownian motion and thermophoresis particle deposition strength.

Table 2 shows the effects of thermal slip, order of chemical reaction, reaction rate, and generation parameters on the dimensionless heat and mass transfer rates. In Table 2, we also that the rates of heat and mass transfer decrease as the thermal slip and order of chemical reaction parameters increase. Thus, the generation and reaction parameters reduce with the heat transfer rate whereas, the parameters increase as the mass transfer rate increase. In this respect, let us note that there is a significant effect on the flow field and accordingly on the heat transfer rate and nanoparticle volume fraction from the plate to the fluid. These effects are observed by the impact of thermophoresis particle deposition in the existence of the heat generation, order of chemical reaction, thermal slip, and chemical reaction parameters with Brownian motion. This demonstrates that all the previous parameters will significantly influence the heating and cooling processes in nano-fluids. This work has aided both scientists and engineers understand the most significant working principle of the deposition process.

A vast range of scientific and engineering applications are being extended in fluid mechanics, heat transfer, and nanoparticle volume fraction. Because of this wide-ranged scope, free convective flow through porous media became a great study area undergoing rapid growth in the aforesaid fields. Finally, we mention one promising scientific implementations of nanoparticles, which is the use of heat transfer fluids with the dispersion of nanoparticles to challenge cooling problems in thermal systems.

7.   Conclusions

Our present paper includes the numerical and physical study of the 2D steady free convective flow of a nano-fluid. The flow is assumed along with a horizontal plate that is faced upward and saturated in a porous medium. Our study considered thermal slip, chemical reaction, and heat emission/absorption under the Brownian motion and thermophoresis. Applying Lie group analysis, firstly, we observe the symmetries of the basic equations, following which we transform these equations to ODEs. We then worked with the resulting equations to obtain the numerical solution using Runge-Kutta-Fehlberg's fourth-fifth order numerical technique with Maple 13. Finally, to recapitulate our most prominent findings, we can write the following:

1) The velocity decreases as the thermal slip and heat absorption increase whilst it increases as the heat generation, order of chemical reaction, and reaction parameters increase.

2) The temperature decreases as the thermal slip and heat absorption increase, whilst it increases as the order of chemical reaction and heat generation parameters increase.

3) The concentration decreases as the thermal slip, reaction parameter, and Lewis number increase, whilst it increases as the order of the chemical reaction increase.

4) The heat and mass transfer rates reduce by increasing the thermal slip and the order of the chemical reaction parameters.

5) The generation and reaction parameters lead to the reduction of heat transfer rate, whilst they cause an increase in the mass transfer rate.

For further extensions of this paper, we can study various convective boundary conditions on the boundary layer free convection flow past a horizontal plate embedded in a porous medium filled by a nano-fluid containing different types of nanoparticles.

Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions. This work was supported and funded by Kuwait University Research Grant No. [SM02/20].

Conflict of interest

The authors declare no conflict of interest in this paper.