Hybrid Effects on MHD Mixed Convective Boundary Layer Flow through a Sloped Plate in Existence of Nanofluid-Saturated Porous Medium ()
1. Introduction
Due to the remarkable applications of nanofluid in extensive thoroughness of engineering and applied fields, including electronics, material processing, scientific measurement, computer technologies, high-power X-rays, optical devices, transportation, food, nuclear reactors, medicine, and synthesis, nanofluids have garnered a great deal of attention in research over the past ten years. Nanofluids have recently been explored as a means of improving thermal conductivity and comprehending their behavior, which will enable their use in numerous applications where improving straight heat transmission is crucial.
For mass, momentum, and heat transport in nanofluids, Buongiorno [1] has devised a two-component, four-equation nonhomogeneous equilibrium model. Abu-Nada [2] has studied heat transmission across a backward-facing stair using nanofluids. It has been found that high thermal conductivity nanoparticles exhibit greater increases on the Nusselt number outside of the circulation regions. However, low-thermal conductivity nanoparticles have superior heat transfer increases within the primary and secondary recirculation zones. Convection in a horizontal tier in porous material inclusive of nanofluid-saturated has been investigated via Nield and Kuznetsov [3]. In their investigation, they have taken thermophoresis and Brownian motion into account. In presence of nanofluids, they also discovered that there could be a substantial rise or fall in the crucial thermal Rayleigh number. Furthermore, The Cheng-Minkowycz scenario, which involves convective boundary tier flow across a perpendicular wall in a nanofluid-saturated medium with pores, has been analytically explored by Nield and Kuznetsov [4]. They offered a similar solution after predicting when nanofluid convection will begin using the Darcy model. Employing a variety of nanofluid types, including Cu, Al2O3, and TiO2, Ahmad and Pop [5] have analytically explored the diverse convective flow of boundary tier across the vertical plain wall inserted into a medium composed of pores that is filled using nanofluids. A computational analysis comprising diverse convection in an environment composed of pores occupied by Al2O3 and water has been carried out by Mittal et al. [6].
Results for several parameters were shown graphically and tabulated, and a discussion followed. Yasin et al. [7] have explored numerically the continuous diverse convective flow of boundary tier across a vertical surface into an environment composed of pores that is filled using nanofluids. In a water-based nanofluid, they employed dissimilar categories of metallic or nonmetallic nanoparticles, such as titania (TiO2), alumina (Al2O3), and copper (Cu) to examine the impact of the nanofluid’s solid volume fraction characteristic upon the heat transmission as well as flow properties. The temperature distributions, velocity, and skin friction coefficient were all shown and discussed afterward. They came to the conclusion that a major element in the improvement of heat transmission is the type of nanofluid. Gorla and Hossain [8] have looked into the impacts of Brownian motion as well as thermophoresis on a boundary layer analysis on an isothermal vertical cylinder within an environment composed of pores saturated by nanofluid involving mixed convection. In the boundary layer, they found that the thermophoresis parameters, buoyancy ratio, and Brownian motion had a significant impact on the rate of mass in conjunction with heat transfers.
In the context of an expanded annulus enclosure comprised of pore materials, Saleh [9] has computationally explored the natural convection process regarding nanofluid heat transfer involving a pair of horizontal concentric cylinders by employing the twelve annular fins of 2.4 mm depth connected with an inside cylinder according to constant-state circumstances. Copper (Cu) was used as the nanoparticle. Uddin and Harmand [10] have developed the Darcy-Forchheimer model to describe the irregular natural convective heat transfer flow of nanofluid over an isothermal vertical plate placed within an environment of pore medium. They discovered the fact that rather than the composition of the additional nanoparticles, the primary factor affecting heat transfer in nanofluids is the character underlying the base fluid. In environments where convection is fueled by two distinct density gradients with varying rates of diffusion, such as oceans and magma chambers, double-diffusion is a significant topic in fluid dynamics occurs. The implications of the considerable Soret reaction on mass transfer as well as double-diffusive natural convection heat in the boundary tier area associated with a semi-infinite sloped plane plate within a medium of pores that is non-Darcy and saturated with nanofluids have been addressed by Murthy et al. [11]. They found that greater Soret parameter values increase the mass transfer rates of heat and nanoparticles but decline the regular mass transfer rate at boundary.
Achieving balance or the desired control over the flow has been facilitated by a uniform distribution of the magnetic force in the opposite direction of the flow. In applications such as magnetic flow control or stabilization, it indicates that the magnetic force has successfully opposed the flow to produce stability or control. However, complex interactions between the magnetic forces and the fluid dynamics have been introduced by varying magnetic force distributions. This can impact the system’s overall efficacy or efficiency and require the use of more complex models to precisely forecast the flow behavior. The impacts of both viscous dissipation as well as the field of magnetic attraction on free convective mass and heat transport over a vertical plate within a medium of pores that is non-Darcy and saturated with nanofluids have been deliberated via Reddy et al. [12]. The authors found that while the rate of nanoparticle mass transfer and temperature in conjunction with velocity distributions were improved via increasing the viscous dissipation parameter, the heat transfer rate in association with distribution of fraction of nanoparticle volume was decreased. Furthermore, lower velocity distributions, mass and heat transfer rates for nanoparticles, in addition to higher volume fraction and temperature distributions for nanoparticles, are the outcomes of higher magnetic parameter values. The system relating the flow of mixed convective boundary layer over a sloped plate in a nanofluid-saturated pores medium filled with has been explained using double distinct analytical and numerical methods, the Chebyshev Pseudo-Spectral Differentiation Matrix and the Homotopy Perturbation Method, as studied by Aly and Ebaid [13]. Employing a range of values of the relevant physical features, the temperature and nanoparticle concentration distributions’ numerical results have been shown by them. Yazdi et al. [14] have explored the two-dimensional mixed convective magnetohydrodynamic boundary layer characterized by stagnation-point flow in the existence of heat radiation through a stretching vertical plate within an environment of porous medium containing a nanofluid.
As one gets farther away of the stagnation point, it is thought that the stretching velocity as well as surrounding fluid velocity varies linearly. With copper serving as the base fluid of the aiding as well as opposing flows, the findings have been presented for three distinct nanoparticle categories: copper, titania, and alumina. It has been studied by Srinivasacharya and Surender [15] how solutal together with thermal stratifications affect diverse convective boundary layer flow along a plate that is vertically placed within a porous medium containing nanofluid. They suggested that increasing the thermal as well as solutal stratification parameters causes the temperature to drop but the concentration to rise. Additionally, they provided a graphic representation of how mass transfer features and heat, in conjunction with velocity varied depending on distinct physical parameters. Employing on boundary layer approximations, El-Dawy et al. [16] have reported mixed convection via a vertical plate in a nanofluid-saturated medium of pores. The rate of surface heat transfers and friction factor in terms of numerical results are shown, along with the variations of the buoyancy ratio parameter, viscosity, and Prandtl number in conjunction with nanoparticle volume percentage.
Moreover, MHD flow involving mass transfer and nanofluid with heat-transfer characteristics across porous medium has been documented by Reddy and Chamkha [17]. For non-Newtonian fluid, Khan et al. [18] have reported, in conjunction with uniform fluid suction, the effect of thermal radiation as well as chemical reactions regarding the unsteady flow by way of a vertically stretching porous plate. They discovered that within the boundary tier, the temperature profiles are higher with respect to the heat source parameter as well as lower on account of the heat sink parameter. Within a substance with pores surrounding an accelerating vertically wavy plate, the influence of Soret regarding unsteady MHD diverse convective transfer of mass as well as heat flow has been explored via Prasad et al. [19], considering thermal radiation, chemical reaction, angle of inclination, heat generation, and Casson fluid. The diverse convective fluid flow of nanofluid across and through a medium with pores utilizing a sloped pore plate has been examined by Uddin et al. [20], reasoning that the magnetic field has unique impacts on viscous dissipation. This work has the potential to be expanded by utilizing the unique characteristics of thermophoretic velocity in addition to chemical reactions concerning concentration for various kinds of nanofluid flow. By Falodun et al. [21], the Cattaneo-Christov model has been utilized to explore the importance of viscous dissipation, thermal radiation, as well as heat generation for a hybrid MHD flow. Utilizing a constant heat source, radiation absorption, an inclined porous vertical plate positioned inside a media with pores, and changing circumstances at the boundary, Raghunath et al. [22] have investigated the chemical reaction as well as diffusion thermos consequences for the natural convective transfer of double diffusive flow of Jeffrey nanofluids.
As previously indicated, it makes more sense to look into how momentum features, together with mass as well as heat transfer, and the properties of nanoparticles, affect mixed convective flow through a vertically sloped porous plate within an environment of nanofluid-saturated medium of pores employing boundary-layer approximations. The significant geometries and the impacts described have not yet been investigated, to the greatest extent of the author’s understanding. This study has significant applications in various fields, including petroleum engineering, aerospace engineering, and the cooling of electronic equipment. Additionally, it plays a crucial role in enhancing nuclear reactor safety and has the potential to benefit many other areas as well.
2. Formulation of Mathematical Modeling
The study investigates the interaction of two-dimensional mixed convective flow with magnetohydrodynamics (MHD) passing through a porous plate sloped at an exquisite angle α relative to the vertical direction. More than that, the fluid is recognized a stable, viscous, and incompressible electrically conducting fluid. The physical parameters y and x are selected to be such a way that y is obtained perpendicular towards the sloped porous plate while x is obtained streamwise as of the edge that leads. Moreover, v and u, respectively, are the components of velocity that perpendicular and in the path of the flow. On the other hand, a field of magnetic B0 in a steady condition that is acted to the path of normal of flow and g is the force of gravity generates acceleration. An external flow characterized by a uniform velocity U∞ directed parallel to the sloped porous plate is present. Let
, C, and T indicate in terms of the fluid’s volume fraction of nanoparticles, concentration, and temperature, correspondingly.
The ambient temperature T∞ and concentration C∞ as well as the nanoparticle volume fraction
are less than those relating to the wall temperature Tw and concentration Cw as well as the nanoparticle volume fraction
, respectively. Figure 1 presents a schematic perspective of the coordinate system and flow setup below.
Figure 1. The Graphic perspective of coordinate system together with flow setup.
Consistent, two-dimensional, laminar boundary layer flow is deliberated as a result of the following equations utilizing the aforementioned characteristics:
(1)
(2)
(3)
(4)
(5)
within the aforementioned equations,
denotes the kinematic viscosity, k* denotes the permeability of the porous medium, σ represents the electrical conductivity, ρ designates density, and k indicates thermal conductivity of the fluid. For a comprehensive understanding of the governing equations, one may review the works of Buongiorno [1], Nield and Kuznetsov [4], and Murthy et al. [11].
Additionally, βC, βT, and βN are the volumetric coefficients of mass fraction, thermal expansion, and nanoparticle fraction, respectively. It is possible to express that the Brownian and thermophoretic diffusion coefficients are DB and DT, whereas the mass diffusivity is DM and the specific heat at constant pressure is Cp, the heat generation constant is Q0 and the thermophoretic velocity is well-defined by
(6)
wherein
is the coefficient of thermophoretic, which is calculated as follows by Talbot et al. [23] and
is some reference temperature as:
(7)
where kn represents the Knudsen number, kg and kp represent the fluid’s and the disseminated particle’s respective thermal conductivities, and Ct, Cs, Cm, C1, C2, and C3 are constants.
The following are the suitable boundary conditions for this consideration’s temperature, velocity, concentration, and volume fraction of nanoparticles:
(8)
(9)
Furthermore, the porous plate’s permeability is denoted by Vw(x), whose sign designates either blowing (>0) or suction (<0), and the free stream velocity is represented by U∞. The subscripts w as well as ∞, correspondingly, indicate the boundaries of the wall as well as the boundary layer. For the purpose of facilitating the analysis, the dimensionless components established by Cebeci et al. [24] are incorporated, so the differential equations that govern the analysis are rendered nondimensional through the appropriate transformations:
(10)
in which
is the stream function that is automatically satisfied by Equation (1) for continuity and is determined through:
(11)
These new variables allow us to express the velocity components as follows:
(12)
In this case, the prime denotes usual differentiation on the subject of similarity variable η. Corresponding modified equations for momentum, energy, concentration, and nanoparticle volume fraction, together with associated boundary situations, can be embodied as follows using dimensionless variables:
(13)
(14)
(15)
(16)
and under the boundary circumstances:
(17)
(18)
Here, however,
indicates wall injection but
designates wall suction, according to the wall mass transfer coefficient:
(19)
The definition of the associated dimensionless groups that show up in the governing equations’ nondimensional version consists of the following:
(20)
where the local Grashof numbers are Grt for local thermal Grashof, Grn for local nanoparticle Grashof, Grc for local mass Grashof, Re for local Reynolds, and Rit for local thermal Richardson, Ric for local mass, and Rin for local nanoparticle Richardson. The parameters that are shown are as follows: M stands for magnetic field, K for permeability, Ec for Eckert number, Q for heat generation, Pr for Prandtl number, and Sc for Schmidt number, τ for thermophoretic parameter, Scn for nanoparticle Schmidt number, then N for modified nanoparticle parameter.
Utilizing the wall shear stress definition
in conjunction with Fourier’s law
, Fick’s law
, and
, it is possible to determine the nondimensional forms of
as local skin-friction coefficient,
as local Nusselt number,
as local Sherwood number, and
as local nanoparticle Sherwood number as well as their corresponding nondimensional forms. However, the local Reynolds number stands for
.
3. Numerical Approach for Solution
The transferred governing equations can be solved using several different tactics. Utilizing the appropriate similarity transformations, non-linear equations of operation are first converted as contemporaneous ordinary differential equations. The boundary value problem is subsequently further converted as the initial value problem through the above equations employing Nachtsheim and Swigert’s shooting technique [25]. Utilizing the Runge-Kutta sixth-order approach, the resulting initial value problem is resolved.
The local skin-friction coefficient, local Nusselt number, and local general together with nanoparticle Sherwood numbers are computed from the numerical process of computation, and their numerical values are graphically presented. Additionally, the fields of velocity, concentration and nanoparticle volume fraction as well as the temperature are incorporated. The leading numerical tactic is reference to Nachtsheim and Swigert [25], which are included throughout the detailed description of the numerical approaches.
4. Validation of the Model
The current work, which does not use a nano-particle equation, is contrasted to the earlier released research by Reddy and Reddy [26] for the purpose of assessing the accuracy of the computational findings, as seen in Figure 2 below. The velocity distribution of Reddy and Reddy [26] and the current numerical results are shown to have a favorable correspondence in Figure 2. The next sections lead to show and analyze the numerical data as a result of this positive comparison.
Figure 2. Similarity among the variation in velocity employed in Rin = 0, Q = 0.5, K = 0.5, Rit = Grt = 2, Ric = Grm = 2, M = M1 = 0.5, M2 = 0, α = 300, fw = 0, Ec = 0, Pr = 0.71, and Sc = 0.6.
5. Findings and Discussions
A comprehensive collection of numerical outcomes derived from the solution of usual differential equations that are nonlinear is visually displayed. Diverse values associated with the corresponding physical parameters are used to obtain the numerical outcomes due to the distributions of local skin friction in conjunction with the velocity, local Nusselt number in conjunction with temperature, local Sherwood number in conjunction with concentration, and local nanoparticle Sherwood number in conjunction with nanoparticle volume fraction. Unless otherwise noted, Rit = 1, Ric = 1, Rin = 1, K = 0.01, U∞/ν = 1, M1 = M2 = 0.02, α = 30˚, Q = 0.5, Sc = 0.6, Pr = 0.71, fw = 1, Ec = 1, Scn = 0.6, N = 0.5 and τ = 0.1 are the set of deliberated values associated with corresponding numerous physical parameters. It is assumed both the corresponding numerical values for Schmidt number (Sc) in conjunction with Prandtl number (Pr) are 0.6 and 0.71, respectively, for water vapor (H2O) and air, respectively.
Keeping other flow field parameters constant, Figure 3 and Figure 4 show the effects of numerical values 0, 1, and 2 for the suction parameter fw on the velocity in addition to local skin friction coefficient, temperature in addition to local Nusselt number, concentration in addition to local Sherwood number, and nanoparticle distributions in addition to local Sherwood number of the nanoparticles. Figure 3(a) demonstrates that when fluid suction rises, the flow field’s velocity drops. This is a result of the plate cooling and the fluid’s viscosity rising by means of the fluid suction within the plate rises. As a result, when the fluid suction increases, the flow field’s velocity drops. Regarding the temperature variation shown within Figure 3(b), it is discovered when fluid suction is present, the flow field’s temperature drops as the plate cools because the flow field produces enhancing fluid suction within the plate. In a similar way, the concentration at all points falls as the fluid suction grows, as can be seen from the concentration variation through Figure 3(c). This happens because increased fluid suction causes the plate to cool.
As the fluid suction is increased, the nanoparticle volume fraction increases at
because the difference in volume fraction between the plate and outer edge decreases, and it decreases at
because the wall slope with respect to associated nanoparticle volume fraction increases slightly. This is seen in Figure 3(d). The local skin friction coefficient in addition to the local Nusselt number and the general in addition to nanoparticles local Sherwood numbers are revealed versus the streamwise distance x in Figure 4. By means of the fluid suction grows, so too does the local skin friction coefficient Cf, which raises that of the flowing fluid’s viscosity as seen in Figure 4(a). However, when the fluid suction parameter rises, the local Nusselt and Sherwood numbers increase as well because of a rise in temperature and concentration variances, correspondingly. These observations can be seen in Figure 4(b) and Figure 4(c), while Figure 4(d) displays how the local nanoparticle Sherwood number declines.
In order to facilitate the implications of Schmidt number, the corresponding values of Sc are taken as 0.94, 0.60, and 0.22 for carbon dioxide (CO2), water vapor (H2O), and hydrogen (H2), respectively, while maintaining similar values for any other flow field elements. The data laid out in Figure 5 and Figure 6 demonstrate the consequences of varying Schmidt numbers Sc on various parameters such as
Figure 3. Illustration of (a) velocity, (b) temperature, (c) concentration, and (d) nanoparticle volume fraction variation based on various fluid suction parameter values fw.
Figure 4. Influence of the fluid suction parameter fw upon the (a) local skin friction coefficient Cf , (b) local Nusselt number Nu, (c) local Sherwood number Sh, and (d) nanoparticle Sherwood number Shn as a function of the streamwise distance x.
Figure 5. Illustration of (a) velocity, (b) temperature, (c) concentration, and (d) nanoparticle volume fraction variation based on various Schmidt number values Sc.
Figure 6. Influence of Schmidt number Sc upon the (a) local skin friction coefficient Cf, (b) local Nusselt number Nu, (c) local Sherwood number Sh, and (d) nanoparticle Sherwood number Shn as a function of the streamwise distance x.
velocity in addition to local skin friction coefficient, temperature in addition to local Nusselt number, concentration in addition to local Sherwood number, and nanoparticle variation in addition to local nanoparticle Sherwood number in regard to the streamwise distance x. The Schmidt number is a ratio of convenient expression for the momentum to diffusivity of mass. Consequently, the Schmidt number is used to quantify the corresponding effectiveness for momentum and mass transmission through diffusion in hydrodynamic and concentration boundary tiers. For the reason of the existence of heavier diffusing species, Figure 5(a) demonstrates the manner in which the flow field’s velocity drops while the Schmidt number rises. On the contrary, in comparison to the flow field’s velocity, the temperature variation—shown in Figure 5(b)—varies insignificantly. It is shown that when the diffusing species gets heavier, the flow field’s concentration variation decreases as Figure 5(c), resulting in a reduction in the flow field’s concentration at the concentration boundary tier. However, in Figure 5(d), as the Schmidt number rises, the nanoparticle volume fraction does as well. The local skin friction coefficient, local Nusselt in conjunction with Sherwood numbers, and local nanoparticle Sherwood number compared with as a function of the streamwise distance x are illustrated throughout Figure 6 to explore the implications of the Schmidt number on the flow field. Figure 6(a) illustrates the manner in which as predicted, an increase in the Schmidt number leads to decline in the local skin friction coefficient. Additionally, Figure 6(b) demonstrates that the local Nusselt number falls with a rise in the overall Schmidt number. However, as anticipated by Figure 6(c), the local Sherwood number rises in the case of an elevated Schmidt number, in contrast to Figure 6(d), the local nanoparticle Sherwood number drops as the Schmidt number rises.
6. Conclusions
This work adopts an evenly distributed magnetic field to explore hybrid effects employing varied convective flow via an inclined vertical porous plate in the presence of a porous medium. The results demonstrate that the fluid suction parameter and Schmidt number have an impact on the flow field. When the fluid suction is increased, the flow field’s velocity, temperature, and concentration all decrease; on the other hand, the nanoparticle volume fraction decreases at
and increases at
. On the contrary, the general local Sherwood number rises while the local nanoparticle Sherwood number falls when the fluid suction component rises, whereas the local skin friction coefficient as well as the local Nusselt number all rise. The volume fraction of nanoparticles grows with increasing Schmidt number, despite the fact that the temperature, concentration, and velocity distributions of the flow field fall. With increasing Schmidt numbers, the local Sherwood number and skin friction coefficient rise, whereas the local Sherwood number for nanoparticles and the local Nusselt number decline.
The outcomes obtained from this investigation, in conjunction with data from physical science, will be beneficial to the engineering community, scholars, and experimental scientists in discovering how nanofluid circulation operates in complex geometries. Additional investigations may be performed on broadening the features of nanofluid circulation over higher-level geometry, considering the current study has an extensive variety of significant implications. By considering the results of the present investigation, the drawbacks involving the present work can be explored for potential future studies in fields including three-dimensional nature flow, inconstant permeability in a medium of pores, and instability in conjunction with turbulent flow.