Vibrational Radiative Double Diffusion in Buongiorno’s Nanofluid Model Within Inclined Chambers Filled with Non-Darcy Porous Elements

Abstract

Vibrational double diffusion has gained increasing attention in recent studies due to its role in enhancing mixing, disrupting thermal boundary layers, and stabilizing convection structures, especially in nanofluids and porous media. This study focuses on the case of two-phase nanofluid flow in the presence of vibrational effects. The flow domain is a fined chamber that is filled with a non-Darcy porous medium. Two concentration formulations are proposed for the species concentration and nanoparticle concentration. The thermal radiation is in both the x- and y-directions, while the flow domain is considered to be inclined. The solution technique depends on an effective finite volume method. The periodic behaviors of the stream function, Nusselt numbers, and Sherwood numbers against the progressing time are presented and interpreted. From the major results, a significant reduction in harmonic behaviors of the stream function is obtained as the lengths of the fins are raised while the gradients of the temperature and concentration are improved. Also, a higher rate of heat and mass transfer is obtained when the vibration frequency is raised. Furthermore, for fixed values of the Rayleigh number and vibration frequency (Ra = 10⁴, σ = 500), the heat transfer coefficient improves by 27.2% as the fin length increases from 0.1 to 0.25.

1. Introduction

Nanofluids are novel fluids whose properties have been enhanced using modern techniques, such as dispersing nanoparticles in a base fluid. These properties have attracted significant attention in recent years and offer promising applications in various industrial and engineering applications, including heat exchangers, biomedical systems, and electronics cooling. Among the different nanofluid models, Buongiorno’s model is based on the effects of Brownian motion and thermophoresis. These effects play a crucial role in determining heat and mass transfer characteristics. On the other side, the study of heat and mass transfer in nanofluids becomes more complex when considering additional transport mechanisms such as vibrational effects, radiative heat transfer, and double-diffusion processes. In fact, vibrational effects can significantly influence fluid flow and heat transfer in systems subjected to external oscillations or vibrations, such as in aerospace and microscale applications. Additionally, radiative heat transfer is essential in high-temperature environments where thermal radiation contributes substantially to the overall energy exchange. Furthermore, double diffusion, driven by simultaneous temperature and concentration gradients, is critical in understanding buoyancy-driven flows in nanofluids [1,2,3,4,5]. In recent times, the study of double-diffusive convection in nanofluids within inclined porous chambers has garnered significant attention due to its critical applications in various engineering and industrial fields. Shoaib et al. [6] reported numerically the double-diffusive free convection flow of nanofluids over an inclined plate within a porous medium, incorporating thermophoresis and Brownian motion effects. They employed an AI-based Levenberg–Marquardt scheme with a backpropagated neural network to analyze the flow dynamics, revealing that the inclusion of nanoparticles significantly alters the heat and mass transfer rates. Akram et al. [7] presented a work on the double-diffusive convection of magneto-fourth-grade nanofluids with peristaltic propulsion through an inclined asymmetric channel. Their findings indicated that the application of an inclined magnetic field and the consideration of double-diffusivity effects play a crucial role in the modulation of heat and mass transfer rates in such complex fluid systems. Further, the influence of variable viscosity and double diffusion on the convective stability of nanofluid flow in an inclined porous channel was considered by researchers in [8]. Utilizing the Darcy–Brinkman model, they observed that variations in viscosity and double-diffusive effects significantly impact the stability and transition of flow regimes within the porous channel. More recently, a numerical simulation of double-diffusive mixed convective nanofluid flow in a square porous lid-driven cavity was explored by researchers in [9]. They established that the interaction between thermal and solutal buoyancy forces, along with the cavity’s inclination, markedly influences the heat and mass transfer performance, providing insights into optimizing such systems for enhanced efficiency.

When analyzing these transport mechanisms within an inclined enclosure, the inclination angle alters the natural convection patterns, affecting heat and mass transfer rates. Several studies have investigated the impact of inclination on convection in enclosures, highlighting its importance in optimizing thermal performance in engineering applications [10]. Furthermore, the presence of a porous medium significantly modifies the flow dynamics by introducing resistance to fluid motion. The non-Darcy porous medium, which accounts for inertial and boundary effects, provides a more accurate representation of real-world porous structures compared to classical Darcy models [11]. Additionally, Asmadi et al. [12] conducted a numerical investigation into the buoyancy-driven heat transfer of a hybrid nanofluid confined in a tilted U-shaped cavity. Their findings indicated that the heat transfer performance is optimized at inclination angles between 40° and 60°, highlighting the critical role of cavity orientation in thermal management. Similarly, Anuar et al. [13] explored the boundary layer flow and heat transfer of a hybrid Ag-MgO/water nanofluid over an inclined permeable stretching/shrinking surface. Their study revealed that increasing the inclination angle and suction parameters enhances the local Nusselt number, emphasizing the influence of inclination on convective heat transfer rates. Furthermore, Mohamed [14] investigated natural convection within an inclined porous cavity using a non-Darcian flow model. The research demonstrated that the initiation of multicellular flow and counter-rotating cells is strongly dependent on the aspect ratio and inclination angle, underscoring the importance of geometric configuration in heat transfer processes. Luo and Yang [15] present an analytical study of fluid flow and heat transfer within a closed cavity whose top and bottom lids move in opposite directions. These lids may also have different temperatures, creating a thermal gradient across the cavity. Luo and Yang [16] present both numerical and experimental studies on electrokinetic instability (EKI) to achieve the mixing of multiple solutions with varying electrical conductivities in a cross-shaped microchannel. Chen et al. [17] investigated how varying the aspect ratio (width-to-height) of a two-sided lid-driven cavity affects the multiplicity of steady-state flow solutions. In this configuration, the top and bottom lids of the cavity move in opposite directions, creating complex internal flow patterns. Lou et al. [18] investigated how hydrodynamic flow affects thermal management in flexible heatsink devices, specifically those with out-of-plane complex microchannel designs. Hafed et al. [19] studied the bioconvective blood flow characteristics of tetra composition nanofluids as they flow through a stenotic artery. Their investigation focused on the effects of Arrhenius energy, which relates to the activation energy needed for chemical reactions in the system. Ahmed et al. [20] studied the role of two isothermal cylinders in the three-dimensional flow and melting process of phase-change materials (PCMs). Most likely, the researchers investigated the effect of these cylinders as heat sources on the melting behavior of these materials, focusing on the three-dimensional flow patterns that form during the melting process. Hafed et al. [21] conducted a study analyzing the behavior of a viscoelastic nanofluid flowing over a stretching surface, taking into account several physical effects such as Arrhenius activation energy, viscous dissipation, temperature-dependent fluid properties, and slip velocity at the boundary surface.

The inpolygon function in MATLAB 2023 is a computational geometry tool used to determine whether a set of points lies inside or on the boundary of a specified polygon. It is particularly useful in numerical simulations for defining complex geometries or distinguishing between internal and external regions of a domain. A recent study by Ahmed et al. [22] demonstrated the application of this technique in analyzing shear-driven flow within a highly complex wavy domain. Building upon these foundational works, the present study offers a novel contribution to the field of convective transport in porous media by investigating two-phase nanofluid flow under vibrational effects within a finned, inclined chamber saturated with a non-Darcy porous medium. Unlike previous studies, this work incorporates dual concentration formulations to separately account for species and nanoparticle concentrations, as well as bidirectional thermal radiation, enhancing the physical accuracy of the model. An effective finite volume method is employed to capture the transient and periodic behaviors of key transport quantities. Notably, this study demonstrates that increasing fin length and vibration frequency can significantly improve the gradients of temperature and concentration, leading to enhanced heat and mass transfer. The solution methodology is based on identifying the inner and outer boundaries using the inpolygon function and the finite volume method. This studied case was chosen to address the complex interplay between vibrational effects, porous media, and nanofluid behavior, which is relevant to a range of engineering and energy-related applications. Specifically, finned chambers filled with nanofluids in porous environments are widely used in thermal management systems such as heat exchangers, nuclear reactors, solar collectors, and electronic cooling devices. The addition of vibrational effects can further enhance mixing and heat transfer, making the system more efficient under dynamic conditions. During this investigation, we hope to present an answer to the following questions:

• How does the inclination angle of the enclosure affect the convective heat transfer and nanoparticle migration in the presence of vibrational forces?
• What are the impacts of radiative heat transfer on the thermal boundary layers and nanoparticle distribution in the nanofluid?
• How does the non-Darcy porous medium influence the overall heat transfer rate compared to a Darcy-based model?
• What is the effect of varying the Brownian motion and thermophoresis parameters on the double-diffusive convection within the enclosure?
• How do different vibration frequencies and amplitudes alter the flow and thermal fields within the nanofluid system?
• How does the combined effect of these factors contribute to the optimization of heat transfer performance in practical engineering applications such as energy systems, thermal insulation, and microfluidic devices?

2. Assumption and Governing System

The time-dependent nanofluid flow within an irregular chamber was analyzed, as shown in Figure 1. The following assumptions were made:

• All outer boundaries are thermally insulated.
• Two heated fins, each with a length of $H_1$ and a width of 0.2, are attached to the lower boundaries at $x=0.5 \mathrm{L}$ and $x=1.2 \mathrm{L}$.
• Two cold fins, also with a length of $H_1$ and a width of 0.2, are attached to the upper boundaries at x = 0.5 L and x = 1.2 L.
• The flow domain is inclined at an angle $\xi$.
• A vibration source is modeled as $g\left[1+\lambda \sin \left({\sigma }^{’}t\right)\right]$.
• The flow region is filled with a non-Darcy porous medium.
• Two concentrations—one for species and one for nanoparticles—are considered, with Buongiorno’s nanofluid model applied.
• Thermal radiation is accounted for in both the x- and y-directions.
• The flow is assumed to be two-dimensional and laminar.

When all the aforementioned assumptions are recognized, the governing system is expressed as [14,23]:

$${\displaystyle \frac{𝜕u}{𝜕x}}+{\displaystyle \frac{𝜕v}{𝜕y}}=0$$

(1)

$$\begin{array}{c}{\rho }_f\left[{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{𝜕u}{𝜕t}}+{\displaystyle \frac{1}{{\epsilon }^2}}\left[u{\displaystyle \frac{𝜕u}{𝜕x}}+v{\displaystyle \frac{𝜕u}{𝜕y}}\right]\right]\hfill \\ =-{\displaystyle \frac{𝜕P}{𝜕x}}+{\displaystyle \frac{{\mu }_{eff}}{ϵ}}\left({\displaystyle \frac{{𝜕}^{2}u}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^2 u}{𝜕y^2}}\right)-{\displaystyle \frac{{\mu }_f}{K}}u-{\displaystyle \frac{{\rho }_f{C}_F}{\sqrt{K}}}\sqrt{u^2+v^2} u\hfill \\ +[\left(1-{\phi }_0\right){\left(\rho {\beta }_T\right)}_f g \left(T-T_c\right)sin\xi +\left(1-{\phi }_0\right){\left(\rho {\beta }_C\right)}_{f}g \left(C^{’}-C_c\right)sin\xi \hfill \\ +\left({\rho }_p-{\rho }_f\right)\left(\phi -{\phi }_0\right)g sin\xi ]\left[1+\lambda \sin \left({\sigma }^{’}t\right)\right]\hfill \end{array}$$

(2)

$$\begin{array}{c}{\rho }_f\left[{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{𝜕v}{𝜕t}}+{\displaystyle \frac{1}{{\epsilon }^2}}\left[u{\displaystyle \frac{𝜕v}{𝜕x}}+v{\displaystyle \frac{𝜕v}{𝜕y}}\right]\right]\hfill \\ =-{\displaystyle \frac{𝜕P}{𝜕y}}+{\displaystyle \frac{{\mu }_{eff}}{ϵ}}\left({\displaystyle \frac{{𝜕}^{2}v}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^2 v}{𝜕y^2}}\right)-{\displaystyle \frac{{\mu }_f}{K}}v-{\displaystyle \frac{{{\rho }_{f}C}_F}{\sqrt{K}}}\sqrt{u^2+v^2} v\hfill \\ +[\left(1-{\phi }_0\right){\left(\rho {\beta }_T\right)}_f g \left(T-T_c\right)cos\xi +\left(1-{\phi }_0\right){\left(\rho {\beta }_C\right)}_{f}g \left(C^{’}-C_c\right)cos\xi \hfill \\ +\left({\rho }_p-{\rho }_f\right)\left(\phi -{\phi }_0\right)g cos\xi \left[1+\lambda \sin \left({\sigma }^{’}t\right)\right]\hfill \end{array}$$

(3)

$$\begin{array}{c}{\left[\left(1-\epsilon \right){\left(\rho c_p\right)}_p+\epsilon {\left(\rho c_p\right)}_f\right]{\displaystyle \frac{𝜕T}{𝜕t}}+\left(\rho C_p\right)}_f\left(u{\displaystyle \frac{𝜕T}{𝜕x}}+v{\displaystyle \frac{𝜕T}{𝜕y}}\right)\hfill \\ ={\alpha }_f\left({\displaystyle \frac{ {𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{ {𝜕}^{2}T}{𝜕y^2}}\right)+\epsilon {\left({\rho C}_p\right)}_p\left[D_B\left({\displaystyle \frac{𝜕\phi }{𝜕x}} {\displaystyle \frac{𝜕T}{𝜕x}}+{\displaystyle \frac{𝜕\phi }{𝜕y}} {\displaystyle \frac{𝜕T}{𝜕y}}\right)+{\displaystyle \frac{D_T}{T_c}}\left({\left({\displaystyle \frac{𝜕T}{𝜕X}}\right)}^2+{\left({\displaystyle \frac{𝜕T}{𝜕y}}\right)}^2\right)\right]\hfill \\ -\left({\displaystyle \frac{𝜕q_{rx}}{𝜕x}}+{\displaystyle \frac{𝜕q_{ry}}{𝜕y}}\right)\hfill \end{array}$$

(4)

$$\epsilon {\displaystyle \frac{𝜕C’}{𝜕t}}+u{\displaystyle \frac{𝜕C’}{𝜕x}}+v{\displaystyle \frac{𝜕C’}{𝜕y}}=D_f\left({\displaystyle \frac{ {𝜕}^{2}C’}{𝜕x^2}}+{\displaystyle \frac{ {𝜕}^{2}C’}{𝜕y^2}}\right)$$

(5)

$$\epsilon {\displaystyle \frac{𝜕\phi }{𝜕t}}+\mathrm{u}{\displaystyle \frac{𝜕\phi }{𝜕x}}+v{\displaystyle \frac{𝜕\phi }{𝜕y}}=D_B\left({\displaystyle \frac{ {𝜕}^2\phi }{𝜕x^2}}+{\displaystyle \frac{ {𝜕}^2\phi }{𝜕y^2}}\right)+{\displaystyle \frac{D_T}{T_c}}\left({\displaystyle \frac{ {𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{ {𝜕}^{2}T}{𝜕y^2}}\right)$$

(6)

In the previous system, the following symbols are defined as:

• ${\rho }_f$: fluid density.
• $\epsilon$: porosity of the medium that can be defined as the fraction of void space in the porous material.
• $u,v$: velocity components in the x-and y- directions, respectively.
• $P$: pressure.
• ${\mu }_{eff}$: effective viscosity, accounting for both fluid and porous medium effects.
• ${\mu }_f$: dynamic viscosity of the fluid.
• $K$: permeability of the porous medium that measures how easily fluid flows through the medium.
• $C_F$: Forchheimer coefficient that represents non-linear drag effects in high-velocity flows.
• ${\beta }_T, {\beta }_C$: thermal and solutal expansion coefficients, respectively; those refer to the effect of temperature and concentration variations on fluid density.
• $g$: acceleration due to gravity.
• $T, C’:$ temperature and concentration of the fluid.
• $\phi , {\phi }_0$: nanoparticle volume fraction and its reference value.
• ${\rho }_p, {\rho }_f$: densities of nanoparticles and fluid, respectively.
• $\lambda , {\sigma }^{’}$: parameters related to time-dependent oscillations.
• $\xi$: inclination angle.

We propose the following dimensionless quantities, along with approximations for the radiation term:

$$\begin{array}{c}X={\displaystyle \frac{x}{L}}, Y={\displaystyle \frac{y}{L}}, U={\displaystyle \frac{uL}{{\alpha }_f}}, V={\displaystyle \frac{vL}{{\alpha }_f}},\tau ={\displaystyle \frac{t{\alpha }_f}{L^2}}\\ \sigma ={\displaystyle \frac{{\sigma }^{’}{L}^2}{{\alpha }_f}}, \theta ={\displaystyle \frac{T-T_0}{∆T}},C={\displaystyle \frac{C^{’}-C_0^{’}}{∆C}},\mathsf{\Phi }={\displaystyle \frac{\phi -{\phi }_0}{∆\phi }},P={\displaystyle \frac{p{L}^2}{{\rho }_{f }{\alpha }_f^2 }},q_{rx}=-{\displaystyle \frac{4{\sigma }^{*}}{3k^{*}}}{\displaystyle \frac{𝜕T^4}{𝜕x}}, q_{ry}=-{\displaystyle \frac{4{\sigma }^{*}}{3k^{*}}}{\displaystyle \frac{𝜕T^4}{𝜕y}} ,T^4\\ \cong 4T_c^{3}T-3T_c^4,q_{rx}=-{\displaystyle \frac{16{\sigma }^{*}{T}_0^3}{3k^{*}}}{\displaystyle \frac{𝜕T}{𝜕x}},q_{ry}=-{\displaystyle \frac{16{\sigma }^{*}{T}_0^3}{3k^{*}}}{\displaystyle \frac{𝜕T}{𝜕y}}.\end{array}$$

Here, it should be mentioned that the Rosseland approximation is applied for the radiation term. This formula describes radiative heat transfer in an optically thick medium, where radiation is treated as a diffusion process. By employing the above expressions, the governing system becomes in the form:

$${\displaystyle \frac{ 𝜕U}{𝜕X}}+{\displaystyle \frac{𝜕V}{𝜕Y}}=0$$

(7)

$$\begin{array}{c}{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{𝜕U}{𝜕\tau }}+{\displaystyle \frac{1}{{\epsilon }^2}}\left[U{\displaystyle \frac{𝜕U}{𝜕X}}+V{\displaystyle \frac{𝜕U}{𝜕Y}}\right]\hfill \\ =-{\displaystyle \frac{𝜕P}{𝜕X}}+{\displaystyle \frac{Pr}{ϵ}}\left({\displaystyle \frac{{𝜕}^{2}U}{𝜕X^2}}+{\displaystyle \frac{{𝜕}^2 U}{𝜕Y^2}}\right)-{\displaystyle \frac{Pr}{Da}}U-{\displaystyle \frac{C_F}{\sqrt{Da}}}\sqrt{U^2+V^2} U\hfill \\ +PrRa\left[\theta sin\xi +N C sin\xi -\mathrm{N}\mathrm{r}\mathsf{\Phi } sin\xi \right]\left[1+\lambda \sin \left({\sigma }^{’}t\right)\right]\hfill \end{array}$$

(8)

$$\begin{array}{c}{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{𝜕V}{𝜕\tau }}+{\displaystyle \frac{1}{{\epsilon }^2}}\left[U{\displaystyle \frac{𝜕V}{𝜕X}}+V{\displaystyle \frac{𝜕V}{𝜕Y}}\right]\hfill \\ =-{\displaystyle \frac{𝜕P}{𝜕Y}}+{\displaystyle \frac{Pr}{ϵ}}\left({\displaystyle \frac{{𝜕}^{2}V}{𝜕X^2}}+{\displaystyle \frac{{𝜕}^{2}V}{𝜕Y^2}}\right)-{\displaystyle \frac{Pr}{Da}}V-{\displaystyle \frac{C_F}{\sqrt{Da}}}\sqrt{U^2+V^2} V\hfill \\ +PrRa\left[ \theta cos\xi +N Ccos\xi -Nr \mathsf{\Phi } cos\xi \right]\left[1+\lambda \sin \left({\sigma }^{’}t\right)\right]\hfill \end{array}$$

(9)

$$\begin{array}{c}\gamma {\displaystyle \frac{𝜕\theta }{𝜕\tau }}+\left(U{\displaystyle \frac{𝜕\theta }{𝜕X}}+V{\displaystyle \frac{𝜕\theta }{𝜕Y}}\right)\hfill \\ =\left({\displaystyle \frac{ {𝜕}^2\theta }{𝜕X^2}}+{\displaystyle \frac{ {𝜕}^2\theta }{𝜕Y^2}}\right)+\epsilon \left[Nb\left({\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕X}} {\displaystyle \frac{𝜕\theta }{𝜕X}}+{\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕Y}} {\displaystyle \frac{𝜕\theta }{𝜕Y}}\right)+Nt\left({\left({\displaystyle \frac{𝜕\theta }{𝜕X}}\right)}^2+{\left({\displaystyle \frac{𝜕\theta }{𝜕Y}}\right)}^2\right)\right]+R\left({\displaystyle \frac{ {𝜕}^2\theta }{𝜕X^2}}+{\displaystyle \frac{ {𝜕}^2\theta }{𝜕Y^2}}\right)\hfill \end{array}$$

(10)

$$\epsilon {\displaystyle \frac{𝜕C}{𝜕\tau }}+U{\displaystyle \frac{𝜕C}{𝜕X}}+V{\displaystyle \frac{𝜕C}{𝜕Y}}={\displaystyle \frac{1}{Lp}}\left({\displaystyle \frac{ {𝜕}^{2}C}{𝜕X^2}}+{\displaystyle \frac{ {𝜕}^{2}C}{𝜕Y^2}}\right)$$

(11)

$$\epsilon {\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕\tau }}+\mathrm{U}{\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕X}}+V{\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕Y}}={\displaystyle \frac{1}{Le}}\left({\displaystyle \frac{ {𝜕}^2\mathsf{\Phi }}{𝜕X^2}}+{\displaystyle \frac{ {𝜕}^2\mathsf{\Phi }}{𝜕Y^2}}\right)+{\displaystyle \frac{Nt}{Nb Le}}\left({\displaystyle \frac{ {𝜕}^2\theta }{𝜕X^2}}+{\displaystyle \frac{ {𝜕}^2\theta }{𝜕Y^2}}\right)$$

(12)

The non-dimensional conditions are given as:

$$U=V=0,\theta =1,C=1,Nb\left({\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕x}}\right)+Nt{\displaystyle \frac{𝜕\theta }{𝜕X}}=0:\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{s}.$$

$$U=V=0, \theta =0, C=0 ,\mathsf{\Phi }=0 \mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{s}.$$

(13)

$$U=V=0,{\displaystyle \frac{𝜕\theta }{𝜕n}}=0,{\displaystyle \frac{𝜕C}{𝜕n}}=0,{\displaystyle \frac{𝜕\mathsf{\Phi }}{𝜕n}}=0\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}.$$

The dimensionless parameters in the previous system are proposed as follows:

$$\begin{array}{c}Pr={\displaystyle \frac{{\nu }_f}{{\alpha }_f}} ,Da={\displaystyle \frac{K}{L^2}}, Ra={\displaystyle \frac{g∆T{L}^3\left(1-{\phi }_0\right){\beta }_T}{{\nu }_f {\alpha }_f}}, N={\displaystyle \frac{{\left(\rho {\beta }_C\right)}_f }{{\left(\rho {\beta }_T\right)}_f}}{\displaystyle \frac{ ∆C}{ ∆T}},\hfill \\ Nr={\displaystyle \frac{\left({\rho }_p-{\rho }_f\right) }{\left(1-{\phi }_0\right){\left(\rho {\beta }_T\right)}_f}}{\displaystyle \frac{ ∆\phi }{ ∆T}} {,\alpha }_f={\displaystyle \frac{k_f}{{\left(\rho C_p\right)}_f}}, \gamma ={\displaystyle \frac{\left[\left(1-\epsilon \right){\left(\rho c_p\right)}_p+\epsilon {\left(\rho c_p\right)}_f\right]}{{\left(\rho c_p\right)}_f}}, Nb\hfill \\ ={\displaystyle \frac{{\left({\rho C}_p\right)}_p{D}_B ∆\phi }{{\left(\rho c_p\right)}_f {\alpha }_f}}, Nt={\displaystyle \frac{{\left({\rho C}_p\right)}_p{D}_T ∆T}{{\left(\rho c_p\right)}_f T_0 {\alpha }_f}}, R={\displaystyle \frac{16{\sigma }^{*}{T}_c^3}{3k^{*}{k}_f}},Lp={\displaystyle \frac{{\alpha }_f}{D_f}} ,Le={\displaystyle \frac{{\alpha }_f}{D_B}}.\hfill \end{array}$$

(14)

These parameters are the Prandtl number, Darcy number, Rayleigh number, buoyancy ratio, nanoparticle buoyancy parameter, thermal diffusivity of fluid, effective heat capacity ratio, Brownian motion parameter, thermophoresis parameter, radiation parameter, porosity-modified thermal diffusivity, and Lewis number.

The local Nusselt number and the local Sherwood number vectors can be specified as follows:

$${Nu}_{loc}=-\left(R+1\right){\displaystyle \frac{𝜕\theta }{𝜕n}}$$

(15)

$${Sh}_{loc}=-{\displaystyle \frac{𝜕C}{𝜕n}}$$

(16)

$$\begin{gathered} \mathrm{The\; mean\; values\; are}: \\ {Nu}_{avg}=-{\int }_0^L{\displaystyle \frac{\left(R+1\right)}{L}}{\displaystyle \frac{𝜕\theta }{𝜕n}}dL \end{gathered}$$

(17)

$${Sh}_{avg}=-{\int }_0^L{\displaystyle \frac{1}{L}}{\displaystyle \frac{𝜕C}{𝜕n}} dL$$

(18)

$$\begin{gathered} \mathrm{The\; time-averaged\; heat\; transfer\; rates\; are\; computed\; as}: \\ {Nu}_{time,a}=-{\displaystyle \frac{\left(1+R\right)}{L}}{\displaystyle \frac{1}{t_{max}}}{\int }_0^{t_{max}}\left({\int }_0^L{\displaystyle \frac{𝜕\theta }{𝜕n}}dL\right)dt \end{gathered}$$

(19)

$${Sh}_{time,a}=-{\displaystyle \frac{1}{L}}{\displaystyle \frac{1}{t_{max}}}{\int }_0^{t_{max}}\left({\int }_0^L{\displaystyle \frac{𝜕C}{𝜕n}}dL\right)dt$$

(20)

3. Numerical Methods

The current technique applies the FVM to solve the governing system that is based on SIMPLE algorithm. The finite volume method (FVM) was selected due to its strong conservation properties, which are essential for accurately capturing mass, momentum, and energy balances in flow through porous media. This local conservation makes it particularly suited for complex domains with variable porosity and boundary interactions. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm was chosen to solve the pressure–velocity coupling efficiently. This approach iteratively corrects pressure and velocity fields to satisfy the continuity equation. A simple analogy is adjusting a plumbing system: If pressure at one point changes, flow elsewhere must be adjusted to maintain balance throughout the network. SIMPLE is widely used in incompressible flow simulations for its robustness and efficiency.

In this method, the convective term in a general transport equation is rewritten in the following general form:

$${\displaystyle \frac{𝜕\left(\mathrm{U}\mathsf{\Omega }\right)}{𝜕\mathrm{X}}}+{\displaystyle \frac{𝜕\left(\mathrm{U}\mathsf{\Omega }\right)}{𝜕\mathrm{Y}}}=0$$

(21)

Applying the divergence theorem in the integral form over a control volume (CV) is given as follows:

$${\int }_V\nabla ·\left(\mathit{U}\mathsf{\Omega }\right)dV={\oint }_A\left(\mathit{U}\mathsf{\Omega }\right).\mathrm{d}\mathrm{A}$$

(22)

Approximating over a Cartesian CV gives the following:

$${\displaystyle \frac{1}{\mathsf{\Delta }X}}\left[u_e{\mathsf{\Omega }}_e-u_w{\mathsf{\Omega }}_w\right]+{\displaystyle \frac{1}{\mathsf{\Delta }Y}}\left[u_n{\mathsf{\Omega }}_n-u_s{\mathsf{\Omega }}_s\right]=0$$

(23)

Computing mass fluxes at all faces and applying the central differences scheme for the diffusion terms gives an algebraic system that is solved using SOR technique. The interior and boundary points are identified using a novel technique based on the inpolygon function in MATLAB, which distinguishes between inner and outer boundaries (Figure 2). This method is implemented through a custom MATLAB code developed in-house, which was used to generate all the results presented in this study. To determine the appropriate grid size, a grid independence test was conducted, as shown in

Table 1. Grid independency study at $Ra={10}^4, \lambda =5, Da=0.001, \sigma =500, H_1=0.2, R=0.1, Nb=Nt=0.3.\mathsf{\xi }=\mathsf{\pi }/4$.

Grid Size	${\left|\mathit{\psi }\right|}_{\mathit{m}\mathit{a}\mathit{x}}$
$91\times 41$	0.6501046
$111\times 61$	0.5754108
$131\times 81$	0.5353933
$151\times 101$	0.5206753
$171\times 121$	0.5168692
$201\times 151$	0.5175501

. In this table, the maximum values of the stream function (|${\left|\psi \right|}_{max}$) are calculated for various grid sizes ($91\times 41, 111\times 61, 131\times 81, 151\times 101, 171\times 121, and 201\times 151)$ at $Ra={10}^4,\lambda =5,Da=0.001,\sigma =500,H_1=0.2,R=0.1,Nb=Nt=0.3.\mathsf{\xi }={\displaystyle \frac{\mathsf{\pi }}{4}}.$ It is observed that the grid size of 201 × 151 provides sufficient accuracy for the present results. In a related context, to verify the accuracy of the present numerical model, a benchmark test was performed by comparing our results with those reported by Sayyou et al. [23]. As shown in

Table 2. Comparisons of ${\mathrm{N}\mathrm{u}}_{\mathrm{a}\mathrm{v}\mathrm{g}}, {\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{S}\mathrm{h}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$ for various Ra and Le at $N=0, \lambda =0$.

		Present Results		Sayyou et al. [23]
$\mathrm{R}\mathrm{a}$	$\mathrm{L}\mathrm{e}$	${\mathrm{N}\mathrm{u}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$	${\mathrm{S}\mathrm{h}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$	${\mathrm{N}\mathrm{u}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$	${\mathrm{S}\mathrm{h}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$
$100$	10	3.15	13.39	3.11	13.29
$200$	10	5.01	20.82	4.96	20.29
$1000$	10	13.50	48.402	13.52	47.63

, for various values of Ra and Le at N $=0$ and $\lambda =$0, excellent agreement is observed, confirming the validity and reliability of the numerical approach employed in this study.

4. Results and Discussion

This part introduced explanations of the obtained behaviors. The key factors, in this study, are the lengths of the heated/cold fins $H_1 {(H}_1=0.1, 0.15, 0.2, 0.25, 0.35)$, the Darcy number $Da$ ($Da=0.01, 0.001, 0.0001, 0.00001)$, the inclination angle $\xi$($\xi =0, \pi /6, \pi /4, \pi /3, \pi /2)$, and the vibration frequency $\sigma (\sigma =200-1500)$. The values of the referenced parameters are $R=0.1, Nb=Nt=0.3$. Here, it is intersected with the periodic profiles of the absolute maximum values of the stream function, of ${\left|\psi \right|}_{max}$, the behaviors of the Nusselt and Sherwood numbers at the heated fins and flow features, and distributions of the temperature, concentration, and nanoparticles within the flow domain.

Figure 2, Figure 3 and Figure 4 display the oscillatory behaviors of ${\left|\psi \right|}_{max}$, profiles of ${Nu}_{hot}$, and ${Sh}_{avg}$ for the variations of the lengths of heated/cold stabilizers of $H_1$ at $Ra={10}^4, \lambda =5, Da=0.001, \sigma =500, \xi ={\displaystyle \frac{\pi }{3}},R=0.1, Nb=Nt=0.3.$ It is noted that the values of ${\left|\psi \right|}_{max}$ are diminishing as $H_1$ increases while the heat and mass transfer rates are augmented. From the physical view, the flow spreads over a larger surface area due to the increase in the fin’s length, which leads to a reduction in the localized maximum stream function value (|ψ|_max). Also, the overall heat and mass transfer rates are improved due to a larger contact area with the surrounding fluid due to the higher lengths of the fins.

Figure 5, Figure 6 and Figure 7 illustrate the harmonic behaviors of ${\left|\psi \right|}_{max}$ against with the variations of the time together with the regular profiles of the ${Nu}_{hot}$ and ${Sh}_{hot}$ for different values of the Darcy number $Da$ at $Ra={10}^4, \lambda =5, Da=0.001, \sigma =500, \xi ={\displaystyle \frac{\pi }{3}},R=0.1, Nb=Nt=0.3$. The findings showed that as the Darcy number drops, the maximum stream function’s harmonic properties become less pronounced. On the contrary, both of the temperature and concentration gradients are enhanced as $Da$ is altered. Physically, as the Darcy number drops (i.e., the porous medium becomes less permeable), the pathways available for fluid flow are more restricted. This limitation disrupts the smooth, harmonic flow pattern that might be observed in a more permeable medium. The result is that the maximum stream function’s harmonic properties become less pronounced, meaning that the flow may become more irregular or less evenly distributed. Also, the aforementioned processes are generally less effective at evening out temperature and concentration differences, resulting in steeper gradients.

In Figure 8, Figure 9 and Figure 10, the fluctuating features of the ${\left|\psi \right|}_{max}$ and profiles of the heat and mass transfer rate for the growing of the vibration frequency $\sigma$ are depicted. It is noted that as $\sigma$ increases, fluctuations in the features of ${\left|\psi \right|}_{max}$ become faster accompanied by a lower amplitude as $\sigma$ is varied. These irregular behaviors cause lower values of ${Nu}_{hot}$ and ${Sh}_{hot}$. In fact, as the vibration frequency $\sigma$ increases, the flow field experiences more rapid oscillations. This means that the periodic or pulsating nature of the stream function (|ψ|_max) changes more quickly. Also, with higher $\sigma ,$ although the oscillations occur more frequently, the peak values of these fluctuations are diminished. This indicates that the strength or intensity of the flow features is reduced. When the fluctuations of the stream function become faster but weaker (lower amplitude), the convective transport mechanisms that enhance heat and mass transfer are disrupted.

Figure 11, Figure 12 and Figure 13 show the periodic features of the maximum stream functions ${\left|\psi \right|}_{max}$ with progressing of the time and profiles of the Nusselt and Sherwood coefficients at the lower edge for the raising of the inclination angle $\xi$. Noticeable enhancements in the values of ${\left|\psi \right|}_{max}$, heat, and mass transfer rates are obtained as $\xi$ is varied. As the inclination angle ξ increases, the alignment of the buoyancy-driven forces with respect to the fin’s geometry changes. This reorientation enhances the overall flow strength, as evidenced by the noticeable increase in |ψ|_max. The periodic features become more pronounced, suggesting that the flow cycles are becoming stronger or more vigorous. This improved mixing enhances the temperature and concentration gradients at the surface, resulting in higher heat and mass transfer rates (${Nu}_{hot}$ and ${Sh}_{avg})$.

Figure 14 and Figure 15 illustrate the streamlines $\psi$, temperature distributions $\theta$, species concentration $C,$ and nanoparticle concentration $\mathsf{\Phi }$ for the alterations of Darcy coefficient $Da$ and lengths of the stabilizers $H_1$, respectively. The findings disclosed that the features of the flow diminish as $Da$ is altered due to the decrease in the permeability of the medium. This flow reduction causes the convection impacts on the features of temperature distributions $\theta$, species concentration $C$, and nanoparticle concentration $\mathsf{\Phi }$ are decreased, resulting in an enhancement in their gradient. In addition, lower flow features are obtained as $H_1$ rises due to the obstruction to the flow due to the higher lengths of the stabilizers while the distributions of the temperature, species concentration, and nanoparticle concentration are improved, clearly.

5. Conclusions

Computational investigations of time-dependent double diffusion within a rectangular chamber filled with porous elements were presented. A vibration source within the flow area was considered, simulated by sinusoidal behavior with various modulation amplitudes and vibration frequencies. In this case, Buongiorno’s nanofluid model, along with the presence of species concentration in the inclined domain, was responsible for the density–buoyancy force. Four stabilizers were attached to the upper and lower boundaries, each with different lengths and heated conditions. The solution method was based on identifying the inner and outer boundaries using the inpolygon function and the finite volume method. The following major outcomes can be summarized:

➢

Harmonic behaviors are obtained for the stream function against the time variations due to the vibrational effects. These behaviors are reduced as either the lengths of the fins increase, or the Darcy number is decreased.

➢

As the vibration frequency rises, the fluctuation in the features of the streamlines is enhanced while their wave’s amplitude is diminishing.

➢

Higher rates for heat and mass transfer are obtained as the inclination angle is altered.

➢

Due to the higher viscous dissipation effects, the raising in Darcy number enhances gradients of the temperature and concentration.

➢

At fixed values of oscillation amplitude $(\lambda =5)$ and oscillation frequency $(\sigma =500),$ increasing the fin lengths from 0.1 to 0.35 results in a $70.98\%$ reduction in flow activity.

5.1. Practical Industrial Applications:

The results are highly relevant to several heat and mass transfer-intensive industries where porous media, vibration, and nanofluids are involved. These include the following:

• Cooling systems in electronics: Vibrational modulation can be used to enhance or control heat transfer in compact heat exchangers filled with porous materials or nanofluids.
• Energy systems and thermal storage: Double-diffusive convection in porous domains is critical in phase-change energy storage, geothermal reservoirs, and solar collectors, where vibrations can occur due to environmental or operational conditions.
• Chemical and biomedical processes: In reactors or separation units filled with porous catalysts or membranes, controlling species transport and temperature profiles using vibration or fin optimization can enhance efficiency and stability.

5.2. Scientific Contributions:

• This study integrates Buongiorno’s nanofluid model with double-diffusive convection in a vibrating porous domain, which has been rarely explored in time-dependent contexts.
• The identification of harmonic flow responses due to vibration adds depth to transient flow dynamics understanding.
• The use of a geometric fin configuration and vibrational control adds a new layer of complexity to previous studies on nanofluid flow in porous cavities.
• This study employs a robust numerical framework (finite volume + geometric domain tracking using inpolygon) that can be extended to more complex geometries.