Search Results (297)

This paper explores the complex dynamics of mixed convective flow in a Casson fluid saturated in a non-Darcy porous medium, focusing on the influence of gyrotactic microorganisms, internal heat generation, and multiple convective mechanisms. Casson fluids, known for their non-Newtonian behavior, are relevant in various industrial and biological contexts where traditional fluid models are insufficient. This study addresses the limitations of the standard Darcy’s law by examining non-Darcy flow, which accounts for nonlinear inertial effects in porous media. The governing equations, derived from conservation laws, are transformed into a system of no linear ordinary differential equations (ODEs) using similarity transformations. These ODEs are solved numerically using a finite differencing method that incorporates central differencing, tridiagonal matrix manipulation, and iterative procedures to ensure accuracy across various convective regimes. The reliability of this method is confirmed through validation with the MATLAB (R2024b) bvp4c scheme. The investigation analyzes the impact of key parameters (such as the Casson fluid parameter, Darcy number, Biot numbers, and heat generation) on velocity, temperature, and microorganism concentration profiles. This study reveals that the Casson fluid parameter significantly improves the velocity, concentration, and motile microorganism profiles while decreasing the temperature profile. Additionally, the Biot number is shown to considerably increase the concentration and dispersion of motile microorganisms, as well as the heat transfer rate. The findings provide valuable insights into non-Newtonian fluid behavior in porous environments, with applications in bioengineering, environmental remediation, and energy systems, such as bioreactor design and geothermal energy extraction. Full article

In double-shield TBM tunnel construction, grouting plays a vital role in consolidating the gravel backfill and maintaining the integrity of the segmental lining. To investigate the permeation behavior of grout within the pea gravel layer, a fluid dynamics model was developed in this study. The model directly simulates the flow of grout through the porous medium by solving the Navier–Stokes equations and employs the level set method to track the evolving interface between the grout and air phases. Unlike conventional continuum approaches, this model incorporates particle-scale heterogeneity, allowing for a more realistic analysis of grout infiltration through the non-uniform pore structures formed by gravel packing. Three different grouting port positions and two boundary conditions are considered in the simulation. The results indicate that under pressure boundary conditions, the grout flow rate increases rapidly in the initial stage, and then decreases and stabilizes, with the flow rate peak increasing as the grout port moves upward. Under velocity boundary conditions, the injection pressure grows slowly in the early stage but accelerates with time. Additionally, the rate of pressure change is faster when the grout port is located lower in the backfilling layer. Through theoretical analysis, the existing analytical formula was extended by introducing a gravitational correction term. When the grouting port is near the upper part of the tunnel, the analytical solution aligns well with the numerical simulation results, but as the grout port moves downward, the discrepancy between the two increases. Full article

This study investigates the consistent and uniform movement of two spherical particles within an infinite porous medium saturated with a couple stress fluid, with a particular focus on the effects of surface slippage. The research reveals that surface slippage significantly reduces the drag force experienced by the particles, thereby influencing their hydrodynamic interactions. Conversely, increases in permeability and particle size similarity tend to enhance both the drag force and the inter-particle interaction forces, affecting the overall dynamics of particle motion. The analysis is conducted within the low-Reynolds-number regime, characteristic of laminar flow dominated by viscous forces, and employs boundary collocation methodologies to derive semi-analytical solutions to the governing differential equations. This approach enables a detailed characterization of the flow behavior and inter-particle forces in intricate fluid environments, including those with porous matrices and complex rheological properties. The findings from this investigation are consistent with prior numerical analyses, notably those conducted by Alotaibi and El-Sapa (2025), and corroborate earlier studies by Shehadeh and Ashmawy (2019), which examined cases of no slippage and permeability effects. Additionally, the results align with earlier research by Shreen et al. (2018) concerning viscous fluids, thereby reinforcing the validity of the conclusions. Overall, the study enhances the understanding of particle-fluid interactions in porous, couple stress-rich media, providing valuable insights into the roles of surface slippage, permeability, and particle size in determining hydrodynamic forces. Full article

In the current paper, an analysis of magnetohydrodynamic Williamson nanofluid boundary layer flow is presented, with multiple slips in a porous medium, using a newly designed human-brain-inspired Ricker wavelet neural network solver. The solver employs a hybrid approach that combines genetic algorithms, serving as a global search method, with sequential quadratic programming, which functions as a local optimization technique. The heat and mass transportation effects are examined through a stretchable surface with radiation, thermal, and velocity slip effects. The primary flow equations, originally expressed as partial differential equations (PDEs), are changed into a dimensionless nonlinear system of ordinary differential equations (ODEs) via similarity transformations. These ODEs are then numerically solved with the proposed computational approach. The current study has significant applications in a variety of practical engineering and industrial scenarios, including thermal energy systems, biomedical cooling devices, and enhanced oil recovery techniques, where the control and optimization of heat and mass transport in complex fluid environments are essential. The numerical outcomes gathered through the designed scheme are compared with reference results acquired through Adam’s numerical method in terms of graphs and tables of absolute errors. The rapid convergence, effectiveness, and stability of the suggested solver are analyzed using various statistical and performance operators. Full article

This article discusses the problem of numerically solving the Navier–Stokes equations, the heat conduction equation, and the transport equation in the orthogonal coordinates of a free curve. Since the numerical solution domain is complex, the curvilinear mesh method was used. To do so, first, a boundary value problem was posed for the elliptic equation to automate the creation of orthogonal curved meshes. By numerically solving this problem, the program code for the curvilinear mesh generator was created. The motion of a liquid or gas through a porous medium was described by numerically solving the Navier–Stokes equations in freely curvilinear orthogonal coordinates. The transformation of the Navier–Stokes equation system, written in the stream function, vorticity variables, and cylindrical coordinates, into arbitrary curvilinear coordinates, was considered in detail by introducing metric coefficients. To solve these equations, the coefficients of which vary rapidly, a three-layer differential scheme was developed. The approximation, stability, and compactness of the differential scheme were previously studied. The considered problem was considered to be the mathematical model of a car catalytic converter, and computational experiments were conducted. Calculations were performed with the developed program code in different geometries of the computational domain and different values of grid size. The Reynolds number was changed from 100 to 10,000, and its effect on the size of the backflow in front of the porous medium was discussed. The software code, which is based on the differential equation of the Navier–Stokes equations written in the orthogonal coordinates of a curved line, and its calculation algorithm can be used for the mathematical and computer modeling of automobile catalytic converters and chemical reactors. Full article

Highlighting the importance of artificial intelligence and machine learning approaches in engineering and fluid mechanics problems, especially in heat transfer applications is main goal of the presented article. With the advancement in Artificial Intelligence (AI) and Machine Learning (ML) techniques, the computational efficiency and accuracy of numerical results are enhanced. The theme of the study is to use machine learning techniques to examine the thermal analysis of MHD boundary layer flow of Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder embedded in a porous medium with heat source/sink and viscous dissipation effects. The considered base fluid is water ($H_{2}O$) and hybrid nanoparticles titanium oxide (${TiO}_2$) and Copper oxide ($CuO$). The governing flow equations are nonlinear PDEs. Non-similar system of PDEs are obtained with efficient conversion variables. The dimensionless PDEs are truncated using a local non-similarity approach up to third level and numerical solution is evaluated using MATLAB built-in-function bvp4c. Artificial Neural Networks (ANNs) simulation approach is used to trained the networks to predict the solution behavior. Thermal boundary layer improves with the enhancement in the value of $Rd$. The accuracy and reliability of ANNs predicted solution is addressed with computation of correlation index and residual analysis. The RMSE is evaluated [0.04892, 0.0007597, 0.0007596, 0.01546, 0.008871, 0.01686] for various scenarios. It is observed that when concentration of hybrid nanoparticles increases then thermal characteristics of the Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder. Full article

This paper deals with laminar forced convection in a rectangular channel through a non-equilibrium thermal gas saturated porous medium. The thermodynamic aspects of this flow, including the entropy generation rate, irreversibility, and exergy, are carefully investigated. The governing conservation equations of momentum, mass, and energy are solved numerically using the finite volume method. The effects of Reynolds number $Re$ (ranging from 100 to 2000), Darcy number $Da$ $\left(\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} {10}^{-6} \mathrm{t}\mathrm{o} {10}^{-1}\right)$, and Biot number $Bi$ (from 10⁻³ to 10³) on the entropy generation, exergy, and irreversibility, for which the Gouy-Stodola relation is employed, are then presented. The results reveal that at low $Re$ and high $Bi$, thermal equilibrium between the two phases is achieved, leading to a reduction in entropy generation and, consequently, less exergy destruction. However, in the limit of high $Re$ and low $Da$, irreversibility is significant due to large velocity gradients, leading to greater exergy destruction. Furthermore, it was observed that the thermal non-equilibrium intensity (LNTE) significantly influences entropy generation, leading to critical exergy destruction. Full article

In this paper, we investigate the effect of singular viscosity on the stability of a thin film of Oldroyd-B viscoelastic fluid flowing along a porous inclined surface under the influence of a normal electric field. First, we derive the governing equations and boundary conditions for the flow of the film and assume that the film satisfies the Beavers–Joseph sliding boundary condition when it flows on a porous inclined surface. Second, through the long-wave approximation, we derive the nonlinear interfacial evolution equation. Then, linear and nonlinear stability analyses are performed for the interfacial evolution equation. The stability analyses show that the singular viscosity has a stabilizing effect on the flow of the film, while the strain delay time of the Oldroyd-B fluid, the electric field, and the parameters of the porous medium all have an unsteady effect on the flow of the film. Interestingly, in the linear stability analysis, the parameters of the porous medium have an unsteady effect on the flow of the film after a certain value is reached and a stabilizing effect before that value is reached. In order to verify these results, we performed numerical simulations of the nonlinear evolution equations using the Fourier spectral method, and the conclusions obtained are in agreement with the results of the linear stability analysis, i.e., the amplitude of the free surface decreases progressively with time in the stable region, whereas it increases progressively with time in the unstable region Full article

This study aims to investigate the effect of a constant magnetic field on heat transfer, flow of fluid, and entropy generation of mixed convection in a lid-driven porous medium enclosure filled with nanofluids (TiO₂-water). Uniform constant heat fluxes are partially applied to the bottom wall of the enclosure, while the remaining parts of the bottom wall are considered to be adiabatic. The vertical walls are maintained at a constant cold temperature and move with a fixed velocity. A sinusoidal wall is assumed to be fixed and kept adiabatic at the top enclosure. Three scenarios are considered corresponding to different directions of the moving isothermal vertical wall (±1). The influence of pertinent parameters on the heat transfer, flow of fluid, and entropy generation in an enclosure are deliberated. The parameters are the Richardson number (${\tilde{R}}_i$ = 1, 10, and 100), the Hartmann number (0 ≤ $\tilde{H}$a ≤ 75 with a 25 step), and the solid volume fraction of nanoparticles (0 ≤ $\tilde{\Phi }$ ≤ 0.15 with a 0.05 step). The Grashof and Darcy numbers are assumed to be constant at 10⁴ and 10⁻³, respectively. The finite element method, utilizing the variational formulation/weak form, is applied to discretize the main governor equations. Triangular elements have been employed within the studied envelope, with the elements adapting as needed. The results showed that the streamfunction and fluid temperature decreased as the solid volume fraction increased. The local $\tilde{\mathrm{N}}\mathrm{u}$ number increased by more than 50% at low values of $\tilde{\mathsf{\Phi }}$ (up to 0.1). This percentage decreases between 25% and 40% when $\tilde{\mathsf{\Phi }}$ is in the range of 0.1 to 0.15. As $\tilde{\mathrm{H}}$a increases from 0 to 75, these percentages increase at low values of the value of $\tilde{\mathrm{R}}\mathrm{i}=1$ and 10. These variations are primarily dependent on the value of the Richardson number. Full article

This study investigates the dynamics of two oscillating rigid spheres moving through an infinite porous medium saturated with Stokes fluid flow, addressing the problem of how fluid properties, permeability, frequency, and slip length influence the system. The objective is to model the interactions between the spheres, which differ in size and velocity as they move along the axis connecting their centers while applying slip boundary conditions to their surfaces. We derive the governing field equations using a semi-analytical method and solve the resulting system of equations numerically through a collocation technique. Our novel quantitative results include insights into the drag force coefficients for both in-phase and out-of-phase oscillations of each hydrophobic sphere, considering parameters such as diameter ratio, permeability, frequency, velocity ratios, slip lengths, and the distances between the spheres. Notably, when the spheres are sufficiently far apart, the normalized drag force coefficients behave as if each sphere is moving independently. Additionally, we present streamlines that illustrate the interactions between the spheres across a range of parameters, highlighting the novelty of our findings. A purely viscous medium and no-slip conditions are used to validate the numerical approach and results. Full article

The transient electroosmotic response in a charged porous medium consisting of a uniform array of parallel circular cylindrical fibers with arbitrary electric double layers filled with an electrolyte solution, for the stepwise application of a transverse electric field, is analyzed. The fluid momentum conservation equation is solved for each cell by using a unit cell model, where a single cylinder is surrounded by a coaxial shell of the electrolyte solution. A closed-form expression for the transient electroosmotic velocity of the bulk fluid in the Laplace transform is obtained as a function of the ratio of the cylinder radius to the Debye screening length and the porosity of the fiber matrix. The effect of the fiber matrix porosity on the continuous growth of the electroosmotic velocity over time is substantial and complicated. For a fiber matrix with larger porosity, the bulk fluid velocity takes longer to reach a certain percentage of its final value. Although the final value of the bulk fluid velocity generally increases with increasing porosity, early velocities may decrease with increasing porosity. For a given fiber matrix porosity, the transient electroosmotic velocity is a monotonically increasing function of the ratio of the cylinder radius to the Debye length. Full article

Bioconvection phenomena play a pivotal role in diverse applications, including the synthesis of biological polymers and advancements in renewable energy technologies. This study develops a comprehensive mathematical model to examine the effects of key parameters, such as the Lewis number (Lb), Peclet number (Pe), volume fraction ($\phi$), and angle of inclination $\left(\alpha \right)$, on the flow and heat transfer characteristics of a nanofluid over an inclined cylinder embedded in a non-Darcy porous medium. The investigated nanofluid comprises nano-encapsulated phase-change materials (NEPCMs) dispersed in water, offering enhanced thermal performance. The governing non-linear partial differential equations are transformed into dimensionless ordinary differential equations using similarity transformations and solved numerically via the Network Simulation Method (NSM) and an implicit Runge–Kutta method implemented through the bvp4c routine in MATLAB R2021a. Validation against the existing literature confirms the accuracy and reliability of the numerical approach, with strong convergence observed. Quantitative analysis reveals that an increase in the Peclet number reduces the shear stress at the cylinder wall by up to 18% while simultaneously enhancing heat transfer by approximately 12%. Similarly, the angle of inclination ($\alpha$) significantly boosts heat transmission rates. Additionally, higher Peclet and Lewis numbers, along with greater nanoparticle volume fractions, amplify the density gradient of microorganisms, intensifying the bioconvection process by nearly 15%. These findings underscore the critical interplay between bioconvection and transport phenomena, providing a framework for optimizing bioconvection-driven heat and mass transfer systems. The insights from this investigation hold substantial implications for industrial processes and renewable energy technologies, paving the way for improved efficiency in applications such as thermal energy storage and advanced cooling systems. Full article

This study investigates the steady, two-dimensional boundary layer flow of a Casson fluid over an inclined nonlinear stretching surface embedded within a Forchheimer porous medium. The governing partial differential equations are transformed into a set of ordinary differential equations through similarity transformations. The analysis incorporates the effects of an external uniform magnetic field, gravitational forces, thermal radiation modeled by the Rosseland approximation, and first-order homogeneous chemical reactions. We consider several dimensionless parameters, including the Casson fluid parameter, magnetic parameter, Darcy and Forchheimer numbers, Prandtl and Schmidt numbers, and the Eckert number to characterize the flow, heat, and mass transfer phenomena. Analytical solutions for the velocity, temperature, and concentration profiles are derived under simplifying assumptions, and expressions for critical physical quantities such as the skin friction coefficient, Nusselt number, and Sherwood number are obtained. Full article

A numerical study was carried out to analyze the phenomenon of natural convection in a porous medium saturated with nanofluid. In the study, the boundary element method was used for computational modeling. The fluid flow through a porous matrix is described using the Darcy–Brinkman–Forchheimer momentum equation. In addition, a mathematical model for nanofluids was used, which follows a single-phase approach and assumes that the nanoparticles within a fluid can be treated as an independent fluid with effective properties. A combination of single- and sub-domain boundary element methods was used to solve the relevant set of partial differential equations. The method was originally developed for pure flow scenarios, but also proves to be effective in the context of fluid flow through porous media. The results are calculated for the case of two- and three-dimensional square cavities. In addition to various values of dimensionless control parameters, including the porous Rayleigh number (Ra_p), Darcy number (Da), porosity (ϕ) and nanoparticle volume fractions (φ), the effects of the inclination angle of the cavity on the overall heat transfer (expressed by the Nusselt number (Nu)) and fluid flow characteristics were investigated. The results indicate a pronounced dependence of the overall heat transfer on the introduction of nanoparticles and inclination angle. The heat transfer in a two-dimensional cavity is increased for higher values of Darcy number in the conduction flow regime, while it is suppressed for lower values of Darcy number in the Darcy flow regime. In the case of a three-dimensional cavity, increasing the volume fraction of nanoparticles leads to a decrease in heat transfer, and furthermore, increasing the inclination angle of the cavity considerably weakens the buoyancy flow. Full article