This study investigates the quasi-steady axisymmetric thermophoretic motion of a spherical particle partially submerged at the flat interface of a semi-infinite Brinkman medium. The analysis is conducted under the assumptions of small Reynolds and P & eacute;clet numbers, while the capillary number is considered sufficiently small to preserve the flatness of the interface. The specific case of a 90 degrees contact angle with the flat surface is examined. To avoid singularities at the contact line, the Knudsen number is assumed to lie within the slip-flow regime. Analytical expressions are derived for the thermophoretic velocity and force acting on the half-submerged particle. Graphical results illustrate the influence of parameters such as Fourier thermal conductivity ratio, Knudsen number, medium permeability, frictional slip, and thermal stress slip. Furthermore, the limiting behavior corresponding to thermophoresis in a classical viscous fluid is discussed. Since the present solution is exact, the case of a 90 degrees contact angle also serves as a benchmark for validating numerical solutions at other contact angles. The findings are relevant to applications involving particle manipulation at fluid-porous interfaces, such as targeted drug delivery across biological membranes, pollutant transport at soil-air boundaries, and the design of microfluidic systems for controlled colloidal assembly.

This study presents an analytical investigation of combined pressure-driven and electroosmotic flow in a hydrophobic cylindrical microannular channel filled with an uncharged hydrogel, modeled as a Brinkman porous medium. Fluid motion is governed by the Darcy–Brinkman equation, which reduces to Darcy flow at low permeability and transitions to Stokes flow at high permeability. Electrokinetic effects are described using the linearized Poisson–Boltzmann equation under the Debye–Hückel approximation, with velocity slip imposed at both annular boundaries and the electrical double layer fully accounted for. A key new finding is the identification of a resonance condition that occurs when the electrokinetic width matches the Brinkman permeability parameter, leading to strong coupling between electroosmotic and pressure-driven transport. Under this condition, electroosmotic flow can rival or even exceed pressure-driven flow in low-permeability regimes, producing flattened velocity profiles and enhanced volumetric flow rates. Pressure-driven transport is found to be insensitive to electrokinetic parameters, whereas electroosmotic flow remains robust under strong porous resistance and is significantly enhanced by wall slip and finite electrical double-layer thickness. Parametric trends of the flow rate and streaming potential are presented and validated against established results in the literature. Since the permeability of hydrogel matrices directly determines the Brinkman resistance parameter, tailoring the hydrogel microstructure provides a practical means of approaching the resonance condition and thereby enhancing electroosmotic transport. These findings provide new physical insight into porous electrokinetic transport and offer practical design guidelines for electroosmotic pumps, hydrogel-based microfluidic devices, heat exchangers, and microelectronic cooling systems operating under high hydraulic resistance.

This work examines thermocapillary migration of a compound droplet containing an off-centered inner core and suspended in an immiscible viscous fluid. The eccentric placement of the core creates a geometrically asymmetric two-interface system. Under low capillary number conditions, both interfaces remain spherical while the core and the surrounding shell undergo coupled but distinct translational motion in response to an imposed uniform temperature gradient. Temperature-induced surface tension variations generate Marangoni stresses along both interfaces, driving motion and inducing complex internal circulation. The intrinsic asymmetry associated with core eccentricity leads to non-uniform interfacial forcing and uneven momentum exchange among the three fluid phases. The coupled Stokes and energy equations governing the system are solved semi-analytically using a spectral collocation approach. The results demonstrate that geometric offset significantly modifies interfacial stress distributions, internal flow structure, and overall migration behavior. The dynamics are strongly influenced by the degree of eccentricity in combination with viscosity ratios, thermal conductivity contrasts, and interfacial tension sensitivities. This study establishes a framework for thermocapillary transport in asymmetric core-shell droplets and offers insight relevant to the development of directionally controllable microcapsules, advanced drug-delivery carriers, and biomimetic encapsulated systems in thermally driven microfluidic technologies.

In this study, we present a semi-analytical investigation of the thermocapillary motion of a concentric compound droplet suspended in an unbounded immiscible fluid near a hydrophobic planar wall. The droplet consists of an inner core encapsulated by an outer liquid shell, with both interfaces assumed spherical under low capillary number conditions. The compound droplet and its core translate along the axis normal to the wall with distinct migration velocities. The analysis is performed in the creeping-flow regime and accounts for coupled thermal and hydrodynamic interactions among the inner and outer interfaces and the nearby wall. Imposed temperature gradients generate Marangoni stresses at both interfaces, driving thermocapillary motion. This motion is strongly influenced by wall-induced distortion of the temperature and velocity fields. Semi-analytical solutions of the Stokes and energy equations are obtained using a collocation technique. These solutions yield accurate and well-converged migration velocities over a wide range of droplet-wall separations. The results demonstrate that the wall significantly modifies the magnitude of thermocapillary migration. This modification occurs through enhanced viscous dissipation and asymmetric interfacial stresses. The coupled effects of thermal conductivity ratios, viscosity contrasts, interfacial tension gradients, and geometric confinement are systematically elucidated. This work extends existing theories by incorporating multi-interface coupling and wall effects within a unified semi-analytical framework. It is directly relevant to the controlled manipulation of encapsulated droplets in microfluidics, targeted drug delivery, microscale thermal management, and cell-mimicking transport near solid boundaries.

The thermophoretic migration of porous particles in confined environments is of considerable importance in microfluidic manipulation, filtration technologies, and targeted transport systems. In the present study, the steady thermophoretic motion of a porous spherical particle near a rigid planar wall is investigated within the Stokes-diffusion limit. The porous interior is modeled as a homogeneous Brinkman medium, thereby accounting for internal flow and permeability effects on thermophoretic transport. A uniform temperature gradient is imposed normal to the wall, inducing particle migration perpendicular to the boundary. A semi-analytical solution is developed using coupled spherical and cylindrical eigenfunction expansions. The wall boundary conditions are satisfied through Fourier-Bessel transform techniques, while the remaining interfacial conditions at the porous particle surface are enforced numerically using a boundary collocation method. The results demonstrate that wall confinement significantly suppresses thermophoretic migration owing to the combined effects of hydrodynamic resistance and thermal-field distortion within the narrow gap region. The reduction in migration velocity becomes more pronounced as the particle approaches the wall and depends strongly on both the permeability parameter and the thermal conductivity ratio. Comparisons with the impermeable-particle limit further clarify the role of internal permeability and show that the classical solid-particle behavior is recovered as a limiting case of the present formulation.

The quasi-steady thermocapillary motion of a spherical liquid droplet embedded within a Brinkman porous medium adjacent to an impermeable plane wall is investigated. A uniform temperature gradient is imposed normal to the wall, leading to thermocapillary-driven migration of the droplet. The governing energy and momentum equations, incorporating the Brinkman model for the porous medium, are solved involving superposition of fundamental solutions and a collocation technique. This analysis is conducted under the assumptions of small P & eacute;clet and Reynolds numbers, ensuring linearity, and a small capillary number, guaranteeing the droplet remains spherical. Thermal equilibrium is assumed between the fluid and solid matrix, and a slip condition governs the flow at the planar wall. The results reveal that the migration velocity U/U0 notably decreases as the droplet approaches the wall, with an approximate 30% reduction in velocity as the spacing parameter increases from a/z0=0.1 to a/z0=0.99. Increasing the Brinkman parameter (alpha) significantly suppresses motion, reducing the velocity by nearly 70%, indicating the strong resistance of the porous medium. Increasing the thermal conductivity ratio (k) or viscosity ratio (sigma) further lower the migration speed by about 20%-40%. The study offers new insights into droplet transport under thermal gradients in confined porous domains. In limiting cases, the model recovers established solutions for either clear fluid domains or wall-free porous environments. This research holds substantial relevance for applications in microfluidics and targeted drug delivery systems, where precise control over droplet transport through complex porous biological environments is crucial.

This study presents a semi-analytical solution, developed using a collocation-based approach, to investigate the flow field induced by the perpendicular motion of a composite spherical particle toward a planar interface separating two immiscible, semi-infinite fluid layers. One fluid exhibits microstructured (micropolar) behavior described by Eringen's theory, while the other behaves as a conventional viscous Newtonian fluid. The composite particle is modeled as a rigid, impermeable spherical core enveloped by a porous Brinkman shell and is assumed to reside in the Newtonian region. The analysis is conducted under low-Reynolds-number and negligible-capillarity conditions, ensuring that interfacial deformation remains minimal. Owing to the linearity of the governing equations, the flow fields in both fluid domains are constructed by superposing fundamental solutions formulated in cylindrical and spherical coordinates. The normalized drag force acting on the composite particle is evaluated over a range of dimensionless parameters, showing excellent numerical convergence and agreement with established limiting cases. Quantitatively, the drag force was found to increase by up to 150%–300% as the particle approaches the interface (from a/z0=0.1 to 0.9), while variations in the porous-layer permeability produced drag reductions of up to 40%–55% for highly permeable shells (α≤0.1). Micropolar effects further modified the drag by 10%–20% depending on the spin-coupling parameter and viscosity ratios. The findings provide valuable insights for the design of advanced microfluidic systems, such as lab-on-a-chip platforms, where the interactions of functionalized or composite microparticles with complex or soft interfaces critically influence particle transport, separation, and sensing performance.

The slow, quasi-steady, axisymmetric translational motion of a solid spherical particle within an eccentric cavity containing a hydrogel medium is analyzed using a semi-analytical approach. The hydrogel is modeled as a porous medium saturated with a microstructured fluid exhibiting micropolar behavior. No-slip and no-spin conditions are applied at both the particle surface and the cavity wall. The hydrodynamic governing equations are solved by constructing general solutions from fundamental solutions formulated in two spherical coordinate systems-one centered on the particle and the other on the cavity. A collocation method is employed to satisfy the boundary conditions at the interfaces. The obtained results show strong agreement with existing literature. This study reveals that the cavity wall, micropolar fluid characteristics, and permeability parameters significantly influence the drag force acting on the particle. These findings provide valuable insights into controlled particle motion in hydrogel environments, with potential applications in targeted drug delivery and biomedical transport systems.

This study presents an analytical investigation of unsteady, time-periodic flow in a hydrogel medium confined within a cylindrical microannulus, driven by both a pressure gradient and an externally applied electric field. The hydrogel is modeled as a Brinkman porous matrix saturated with a micropolar fluid. By coupling the linearized Poisson-Boltzmann equation (Debye-H & uuml;ckel approximation) with the Brinkman-micropolar momentum equations, closed-form expressions are derived for the axial velocity and microrotation as functions of radial position, time, and key dimensionless parameters. The flow is shown to comprise two independent contributions: a pressure-driven component and an electroosmotic component, each influenced by specific physical mechanisms. Quantitatively, increasing the permeability resistance parameter (lambda) from 0 to 10 reduces the volume flow rate by 89.64%, the streaming function by 83.63%, while increasing microrotation strength by 62.05 %. Raising the micropolar coupling number (c) from 0.1 to 0.9 leads to a 38.20% decrease in flow rate, a 165.55% increase in the streaming function, and a 28.43 % rise in microrotation. Frequency effects are especially pronounced: increasing the forcing frequency parameter (a) from 0.5 to 50 results in a 99.98% drop in flow rate, a 100% increase in the streaming function, and a 99.9997% rise in microrotation. The electrokinetic width (k) is a dominant tuning parameter-doubling k from 10 to 20 leads to a 581.94% increase in flow rate, and 315.86% in the streaming function. The analysis also reveals how zeta potential asymmetry (beta not equal 1) enables precise flow control, including reversal. All classical limiting cases-Newtonian, purely pressure-driven, and steady electroosmotic-are exactly recovered, validating the model. These findings provide quantitative guidelines for the design of hydrogel-based microfluidic systems where electrokinetic and microstructural effects critically influence transport.

A methodological blend of analytical and numerical strategies employing collocation techniques is presented to investigate the task of describing the Stokes flow generated by a soft particle (composite sphere) moving perpendicularly to a planar interface of infinite extent, separating two semi-infinite, immiscible viscous fluid domains. The particle consists of a solid core enclosed by a porous membrane allowing fluid passage. The movement of the soft nanoparticle has been examined through a continuum mathematical model. This model incorporates the Stokes and Brinkman equations, accounting for the hydrodynamic fields both outside and within the porous membrane layer, respectively. The motion is investigated under conditions characterized by low Reynolds and capillary numbers, where the interface experiences negligible deformation. The solution combines cylindrical and spherical fundamental solutions via superposition. Initially, the boundary conditions at the fluid–fluid interface are satisfied utilizing Fourier–Bessel transforms, subsequently addressing the conditions at the soft particle's surface through a collocation method. The normalized drag force exerted on the particle is accurately calculated, exhibiting robust convergence across various geometric and physical parameters. These findings are effectively visualized via graphs and tables. We juxtapose our drag force coefficient results with established literature data, particularly focusing on the extreme cases. The findings highlight the substantial impact of the interface on the drag force coefficient. Across the full range of viscosity ratios, the normalized drag force decreases as the relative thickness of the porous layer increases. These results enhance the understanding of practical systems and industrial processes such as sedimentation, flotation, electrophoresis, and agglomeration.

This article deals with the axisymmetric flow of a Newtonian fluid droplet moving along the axis of a thin circular disk under the conditions of a low Reynolds number in the quasi-steady limit. The system is immersed in a second Newtonian fluid. With the same geometry, the motion of a slip spherical particle is treated also. Basic solutions are constructed in cylindrical and spherical coordinates according to the appropriate flow region. A combined analytic-numerical solution procedure in conjunction with a dual integral equation is used. Numerical results show that the normalized drag force coefficient acting on the droplet/ slip particle is obtained with good convergence for various values of viscosity ratio, slip parameter, and clearance parameter. The findings demonstrate that the results of the drag coefficient are in good agreement with the limiting cases available in the literature.

A comprehensive analytical study of the electrophoresis of a suspension of charged spherical particles in an arbitrary electrolyte solution through a porous medium is analyzed using a unit cell model. The unsteady Brinkman equation with the term of the electric force governing the fluid velocity fields is solved by means of the Laplace transform. The porous medium is uniformly charged by fixed charge density Q, and the embedded spherical particle is conductive and charged with constant zeta potential. Two different accurate concepts of the boundary conditions at the outer virtual surface, are presented to solve the governing equations for the flow field. Steady-state electrophoretic velocity is obtained analytically and displayed graphically for different physical parameters. Also, a closed form of the transient electrophoretic velocity versus the dimensionless elapsed time is plotted and discussed for different values of the Debye length parameter, particle volume fraction, density ratio, permeability of the porous medium, and for highly-and non-conducting particles. The steady/ transient electrophoretic velocity is a monotonic decreasing function of the permeability parameter, the particle-to-fluid density ratio, and the particle volume fraction, but it increases with an increase in the Debye length parameter with constant zeta potential. In general, the Kuwabara cell model predicts a smaller value for the electrophoresis velocity than the Happel model, but the difference is not significant. On the other hand, when the transient velocity is normalized by its steady state, the Happel model has smaller values than the Kuwabara model. The effects of the relevant parameters on the transient starting electrokinetic flow in the porous medium are interesting and significant. The results are found to be in excellent agreement with the exact numerical results obtained by Lai and Keh.

A perturbation analysis is presented to study the flow of a Newtonian fluid in a three-dimensional channel with impermeable walls of random tortuosity filled with a porous medium. The amplitude of the bumps is small compared to the mean distance between the two walls of the channel. Basset slip conditions are used on the channel walls. The effect of stochastic curvature of the channel walls on fluid flow is investigated. An expression is obtained for the mean flow rate, up to the second-order of the normalized amplitude, as a function of the permeability of the porous medium, slip parameter, and wavenumbers. A decrease in the flow rate as the permeability parameter increases is discovered. The phase difference and the longitudinal and transverse wavenumbers of corrugations play an important role in the flow rate. The limiting cases of Stokes and Darcy’s flows are recovered.

This paper reports the axisymmetric motion of a viscous droplet or solid spherical particle with a slip-flow surface that moves perpendicular toward an orifice in a plane wall. The motion is studied in the quasi-steady limit under a low Reynolds number. To maintain the spherical shape of the droplet, we assumed that the interfacial tension is very large. The radius of the droplet/particle may be either smaller or larger than the radius of the orifice. A general solution is established from fundamental solutions in both spherical and cylindrical coordinate systems. A semi-analytical approach based on dual integral equations and a collocation scheme is used. Numerical results show that the normalized drag coefficient acting on the droplet/particle is obtained with good convergence for different values of slip parameter, viscosity ratio, and spacing parameters. The findings demonstrate that the collocation results of the drag coefficient are consistent with the limiting cases available in the literature.

This paper reports an analytical study for the thermophoresis of a circular cylindrical aerosol particle embedded in a porous medium of constant porosity based on the Brinkman model. The Knudsen number is supposed to be in the slip-flow regime. The Peclet and Reynolds numbers are small, therefore the convective effects are neglected, and the problem can be considered quasi-steady. The porous medium is supposed to be homogeneous, isotropic and the solid phase is in thermal equilibrium with the fluid through the voids of the medium. In the analysis of motion, at the surface of the particle, we consider the following effects: temperature jump, thermal creep, viscous slip, and thermal stress slip. Formulas for thermophoretic velocity and force are derived. The novelty of the problem is the permeability parameter which characterizing the Brinkman flow. The effect of this parameter is shown through several plots for thermophoretic velocity and force against the thermal properties of the particle and porous medium. The limiting cases of Stokes and Darcy’s flows and the case of no thermal slip are discussed. results are also compared with the corresponding values for the case of spherical particles.

In this study, the time‐dependent electrophoretic motion of a conducting spherical particle embedded in an arbitrary electrolyte solution saturated porous medium is investigated. The porous medium is uniformly charged and the embedded hard particle is charged with constant ζ ‐potential or constant surface charge density. The unsteady modified Brinkman equation with an electric force term, which governs the fluid velocity field, is used to model the porous medium and is solved by Laplace's transform technique. An analytical expression for the electrophoretic velocity of the spherical particle is obtained in Laplace transform domain as a function of the relevant parameters, and its inversion is obtained through numerical techniques. Also, in this study, the steady‐state electrophoretic velocity is obtained analytically as linear functions of ζ ‐potential (or surface density charge) and the fixed charge density. The steady‐state electrophoretic velocity is displayed graphically for various relevant parameters and compered with the available data in the literature. Also, the numerical values of the transient electrophoretic velocity are plotted versus the nondimensional elapsed time and discussed for different values of the Debye length parameter, density ratio, permeability of the porous medium, and for high and nonconducting particles.

An analytical and numerical study for the creeping flow caused by a solid spherical particle with a slip-flow surface is considered in the presence of a fluid–fluid plane interface. The particle rotating about or translating along an axis perpendicular to the interface. The motion is investigated in the limit of low capillary number where in this situation the interface is of negligible deformation. Using a bipolar coordinate system, the stream functions are constructed for both fluid phases as Reynolds number tends to zero. The novelty of this work is allowing the slip on the surface of the particle. The matching boundary conditions at the plane interface and the slip boundary condition on the particle’s surface are applied to the truncated solutions to specify the unknown coefficients. A comparison is made between the results of the analytical solution and the results obtained from a boundary collocation method. The torque and drag force exerted on the particle are calculated using both techniques, which are found in perfect agreement. In addition to compression with collocation techniques, we also studied the predicted changes in the drag force and torque due to the presence of the plane interface and the slippage at the surface of the particle. Our results of the drag force and torque are compared with the available data in the literature for the special cases. The work is motivated by its possible application as an analytical tool in the study of locomotion of microswimmers near an interface such as synthetic swimmers and microorganisms.

An analytical–numerical method based on collocation techniques is presented to investigate the problem of determining Stokes flow caused by a spherical particle moving perpendicularly to a plane interface separating two semi-infinite immiscible fluid phases. Attention is focused on the case when one of the two fluid phases is of a microstructure nature (micropolar fluid). A linear slip boundary conditions, of Basset type, is used on the surface of the particle. The motion is considered in the limit of small Reynolds and capillary numbers where in this case the interface is of insignificant deformation. To solve the axisymmetric creeping motion of the micropolar fluid for velocity and microrotation components, a general solution is constructed from the superposition of some basic solutions in both cylindrical and spherical coordinate systems. Boundary conditions are satisfied first at the fluid–fluid interface using the Fourier Bessel transforms and then on the surface of the particle by the collocation scheme. Numerical results of the normalized drag force acting on the particle are obtained with good convergence for various values of the relevant parameters and presented both in graphical and tabular. Our results of the normalized drag force are compared with the available data in the literature for the limiting cases.

An analytical investigation is considered for the thermophoresis and photophoresis of a spherical aerosol particle embedded in a porous medium. The Knudsen number is assumed to be in the slip-flow regime. The porous medium is assumed to be homogenous, isotropic and the solid matrix is in thermal equilibrium with the fluid through the voids of the medium. At the surface of the particle, a temperature jump, a thermal creep, a viscous slip and thermal stress slip are considered in the analysis of motion. Expressions for thermophoretic and photophoretic velocities and forces are obtained. The effect of the Brinkman number characterizing the permeability of the medium is investigated as functions of the thermal properties of the porous medium and particle. The limiting cases of Stokes and Darcy's flows and no-slip case are discussed.