Thermophoretic Motion of a Sphere Parallel to an Insulated Plane

An analytical study is presented for the thermophoresis of a sphere in a constant applied temperature gradient parallel to an adiabatic plane. The Knudsen number is assumed to be small so that the fluid flow can be described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surface. A method of reflections is used to obtain the asymptotic formulas for the temperature and velocity fields in the quasisteady situation. The thermal insulated plane may be a solid wall (no-slip) and/or a free surface (perfect-slip). The boundary effect on the thermophoretic motion is found to be weaker than that on the axisymmetric thermophoresis of a sphere normal to a plane with constant temperature. In comparison with the motion driven by gravitational force, the interaction between the particle and the boundary is less significant under thermophoresis. Even so, the interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the thermophoretic motion of a particle parallel to a solid plane, the effect of the plane surface is to reduce the translational velocity of the particle. In the case of particle migration parallel to a free surface due to thermophoresis, the translating velocity of a particle can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Not only the translational velocity but also the rotational velocity of the thermophoretic sphere near the plane boundary is formulated analytically. The rotating direction of the particle is strongly dominated by its surface properties and the internal-to-external thermal conductivity. Besides the particle motion, the thickness of the thermophoretic boundary layer is evaluated by considering the thermophoretic mobility. Generally speaking, a free surface exerts less influence on the particle movement than a solid wall. Copyright 2000 Academic Press.     

Photophoresis of an aerosol sphere normal to a plane wall

A combined analytical-numerical study is presented for the quasisteady photophoretic motion of a spherical aerosol particle of arbitrary thermal conductivity and surface properties exposed to a radiative flux perpendicular to a large plane wall. The Knudsen number is assumed to be so small that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the surface of the radiation-absorbing particle. In the limit of small Peclet and Reynolds numbers, the appropriate equations of conservation of energy and momentum for the system are solved using a boundary collocation method and numerical results for the photophoretic velocity of the particle are obtained for various cases. The presence of the neighboring wall causes two basic effects on the particle velocity: first, the local temperature gradient on the particle surface is enhanced or reduced by the wall, thereby speeding up or slowing down the particle; second, the wall increases viscous retardation of the moving particle. The net effect of the wall can decrease or increase the particle velocity, depending upon the relative conductivity and surface properties of the particle as well as the relative particle-wall separation distance. In general, the boundary effect of a plane wall on the photophoresis of an aerosol particle can be quite significant in some situations. In most aerosol systems, the boundary effect on photophoresis is weaker than that on the motion driven by a gravitational field.     

Hydrodynamic interactions of evaporating droplet with droplet of nonvolatile substance

The hydrodynamic interactions of freely evaporating or growing droplet (suspended in gaseous medium) in the supersaturated vapor with the droplet of nonvolatile substance or spherical solid particle are theoretically studied with allowance for effects that are linear with respect to the Knudsen number. The process of interaction between the volatile droplet and the infinite plane surface of nonvolatile liquid or solid is considered as a limiting case. Numerical estimates of the velocities of the steady motion of evaporating droplets of water and castor oil are reported. For the droplet of water and spherical solid particle, the effect of the heat conductivity of the latter on the velocity of particle motion is considered. Analogous estimates are obtained for a water droplet that evaporates near the infinite solid surface of castor oil or solid. The effects of the droplet size and the heat conductivity of wall on the rate of the evaporation of water droplet are analyzed.     

Boundary effects on electrophoresis of a colloidal cylinder with a nonuniform zeta potential distribution

The electrophoretic motion of a long dielectric circular cylinder with a general angular distribution of its surface potential under a transversely imposed electric field in the vicinity of a large plane wall parallel to its axis is analyzed. The thickness of the electric double layers adjacent to the solid surfaces is assumed to be much smaller than the particle radius and the gap width between the surfaces, but the applied electric field can be either perpendicular or parallel to the plane wall. The presence of the confining wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases viscous retardation of the moving particle; (3) an electroosmotic flow of the suspending fluid may exist due to the interaction between the charged wall and the tangentially imposed electric field. Through the use of cylindrical bipolar coordinates, the Laplace and Stokes equations are solved analytically for the two-dimensional electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the quasisteady electrophoretic and angular velocities of the cylindrical particle are obtained. To apply these formulas, one has only to calculate the multipole moments of the zeta potential distribution at the particle surface. It is found that the existence of a plane wall near a nonuniformly charged particle can cause its translation or rotation which does not occur in an unbounded fluid with the same applied electric field.     

Boundary Effects on Diffusiophoresis of Cylindrical Particles in Nonelectrolyte Gradients

The diffusiophoretic motion of a long circular cylinder in a transversely imposed gradient of a nonionic solute near a large plane wall parallel to its axis is analyzed. The range of the interaction between the solute and the solid surfaces is assumed to be small relative to the particle radius and to the gap width between the particle and the wall, but the polarization effect of the mobile solute in the thin diffuse layers adjacent to the solid surfaces caused by the strong adsorption of the solute is incorporated. A normal flux of the solute and a slip velocity of the fluid at the outer edge of the diffuse layers are used as the boundary conditions for the fluid domain outside the diffuse layers. Through the use of cylindrical bipolar coordinates along with these boundary conditions, a set of transport equations governing this problem is solved in the quasisteady situation and the wall effects on the diffusiophoresis of the cylinder are computed for various cases. For the diffusiophoretic motion of a cylinder normal to a plane, the particle mobility decreases monotonically with the decrease of the distance of the particle axis from the wall. The stronger the polarization effect in the diffuse layer, the weaker the wall effect on the diffusiophoresis. The effect of the normal plane on the diffusiophoresis of a cylinder is much more significant than that for a sphere at the same separation. For the diffusiophoresis of a cylinder parallel to a plane, the boundary effect is a complicated function of the relevant parameters (not necessarily varies monotonically with the extent of separation) mainly due to the existence of a diffusio-osmotic flow caused by the tangential fluid velocity at the plane wall. Copyright 2000 Academic Press.     

Interactions between two bubbles on a hot or cold wall

A temperature gradient normal to a planar wall produces two-dimensional motion and aggregation or separation of bubbles on the hot or cold wall, respectively. The origin of the motion is fluid convection driven by the thermal Marangoni stress on the surface of the bubbles. Previous theories for the dynamics of two or more bubbles have been based on an analysis of flow about a single bubble and the resulting convection that entrains its neighbors. Here we extend the theory by solving the quasi-steady equations for the temperature and velocity fields for two bubbles. The result is a quantitative model for the relative velocity between two bubbles as a function of both the distance between them and the gap between each bubble and the surface. Interactions between the bubbles strongly increase the approach velocity, which is counter-intuitive because the hydrodynamic resistance increases as the bubbles approach each other. An asymptotic analysis indicates the thermocapillary force bringing them together or pushing them apart is singular in the separation when the bubbles are close to each other. The two-bubble theory agrees reasonably well with the experimentally measured velocities of pairs of bubbles on hot or cold surfaces, though it slightly overestimates the velocities.     

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions

An analytical study is presented for the quasi-steady electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity when the surface potentials are arbitrarily nonuniform. The applied electric field is constant, and the electric double layers adjacent to the solid surfaces are assumed to be much thinner than the particle radius and the gap width between the surfaces. The presence of the cavity wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases the viscous retardation of the moving particle; and (3) a circulating electroosmotic flow of the suspending fluid exists because of the interaction between the electric field and the charged wall. The Laplace and Stokes equations are solved analytically for the electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the electrophoretic and angular velocities of the particle are obtained. To apply these formulas, one has to calculate only the monopole, dipole, and quadrupole moments of the zeta-potential distributions at the particle and cavity surfaces. It is found that the contribution from the electroosmotic flow developing from the interaction of the imposed electric field with the thin double layer adjacent to the cavity wall and the contribution from the wall-corrected electrophoretic driving force to the particle velocities can be superimposed as a result of the linearity of the problem.     

Thermocapillary convection in double-layer fluid structures within a two-dimensional open cavity

Thermocapillary convection within a differentially-heated open rectangular cavity containing two immiscible liquid layers is considered in the absence of gravitational effects. The temperature and flow fields in the two layers are computed using domain mapping in conjunction with a finite-difference scheme on a staggered grid. The melt-encapsulant and air-encapsulant interfaces are allowed to deform, with the contact lines pinned on the solid boundaries. The presence of a free surface at the top leads to increased convection in the encapsulant phase while retarding thermocapillary flow in the melt. The intensity of thermocapillary convection in the encapsulated layer is reduced as the viscosity of the encapsulant is increased or the thickness of the encapsulant layer is decreased. Choosing an encapsulant with a greater sensitivity of interfacial tension to temperature (as compared to that of the melt phase) can almost completely suppress thermocapillary convection in the melt. Deformations of the melt-encapsulant interface in an open cavity are found to be larger than those in a closed cavity with a rigid top surface, due to higher pressure gradients realized in the encapsulant phase. In contrast to interface deformation behavior reported earlier for a double-layer system in a closed cavity, the shape of the melt-encapsulant interface is qualitatively similar for all values of the viscosity ratio, with the interface dipping into the melt near the cold wall, and into the encapsulant near the hot wall. For the double-layers considered in this study, a free surface at the top of the encapsulant layer was found to be more effective than a rigid top in reducing the intensity of thermocapillary convection in the melt.     

Analysis of particle-wall interactions during particle free fall

In this study, the vertical motion of a particle in a quiescent fluid falling toward a horizontal plane wall is analyzed, based on simplified models. Using the distance between the particle and wall as a parameter, the effects of various forces acting on the particle and the particle motion are examined. Without the colloidal and Brownian forces being included, the velocity of small particles is found to be approximately equal to the inverse of the drag force correction function used in this study as the particle approaches the near-wall region. Colloidal force is added to the particle equation of motion as the particle moves a distance comparable to its size. It is found that the particle might become suspended above or deposited onto the wall, depending on the Hamaker constant, the surface potentials of the particle and wall, and the thickness of the electrical double layer (EDL). For strong EDL repulsive force and weaker van der Waals (VDW) attractive force, the particle will become suspended above the wall at a distance at which the particle velocity is zero. This location is referred to as the equilibrium distance. The equilibrium distance is found to increase with increased in EDL thickness when a repulsive force barrier appears in the colloidal force interaction. For the weak EDL repulsive force and strong VDW attractive force case, the particle can become deposited onto the wall without the Brownian motion effect. The Brownian jump length was found to be very small. Many Brownian jumps would be required in a direction toward the wall for a suspended particle to become deposited.     

Motion of a Colloidal Sphere Covered by a Layer of Adsorbed Polymers Normal to a Plane Surface

A combined analytical–numerical study is presented for the slow motion of a spherical particle coated with a layer of adsorbed polymers perpendicular to an infinite plane, which can be either a solid wall or a free surface. The Reynolds number is assumed to be vanishingly small, and the thickness of the surface polymer layer is assumed to be much smaller than the particle radius and the spacing between the particle and the plane boundary. A method of matched asymptotic expansions in a small parameter λ incorporated with a boundary collocation technique is used to solve the creeping flow equations inside and outside the adsorbed polymer layer, where λ is the ratio of the characteristic thickness of the polymer layer to the particle radius. The results for the hydrodynamic force exerted on the particle in a resistance problem and for the particle velocity in a mobility problem are expressed in terms of the effective hydrodynamic thickness (L) of the polymer layer, which is accurate to O(λ²). The O(λ) term forLnormalized by its value in the absence of the plane boundary is found to be independent of the polymer segment distribution and the volume fraction of the segments. The O(λ²) term forL, however, is a sensitive function of the polymer segment distribution and the volume fraction of the segments. In general, the boundary effects on the motion of a polymer-coated particle can be quite significant.     

An empirically validated analytical model of droplet dynamics in electrowetting on dielectric devices

Explicit analytical models that describe the capillary force on confined droplets actuated in electrowetting on dielectric devices and the reduction in that force by contact angle hysteresis as a function of the three-dimensional shape of the droplet interface are presented. These models are used to develop an analytical model for the transient position and velocity of the droplet. An order of magnitude analysis showed that droplet motion could be modeled using the driving capillary force opposed by contact angle hysteresis, wall shear, and contact line friction. Droplet dynamics were found to be a function of gap height, droplet radius, surface tension, fluid density, the initial and deformed contact angles, contact angle hysteresis, and friction coefficients pertaining to viscous wall friction and contact line friction. The first four parameters describe the device geometry and fluid properties; the remaining parameters were determined experimentally. Images of the droplet during motion were used to determine the evolution of the shape, position, and velocity of the droplet with time. Comparisons between the measured and predicted results show that the proposed model provides good accuracy over a range of practical voltages and droplet aspect ratios.     

Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion

Brownian motion of a particle situated near a wall bounding the fluid in which it is immersed is affected by the wall. Specifically, it is assumed that an incompressible viscous fluid fills a half-space bounded by a plane wall and that the fluid flow satisfies stick boundary conditions at the wall. The fluctuation-dissipation theorem shows that the velocity autocorrelation function of the Brownian particle can be calculated from the frequency-dependent admittance valid locally. It is shown that the t(-3/2) long-time tail of the velocity relaxation function, valid in bulk fluid, is obliterated and replaced by a t(-5/2) long-time tail of positive amplitude for motions parallel to the wall and by a t(-5/2) long-time tail of negative amplitude for motions perpendicular to the wall. The latter finding is at variance with an earlier calculation by Gotoh and Kaneda.     

Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion in a viscous compressible fluid

Brownian motion of a particle situated near a wall bounding the fluid in which it is immersed is affected by the wall. Specifically, it is assumed that a viscous compressible fluid fills a half space bounded by a plane wall, and that the fluid flow satisfies stick boundary conditions at the wall. The fluctuation-dissipation theorem shows that the velocity autocorrelation function of the Brownian particle can be calculated from the frequency-dependent admittance valid locally. The admittance can be found from the linearized Navier-Stokes equations. The t(-3/2) long-time tail of the velocity relaxation function, valid in bulk fluid, is obliterated by the wall and replaced by a t(-5/2) long-time tail of positive amplitude for motions parallel to the wall and by a t(-5/2) long-time tail of negative amplitude for motions perpendicular to the wall. In both cases the amplitude of the t(-5/2) long-time tail turns out to be independent of fluid compressibility and bulk viscosity.     

Hydrodynamics of interaction of particles (including cells) with surfaces

The study of the phenomena related to the motion of particles flowing in the proximity of the wall is pursued for purely cognitive reason as well as for some important practical purposes in various fields of technology, biology and medicine.When small spherical rigid particles move in the direction parallel to the surface their velocity is smaller than that of the fluid and depends on the ratio of the distance from the wall to the particle radius. The velocity of a particle falling down in a vertical cylinder is maximal in an eccentric position. A sphere in contact with the wall remains stationary. Translational velocity of spherical rigid particles the dimension of which are comparable to that of the tube is only slightly dependent of their lateral position. The differences in the flow parameters of deformable particles in comparison with rigid ones depend on the particle and fluid viscosity coefficient. When the particles move perpendicularly toward the wall, their velocity decreases as the particle approaches the surface. The change of particle velocity is inversely proportional to the gap.There are several theories explaining the influence of the channel diameter on the suspension viscosity (sigma phenomenon); a modern approach is based on the analysis of rheological properties of suspensions. The explanations of the Fahraeus effect (i.e. the fact that the concentration of particles flowing in a tube linking two containers are smaller than that in the containers) are based on non-uniform particle distribution in a transverse cross section and on the differences of velocities of particles and medium. The deviation of the velocity profile of a suspension of rigid particles flowing through a tube from the parabolic shape (blunting) does not depend on the flow velocity; as concerns deformable particles, however, this effect is the smaller the greater is the flow velocity.When the Reynolds number for particles is greater than 10^-3, there appears a component of particle velocity perpendicular to the streamline direction.This phenomenon is the cause of the lateral migration of particles. Neutrally buoyant rigid particles migrate to a certain concentrical region situated between the tube axis and the wall (tubular pinch region). Deformable neutrally buoyant particles migrate towards the tube axis, and deformable non-neutrally buoyant particles may move either toward the tube axis or toward the wall.In the research on the influence of the flow delimiting surface on the motion of particles in suspension a considerable progress has recently been made.However, the phenomena in this field are extremely complex. At present, two main types of approach may be distinguished. On a microscopic level direct interactions between particles and surfaces are analyzed. A macroscopic approach consists in treating particle suspension as fluid, and overall influence of the surface on its properties are studied. A comprehensive theory linking these two levels has not yet emerged.     

Thermal Marangoni Convection of a Fluid Film Coating a Deformable Membrane

The thermal Marangoni instability of a fluid film coating a deformable membrane has been investigated by taking into account the deformation of the fluid free surface. Numerical calculations for different thermal boundary conditions are presented. The prestressed membrane is supposed to be very thin and therefore its behavior is similar to that of an isothermal fluid free surface with a surface tension but with a different mechanical boundary condition; that is, the fluid should stick on its surface and thus the fluid velocity is zero. An important assumption is that the membrane has no temperature dependence and therefore that only one Marangoni number exists for the free surface of the fluid. Numerical results are presented for stationary and oscillatory thermocapillary instability in both the sinuous and the varicose modes. It is shown that membrane deformation has important implications on the Marangoni instability of the fluid layer for positive and negative Marangoni numbers. Copyright 2001 Academic Press.     

Sliding Water Droplet on Oil Impregnated Surface and Dust Particle Mitigation

Self-cleaning of surfaces becomes challenging for energy harvesting devices because of the requirements of high optical transmittance of device surfaces. Surface texturing towards hydrophobizing can improve the self-cleaning ability of surfaces, yet lowers the optical transmittance. Introducing optical matching fluid, such as silicon oil, over the hydrophobized surface improves the optical transmittance. However, self-cleaning ability, such as dust mitigation, of the oil-impregnated hydrophobic surfaces needs to be investigated. Hence, solution crystallization of the polycarbonate surface towards creating hydrophobic texture is considered and silicon oil impregnation of the crystallized surface is explored for improved optical transmittance and self-cleaning ability. The condition for silicon oil spreading over the solution treated surface is assessed and silicon oil and water infusions on the dust particles are evaluated. The movement of the water droplet over the silicon oil-impregnated sample is examined utilizing the high-speed facility and the tracker program. The effect of oil film thickness and the tilting angle of the surface on the sliding droplet velocity is estimated for two droplet volumes. The mechanism for the dust particle mitigation from the oil film surface by the sliding water droplet is analyzed. The findings reveal that silicon oil impregnation of the crystallized sample surface improves the optical transmittance significantly. The sliding velocity of the water droplet over the thick film (~700 µm) remains higher than that of the small thickness oil film (~50 µm), which is attributed to the large interfacial resistance created between the moving droplet and the oil on the crystallized surface. The environmental dust particles can be mitigated from the oil film surface by the sliding water droplet. The droplet fluid infusion over the dust particle enables to reorient the particle inside the droplet fluid. As the dust particle settles at the trailing edge of the droplet, the sliding velocity decays on the oil-impregnated sample.     

Thermocapillary Flow and Aggregation of Bubbles on a Solid Wall

Theory suggests that thermocapillary flow about neighboring bubbles in liquids on hot walls pulls the bubbles together. A temperature gradient perpendicular to the wall establishes a surface tension gradient at the bubble-liquid interface, which in turn sustains a shear stress gradient that pumps adjacent fluid away from the wall. Neighboring bubbles are mutually entrained in this flow and also respond thermophoretically to lateral temperature gradients in the temperature near field. The theory predicts that the aggregation velocity scales with the temperature gradient, the radius of the bubbles, the derivative of the surface tension with respect to temperature, and the reciprocal of the liquid's viscosity. Bubble aggregation experiments under controlled conditions were performed to test the theory. Scaling the experimental bubble trajectories according to the theory substantially collapses all of the data onto a master curve when the interbubble separation is greater than 3 radii, which suggests that the theory is correct. Calculated velocities agree with the experimental results when hindrance of bubble motion due to the wall is included. Values for the parameter that describes the hindrance effect are obtained from fitting the data to the theory, from independent measurements, and from direct hydrodynamic calculation. The results of the three determinations agree within 15% of the possible range of the value of the parameter. Copyright 2000 Academic Press.     

Revisit of wall‐induced lateral migration in particle electrophoresis through a straight rectangular microchannel: Effects of particle zeta potential

Previous studies have reported a lateral migration in particle electrophoresis through a straight rectangular microchannel. This phenomenon arises from the inherent wall‐induced electrical lift that can be exploited to focus and separate particles for microfluidic applications. Such a dielectrophoretic‐like force has been recently found to vary with the buffer concentration. We demonstrate in this work that the particle zeta potential also has a significant effect on the wall‐induced electrical lift. We perform an experimental study of the lateral migration of equal‐sized polystyrene particles with varying surface charges under identical electrokinetic flow conditions. Surprisingly, an enhanced focusing is observed for particles with a faster electrokinetic motion, which indicates a substantially larger electrical lift for particles with a smaller zeta potential. We speculate this phenomenon may be correlated with the particle surface conduction that is a strong function of particle and fluid properties.     

On the evaporation and motion of a volatile droplet near a volatile liquid surface

A problem concerning the free evaporation or condensation growth of a droplet near an infinite planar surface of the same liquid is solved. The behavior of the droplet is considered at vapor temperature and concentration gradients preset at an infinite distance from it. The boundary conditions take into account effects that are linear with respect to the Knudsen number. Equations are derived for the rate of variations in the radius of the droplet and the velocity of its steady motion induced by nonuniform temperature and concentration of the vapor. Dependences of the rate of variations in the radius and the velocity of the steady motion of the droplet on the distance from the planar surface are presented for a droplet 1 ??m in radius suspended in air.