Hydrodynamic Forces Between Fluid Boundaries

The situation is more complicated if the sphere approaching a planar surface is deformable, such as a bubble in a liquid or an oil drop in water. We can also think of the inverse situation a solid particle interacting with liquid surface such as a bubble or drop. A particle approaching a liquid-fluid interface will lead to a deformation of the interface. Then, there are three possibilities The particle is repelled by the interface and remains in the first liquid, it goes into the interface and forms a stable three-phase contact, or it crosses the interface and enters the fluid phase completely. The second fluid can be a gas. An example, is interaction of particles with bubbles in a liquid [685]. This interaction is of fundamental importance for flotation [591]. Another example is bubble interacting in a liquid. The hydrodynamic interaction between fluid interfaces is more complicated than between rigid interfaces because we have to take a deformation into account. 

In this section, we discuss the interaction between a drop and a bubble with a planar solid surface. In Chapter 7, we also include drops interacting with drops or bubbles and particles interacting with drops and bubbles. 

The drainage of liquid films between a solid-liquid and a liquid-fluid interface has been studied experimentally and theoretically [690-695, 702]. The formation of the dimple and the close distance at the rim hinders the liquid in the center to flow out As a consequence, surface forces indirectly influence the drainage time. A strong repulsion leads to a large film thickness in the rim and a fast drainage. Weak repulsion leads to a closely approaching rim and a slow drainage [697]. 

To understand hydrodynamic interactions, the boundary conditions need to be known. In contrast to solid surfaces, liquid-fluid interfaces are mobile. For a mobile interface between two fluid phase A and B, the no-slip boundary condition translates into = V . In particular in a direction tangential to a liquid-fluid interface, the velocities at both sides of the interface must be the same [702-705]  

the interface is assumed to be in 3c- and y-directions, and z is in normal threciion. Equation (6.69) tells us that the tangential velocities change continuously. In addition, the tangential stresses have to be balanced. In the absence of surface viscosity and surface tension gradients, this leads to the condition 

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In this chapter, we discuss the principles of how to calculate fluid flow. As we shall see, hydrodynamics is governed by a partial differential equation, the Navier -Stokes equation. It can be solved analytically only for a few simple cases. A systematic introduction into hydrodynamics is beyond the scope of this book. For an instructive introduction, we recommend Refs [625, 626]. New methods for the calculation of hydrodynamic interactions in dispersions are described in Ref. [627]. As one important example, we derive the hydrodynamic force between a rigid sphere and a plane in an incompressible liquid. Finally, hydrodynamic interactions between fluid boundaries are discussed. 

Compared to small molecules the description of convective diffusion of particles of finite size in a fluid near a solid boundary has to account for both the interaction forces between particles and collector (such as van der Waals and double-layer forces) and for the hydrodynamic interactions between particles and fluid. The effect of the London-van der Waals forces and doublelayer attractive forces is important if the range over which they act is comparable to the thickness over which the convective diffusion affects the transport of the particles. If, however, because of the competition between the double-layer repulsive forces and London attractive forces, a potential barrier is generated, then the effect of the interaction forces is important even when they act over distances much shorter than the thickness of the diffusion boundary layer. For... 

Hydrodynamic boundary layer — is the region of fluid flow at or near a solid surface where the shear stresses are significantly different to those observed in bulk. The interaction between fluid and solid results in a retardation of the fluid flow which gives rise to a boundary layer of slower moving material. As the distance from the surface increases the fluid becomes less affected by these forces and the fluid velocity approaches the freestream velocity. The thickness of the boundary layer is commonly defined as the distance from the surface where the velocity is 99% of the freestream velocity. The hydrodynamic boundary layer is significant in electrochemical measurements whether the convection is forced or natural the effect of the size of the boundary layer has been studied using hydrodynamic measurements such as the rotating disk electrode [i] and - flow-cells [ii]. 

Mazur (1982) and Mazur and van Saarloos (1982) developed the so-called method of induced forces in order to examine hydrodynamic interactions among many spheres. These forces are expanded in irreducible induced-force multipoles and in a hierarchy of equations obtained for these multipoles when the boundary conditions on each sphere were employed. Mobilities are subsequently derived as a power series-expansion in p 1. In principle, calculations may be performed to any order, having been carried out by the above authors through terms of 0(p 7) for a suspension in a quiescent fluid. To that order, hydrodynamic interactions between two, three, and four spheres all contribute to the final result. This work is reviewed by Mazur (1987). 

The mechanisms of particle removal have been studied in the past few years. Reports show that the particles adhere to a surface primarily by van der Waals forces, electrostatic attraction, or capillary action.2 The cleaning is by hydrodynamic lubrication. The thickness of the hydrodynamic fluid layer, as estimated, was around 3.7 pm.1 On the contrary, numerical analysis concluded that the lift force in the hydrodynamic boundary layer of fluid was too small to lift particles off the surface.3 The possible removal force comes likely from the drag force between the brush and the wafer surface. Major... 

When the distance h between a spherical particle of radius a and a solid boundary becomes sufficiently small (hla 1), hydrodynamic interactions between the particle and wall hinder the Brownian motion of the particle. Such effects are critical to near-wall measurements and the accuracy of velocimetry techniques, which rely on an accurate accounting of particle displacements to infer fluid velocity. By applying the evanescent wave-based 3D PTV techniques to freely suspended fluorescent particles, anisotropic hindered Brownian motion has been quantified for particle gap sizes hla 1 with 200 nm diameter tracers [8] and hla 1 with 3 pm diameter tracers [9]. These results confirm the increase of hydrodynamic drag when a particle approaches a solid boundary, and such correction shall be applied to not only Brownian motion but also other translational motion of particles where the drag force is of concern. 

Although the momentum exchange between fluid and solid occurs instantaneously at the half time step, in calculating U,(t -h h) we have made the assumption that the hydrodynamic force is distributed over the time step. We actually attempted to derive an update for the velocity assuming that the hydrodynamic force acts over a very small fraction of the time step, but this has not led to a sensible result as yet. It is not entirely clear if the assumption that the hydrodynamic force acts over the whole time step is valid, and does not, for example, produce an artificial dissipation. To resolve this question will require a detailed analysis of the fully coupled system, along the lines given in Sect. 4.5 for the simpler case of frictional coupling. A similar analysis for solid-fluid boundary conditions is an open area for further research. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

External mass tran.sfer between the external surfaces of the adsorbent particles and the surrounding fluid phase. The driving force is the concentration difference across the boundary layer that surrounds each particle, and the latter is affected by the hydrodynamic conditions outside the particles. 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

Just as the hydrodynamic boundary layer was defined as that region of the flow where viscous forces are felt, a thermal boundary layer may be defined as that region where temperature gradients are present in the flow. These temperature gradients would result from a heat-exchange process between the fluid and the wall. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

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