Liquid-solid hydrodynamics and slip

The following treatment, adapted from the work of Van Oene et al. is used to illustrate the assumptions and general conclusions of the surface chemical approach in which the no slip condition is ignored the hydrodynamic boundary condition that for a liquid moving over a solid surface there can be no motion of the liquid immediately adjacent to the liquid/solid interface. The work of Van Oene is similar to that of Schon-horn et al. and at the end of this section we will compare the results of the two investigations. 

The kinematics and dynamics boundary conditions at the interfaces close the hydrodynamic problem (l)-(2). On the solid-liquid boundary the non-slip boundary conditions are applied -the liquid velocity close to the particle boundary is equal to the velocity of particle motion. In the case of pure liquid phases the non-slip boundary condition is replaced by the dynamic boundary condition. The tangential hydrodynamic forces of the contiguous bulk phases, nx(P+Pb) n, are equal from both sides of the interface, where n is the unit normal of the mathematical dividing surface. The capillary pressure compensates the difference between the... 

Figure 1. Sketch of the hydrodynamic flow close to a superhydrophobic surface in the Fakir state in the limit of vanishing solid fraction. V is the liquid velocity far away from the solid-gas interface, L the lateral period of the roughness pattern and a the width of a single solid post. The local slip length b, which is 0 at the liquid-solid interface because of the viscous dissipation, tends to infinity at the liquid-gas interface.

Abstract. The lectures review the statics and dynamics of the gas-liquid-solid contact line, with the emphasis on the role of intermolecular forces and mesoscopic dynamics in the immediate vicinity of the three-phase boundary. We discuss paradoxes of the existing hydrodynamic theories and ways to resoluve them by taking account of intermoleculr forces, activated slip in the first molecular layer, diffuse character of the gas-liquid interface and interphase transport. 

The important hydrodynamic variables are the relative velocity. Vs, between the solids and the liquid (also know as slip velocity) and the rate of renewal of the liquid layer near the solid surface. The relative velocity, Vg, obviously varies from point to point within the vessel, and the average value is difficult to estimate. So, in practice, the relative velocity. Vs, is assumed equal to the free settling velocity, Vt. The renewal of the boundary layer depends on the intensity of turbulence around the solid particle as well as the convective velocity distribution in the vessel. 

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

Recent times have seen much discussion of the choice of hydrodynamic boundary conditions that can be employed in a description of the solid-liquid interface. For some time, the no-slip approximation was deemed acceptable and has constituted something of a dogma in many fields concerned with fluid mechanics. This assumption is based on observations made at a macroscopic level, where the mean free path of the hquid being considered is much smaller... 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

Film lubrication has the general assumption that the liquid layer directly in contact with a solid surface moves at the same speed as the surface itself. However, recent work has shown that simple liquids can slip against very smooth, lyopho-bic surfaces including monolayer-coated mica, sapphire and silicon [50]. Liquid slip was believed to cause (i) a significantly reduced hydrodynamic squeeze force... 

The process of convective diffusion to the liquid-liquid (liquid-gas) interface substantially differs from the diffusion to the fluid-solid interface. This is due to the difference between the hydrodynamic conditions on the interfaces. The fluid velocity on the surface of a solid is always zero by virtue of the no-slip condition. On the contrary, the interface between two fluids can move, and the tangential velocity on the interface differs from zero. 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

The propenies of the solid-liquid or liquid-liquid interfaces are also important. often so important that their influence overshadows any other factor. The contribution of hydrodynamic driven phenomena such as slip Row, secondary flow, edge and end effects, viscous heating, and inertia may also play an important role (see Sec. IV). Good experimental and calculation procedures should ensure either that these factors are absent or that the data are corrected to eliminate their contribution. These will be discussed in Sec. IV. In the following sections, the main physicochemical factors that influence rheological behavior and viscosity are discussed. For the sake of clarity, a distinction will be made between the factors that are related to physical propeities such as composition and particle size, and physicochemical aspects, especially inteifacial properties. 

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