Hydrodynamic theory with slip

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. 

Perfect adhesion, or freedom from slip at the boundary, is assumed in the modern hydrodynamic theory. Without such tangential adhesion the necessary fluid pressures could not be maintained. Imagine two lubricants of equal viscosity, differing with respect to the maximum tangential stress which can be supported at the boundary. Then if the operating conditions are such as to exceed the critical value for either or both lubricants there will be a difference in the frictional resistance, and we have another explanation of oiliness. 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

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 physics of motion in a layer adjacent to the solid surface is quite different from the bulk motion described by the Stokes equation. This generates effective slip at a microscopic scale comparable with intermolecular distances. The presence of a slip in dense fluids it is confirmed by molecular dynamics simulations [17, 18] as well as experiment [19]. The two alternatives are shp conditions of hydrodynamic and kinetic type. The version of the slip condition most commonly used in fluid-mechairical theory is a linear relation between the velocity component along the solid surface and the shear stress... 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

Experiments in thick channels - have established that hydrodynamic flows are generally slower than one would expect from theory. Current analytical models of the superhydrophobic effective slip are based on the idealized model of a heterogeneous surface with patches of boundary conditions and mostly neglect a number of dissipation mechanisms in the gas phase and at the interface. The effects associated with different aspects of the gas flow and meniscus curvature must be included in the models. Regardless of recent semianalytical and numerical analyses,the goal should remain to find simple analytical formulas, with as few adjustable parameters as possible, to fit experimental data. 

Drainage experiments conducted using the AFM and SEA, which have the ability to probe fluid films of nano- and molecular thicknesses, may be able to yield detailed information on the superhydrophobic slip. A lot of data is already available. " The recent theory of hydrodynamic interaction between disks shed some light on what could happen qualitatively, but cannot be used for a quantitative analysis of the hydrodynamic data obtained with the AFM and SEA. The same remark concerns the use of a theory of a film drainage between smooth hydrophobic surfaces, which is not fully applicable to quantify a superhydrophobic slip. We believe that a challenge for a theory would be to extend a theoretical modeling of experimentally relevant sphere versus plane geometry. This will open many possibilities for new experiments and could revolutionize the field. 

---