Finite Surface Viscosity

The Reynold s model and equation have also been modified to account for surface effects on film drainage rate [29]. To accomplish this, the assumption (boundary condition) of a surface velocity of zero must be relaxed. This change also decreases the surface viscosity from an infinite to a finite value. This process yields Equation 2. Allowing for a finite surface velocity increases the film drainage rate from that which would be expected under the Reynold s conditions. 

When there is finite surface viscosity, the surface of the film is mobile. Lubrication theory for our vertical draining film gives a system of two partial differential equations describing the evolution of the film surface k z,t) and the velocity of the fluid at the surface w (z,t). These equations are... 

The factor in Eq. (36) accounts for the effect of finite surface viscosity (j/s) and has been computed by Desai and Kumar [61] as a function of the inverse of the dimensionless surface viscosity (y = 0.4387/i /5jiIt must be emphasized that their results for the calculation of Cy are valid only for foams because they neglected the viscosity of the dispersed phase. Equation (36) can be used for liquid liquid concentrated emulsions only when the surfaces are immobile (i.e., when Cy = 1). The pressure gradient (dp/dz) can be computed as follows. 

How must the expressions derived in the sections above be modified to take into account the variation in rj and the finite distance over which it increases The answer is that rj — the viscosity within the double layer —must be written as a function of location. Our objective in discussing this variation is not to examine in detail the efforts that have been directed along these lines. Instead, it is to arrive at a better understanding of the relationship between f and the potential at the inner limit of the diffuse double layer and a better appreciation of the physical significance of the surface of shear. 

Viscosity and density of the component phases can be measured with confidence by conventional methods, as can the interfacial tension between a pure liquid and a gas. The interfacial tension of a system involving a solution or micellar dispersion becomes less satisfactory, because the interfacial free energy depends on the concentration of solute at the interface. Dynamic methods and even some of the so-called static methods involve the creation of new surfaces. Since the establishment of equilibrium between this surface and the solute in the body of the solution requires a finite amount of time, the value measured will be in error if the measurement is made more rapidly than the solute can diffuse to the fresh surface. Eckenfelder and Barnhart (Am. Inst. Chem. Engrs., 42d national meeting, Repr. 30, Atlanta, 1960) found that measurements of the surface tension of sodium lauryl sulfate solutions by maximum bubble pressure were higher than those by DuNuoy tensiometer by 40 to 90 percent, the larger factor corresponding to a concentration of about 100 ppm, and the smaller to a concentration of 2500 ppm of sulfate. 

These correspond, respectively, to polymer or electrolyte entrapped within surface features, the polymer film, and the solution. The first of these is a minor effect when using polished crystals the surface mechanical impedance of this contribution is Z, = wp where j = V-l, o> = 2nf0, and p is the areal mass density of the entrapped material. For finite and semiinfinite viscoelastic layers, the surface mechanical impedance is given by Z, = (GPf)m and Zv = (Gpf)1/2 tanh(y/i/), respectively, where prf and hf are the film density and thickness and y = /w(p/G)l/2. For the solution, Zs = (tapsT]J2)m (1 + j), where p, and tj, are the density and viscosity of the solution. When rigid mass, finite viscoelastic film and semi-infinite liquid loadings are all present, as in the experiment of Fig. 13.7, one can show that [42] ... 

Indeed, the shear stress at the solid surface is txz=T (S 8z)z=q (where T (, is the melt viscosity and (8USz)z=0 the shear rate at the interface). If there is a finite slip velocity Vs at the interface, the shear stress at the solid surface can also be evaluated as txz=P Fs, where 3 is the friction coefficient between the fluid molecules in contact with the surface and the solid surface [139]. Introducing the extrapolation length b of the velocity profile to zero (b=Vs/(8vy8z)z=0, see Fig. 18), one obtains (3=r bA). Thus, any determination of b will yield (3, the friction coefficient between the surface and the fluid. This friction coefficient is a crucial characteristics of the interface it is obviously directly related to the molecular interactions between the fluid and the solid surface, and it connects these interactions at the molecular level to the rheological properties of the system. 

A three dimensional turbulent flow field in unbaffled tank with turbine stirrer or 6-paddle stirrer was numerically simulated by the method of finite volume elements [80], whereas in the case of free surface the vortex profile was also determined using iterative techniques. The prediction of the velocity and turbulence fields in the whole tank and the stirrer power was compared with literature data and their own results. Of the two simulation techniques used, turbulent eddy-viscosity/zc-e turbulence model and the DS model (differential 2. order shear stress), only the latter produced satisfactory results. In particular it proved that fluctuating Coriolis forces have to be taken into account by source terms in the transport equation for the Reynolds shear stress. 

An alternative point of view is that vorticity accumulates at the rear of the body, which leads to a large recirculating eddy structure, and as a consequence, the flow in the vicinity of the body surface is forced to detach from the surface. This is quite a different mechanism from the first one because it assumes that the primary process leading to separation is the production and accumulation of vorticity rather than the local dynamics within the boundary layer.25 However, viscosity still plays a critical role for a solid body in the production of vorticity. In fact, for any finite Reynolds number, there is probably some element of truth in both explanations. Furthermore, it is unlikely that experimental evidence (or evidence based on numerical solutions of the complete Navier Stokes equations) will be able to distinguish between these ideas, because such evidence for steady flows will inevitably be limited to moderate Reynolds numbers. 

Equation 62 is very attractive for the study of adhesion between polymers and sohd surfaces, since it allows for the determination of the viscoelastic constants of the adhesive in the immediate vicinity of the contact. Unfortunately, the QCM does not work well with semi-infinite media when the viscosity, r], is larger than about 50 cP. The bandwidth in this case is too large. Most polymers exceed this limit. If, however, the contact area can be confined to a small spot in the center of the crystal the measurement becomes feasible [74]. Such a small contact area can, for instance, be estabhshed with a JKR tester [75]. The area of contact can be determined by optical microscopy. Of course, this kind of sample is laterally heterogeneous and the apphcabihty of simple models may be questioned. Experiment shows that the finite contact area can be reasonably well accounted for by modifying as ... 

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

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