Surfactant 123 Velocity field

Warholic MD, Schmidt GM, Hanratty TJ (1999) The influence of a drag-reducing surfactant on a turbulent velocity field. J Fluid Mech 388 1-20... 

At present no quantitative theory of drop motion in liquids at large Reynolds numbers exists. For bubbles, however, the situation is much simpler because gas viscosities are negligible with respect to liquid viscosities. For Re < 500 800, the bubble retains its spherical shape and the velocity field is that for an ideal liquid, (Levich, 1962) provided that no surfactant is present on the bubble surface. 

T), may depend on pol3nner or ion concentration, temperature etc is the dispersion of component i in the aqueous phase and q are the source/sink terms for component i through chemical reaction and injection/ production respectively. Polymer adsorption, as described by the term in equation (2), may feed back onto the mobility term in equation (1) through permeability reduction. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (5-8), in order to find the velocity fields for each of the phases present ie aqueous, oleic and micellar (if there is a surfactant present). If thermal effects are also to be included, then a heat balance equation is also required. The SCORPIO simulator (26, 27), which is used in our studies allows for all of these effects. 

The hydrodynamics of the experimental system can be described theoretically. Such approach is very important for correct interpretation of the experimental results, and for their extrapolation for the conditions not attainable in the existing experimental system. With the mathematical model the parametric study of the system is also possible, what can reveal the most important factors responsible for the occurrence of the specific transport phenomena. The model was presented in details elsewhere [2]. It was based on the equations of the momentum and mass transfer in the simplified two-dimensional geometry of the air-water-surfactant system. Those basic equations were supplemented with the equation of state for the phopsholipid monolayer. The resultant set of equations with the appropriate initial and boundary conditions was solved numerically and led to temporal profiles of the surface density of the surfactant, T [mol m ], surface tension, a [N m ], and velocity of the interface. Vs [m s ]. The surface tension variation and velocity field obtained from the computations can be compared with the results of experiments conducted with the LFB. 

Equations (25)-(27) along with the boundary conditions (29)-(32) must be solved subject to initial conditions at = 0 for the velocity field (which should be solenoidal for consistency with (27)) and the surface surfactant concentration F(z, 0). As can be seen the problem is difficult due to the nonlinear coupling present. An additional difficulty which is of interest here, is the possibility of jet pinching which manifests itself as a finite-time singularity of the system we will describe how results from the analysis of such events using asymptotic methods can be used in practical applications. Finally, note that if F = 0, we have the case of clean interfaces with constant surface tension. 

Let the velocity field be u = uer + ve + Oe = (u, v, 0) and denote the pressure by P. If we nondimesionalize lengths with the bubble radius a, velocities with f/oo> bulk concentration of surfactant with Coo, and pressure with then the dimensionless equations in the bulk, written in vector notation (for the component form of the equations see the [2], [18]), are the Navier-Stokes equations and a convection diffusion equation for the concentration ... 

Koch S, Schneider T, Kilter W (1998) The velocity field of dilute cationic surfactant solutions in a couette-viscometer. J Non-Newtonian Ruid Mech 78(l) 47-59... 

Equation (125) is coupled to the equations governing the liquid velocity field and the volume concentration of surfactant in the liquid to give the distribution of surface concentration of surfactant. In dimensionless form, the volume concentration equation is as follows ... 

The coupling between the stress field and the diffusion was first noticed and studied by Brochard and de Gennes [5] for polymer solutions far from the critical point. Relating to this, shear effects on complex fluids have recently attracted much attention because of its unusual nature known as Reynolds effect for example, a shear flow that intuitively helps the mixing of components induces phase separation in polymer solutions [6]. There are many examples of such effects in complex fluids such as surfactant systems, block copolymers, and charged colloidal systems. This is caused by the coupling between shear velocity fields and the elastic internal degrees of freedom of complex fluids. To explain this unique feature of polymer solution, there have been considerable theoretical efforts [5, 7-12]. 

An electric field-induced bending of gel makes a worm-like motion feasible [18]. A weakly crosslinked PAMPS gel in a surfactant solution bends toward the anode under dc electric fields. Both ends of the gel are placed on front and rear hooks and are then hung on a plastic ratchet bar. When a varying electric field of 10 V/cm and 0.5 Hz is applied, the gel moves forward in the solution with a bending motion at a velocity of 25 cm/min. 

The surface viscosity effect on terminal velocity results in a calculated drag curve that is closer to the one for rigid spheres (K5). The deep dip exhibited by the drag curve for drops in pure liquid fields is replaced by a smooth transition without a deep valley. The damping of internal circulation reduces the rate of mass transfer. Even a few parts per million of the surfactant are sometimes sufficient to cause a very radical change. 

McCready et al., 1986). The surface renewal theory can be made to fit the transfer data at fluid-fluid interfaces. The exception to this is bubbles with a diameter less than approximately 0.5 mm. Even though there is a fluid on both sides, surface tension causes these small bubbles to behave as though they have a solid-fluid interface. There is also some debate about this 1 /2 power relationship at free surfaces exposed to low shear, such as wind-wave flumes at low wind velocity (Jahne et al., 1987) and tanks with surfactants and low turbulence generation (Asher et al., 1996). The difficulty is that these results are influenced by the small facilities used to measure Kl, where surfactants wiU be more able to restrict free-surface turbulence and the impact on field scale gas transfer has not been demonstrated. 

Micellar catalysis is a broad field (Fendler and Fendler, 1975 Rathman, 1996 Rispens and Engberts, 2001), and caution is needed when using this term. In fact, whereas the broad term catalysis is justihed when referring to an increase of the velocity of reachon, this does not always mean that the velocity constant is increased (namely that there is a decrease of the specific activation energy). Rather, the velocity effect can be due to a concentration effect operated by the surface of the micelles. This is also the case for the autocatalytic self-reproduction of micelles discussed in the previous chapter, where the lipophilic precursor of the surfactant is concentrated on the hydrophobic surface of the fatty acid micelles (Bachmann et al., 1992), a feature that has given rise to some controversy (Mavelli and Luisi, 1996 Buhse etal, 1991 1998 Mavelli, 2004). 

Charged solutes in electrolyte solutions that are electrokinetically driven through channels with nanoscale widths exhibit unique transport characteristics that may enable rapid and efficient separations under a variety of physiological and environmental conditions. Many biomolecules, including DNA, proteins, and peptides, are charged or can be complexed with charged surfactant molecules. Manipulating the velocity of biomolecules by variation in flow pressure or electric fields in channels of nanoscopic widths will enable efficient separations that are not possible in micro- or macroscopic channels. 

Use the general representation of solutions for creeping flows in terms of vector harmonic functions to solve for the velocity and pressure fields in the two fluids, as well as the deformation and surfactant concentration distribution functions, at steady state. You should find... 

The theoretical description of a diffusion process of a surfactant to, or from, the surface of a floating bubble is impossible without information on the floating velocity and the hydrodynamic field around the bubble. The first of these quantities can be found comparatively easily experimentally, whereas the Navier-Stokes equation is used to define the hydrodynamic field around the floating bubble. A solution of the equation must satisfy all boundary conditions at the bubble surface. It should be stated that a general analytical solution of this... 

In contrast to free bubble surfaces, the role of boundary layers for the strong or complete retardation of a bubble surface is of substantial interest. Levich (1962), who introduced the notion of the hydrodynamic potential field of bubbles in the absence of surfactant pointed out that the velocity drop across the boundary layer is small. For completely or strongly retarded surfaces, the velocity distribution beyond the hydrodynamic boundary layer is a potential one. 

Equations are based on experimental observations of the effect of surfactant concentration, oil saturation, and velocity. The values in parentheses were used in an actual field simulation (44). 

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