Surfactant transport equation

In general, the surfactant is distributed along the interface by a combination of convection and diffusion, as well as transport to and from the interface from the bulk solvents. However, in many cases, the solubility of a surfactant in the two solvents is very low, and a good approximation is that the transport from the solvents is negligible. In this case, it is said that the surfactant is an insoluble surfactant, and the total quantity of surfactant on the interface is conserved. We have notpreviously derived a bulk-phase conservation equation to describe the transport of a solute in a solvent. Hence in this section we adopt the insoluble surfactant case, and follow Stone50 in deriving a surfactant transport equation that relates only to convection and diffusion processes on the interface. 

Various functional forms of FM have been investigated during this study explicit details will be given later. The primary advantage of this foam modelling approach lies in its simplicity use is made of the phase flow equations plus the surfactant transport equation. In addition, the functional form of FM should mimic suitable foaming conditions. 

To take into account the influence of surfactant adsorption. Equations 5.240 and 5.241 are to be complemented with transport equations for each of the species (A = 1,2,...,AT) in the bulk... 

We have previously written an expression for j n in Eq. (2-150), but this expression is in terms of the local bulk concentration evaluated at the interface, c, and thus to determine c we would need to solve bulk-phase transport equations. We will not pursue that subject here. However, when we use this material to solve flow problems, we will consider several cases for which it is not necessary to solve the full convection-diffusion equation for c. We will see that the concentration of surfactant tends to become nonuniform in the presence of flow -i.e., when u n and u v are nonzero at the interface. This tendency is counteracted by surface diffusion. When mass transfer of surfactant to and from the bulk fluids is added, this will often tend to act as an additional mechanism for maintenance of a uniform concentration T. This is because the rate of desorption from the interface will tend to be largest where T is largest, and the rate of adsorption largest where T is smallest. 

H. A. Stone, A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface, Phys. Fluids A 2, 111-12 (1990). 

The description of the adsorption kinetics of a surfactant mixture is made in an analogous way as for a single surfactant solution. Instead of Eq. (4.9) a set of transport equations (4.14) has to be used, one for each of the r different surfactants. The initial and boundary conditions are defined for each component, in analogy to Eqs (4.11), (4.15) and (4.17). A system of r integral equations result, either in the form... 

The present state of resecirch allows to describe the adsorption kinetics of surfactants at liquid interfaces in most cases quantitatively. The first model for interfaces with constant area was derived by Wend Tordai [3]. It is based on the assumption that the time dependence of interfacial tension, which is directly related to the interfacial concentration T of adsorbed molecules via an equation of state, is mainly caused by the surfactant transport to the interface. In the absence of any external influences this transport is controlled by diffusion and the result, the so-called diffusion controlled adsorption kinetics model, has the following form... 

The results of the preceding section allow us now to move on to describe the surfactant transport from the depth of the bulk phase to the interface or in the opposite direction. If any adsorption barriers are absent, this process determines the adsorption and desorption rates. The main step in the solution of this problem consists in the formulation of the surfactant diffusion equations for micellar solutions. The problem of surfactant diffusion to the interface was considered and solved for the first time by Lucassen for small perturbations [94]. He used the simplified model (5.146) where micelles were assumed to be monodisperse and the micellisation process was regarded as consisting of one step. Later Miller solved numerically the problem of adsorption on a fresh liquid surface using the same assumptions [146], Joos and van Hunsel applied also the same model to the interpretation of dynamic surface tension of... 

Surfactant transport is also important in the vertical draining film surfactant concentration gradients may develop, which may in turn strongly affect the fluid flow via the Marangoni effect. When surfactant transport is considered, lubrication theory then gives three nonlinear partial differential equations for the free surface shape k(z,t), the surface velocity w z,t) and the surface concentration of surfactant (, ). The mathematical problem to be solved for these dependent variables is ... 

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. 

This condition should be applied to the immobilized portion of the bubble surface as discussed by McLaughlin [68]. As shown by McLaughlin [68], an immobilized surfactant cap forms on a bubble in steady motion provided that convective transport of the surfactant on the bubble dominates diffusion and that the surfactant is sparingly soluble. More generally, one should compute the surface concentration of surfactant. The transport equation for the surface concentration of surfactant is given by Stone [69] and Stone and Leal... 

The bulk concentration of surfactant in the liquid, C, the maximum surface concentration of surfactant, T, and the surface tension of clean water, jq, may be used to make the equations governing the surfactant dimensionless. The Peclet number in the bulk liquid, Pe, and on the surface of the bubble, Pe, take the following forms Pe = 2UtrelD and Pe = 2Uir lD, where D, D, and denote the diffusivity of the surfactant in the liquid, the dilfusivity on the surface of the bubble, and the equivalent spherical radius of the bubble, respectively. In dimensionless form, the transport equation for the surface concentration of surfactant, T, takes the following form ... 

The volume concentration of surfactant in the liquid satisfies the following transport equation ... 

Equation 4.288 provides a boundary condition for the normally resolved flux, From another viewpoint. Equation 4.288 represents a 2D analogue of Equation 4.287. The interfacial flux, ji, can also contain contributions from the interfacial molecular diffusion, electrodiffusion, and thermodiffusion. A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface is given by Brenner and Leal [734-737], Davis et al. [669], and Stone [738]. If the molecules are charged, the bulk and surfaces electrodiffusion fluxes can be expressed in the form [651,739,740] ... 

Abstract. This article describes a hydrodynamic model of collaborative flnids (oil, water) flow in porons media for enhanced oil recovery, which takes into account the influence of temperature, polymer and surfactant concentration changes on water and oil viscosity. For the mathematical description of oil displacement process by polymer and surfactant injection in a porous medium, we used the balance equations for the oil and water phase, the transport equation of the polymer/surfactant/salt and heat transfer equation. Also, consider the change of permeabihty for an aqueous phase, depending on the polymer adsorption and residual resistance factor. Results of the numerical investigation on three-dimensional domain are presented in this article and distributions of pressure, saturation, concentrations of poly mer/surfactant/salt and temperature are determined. The results of polymer/surfactant flooding are verified by comparing with the results obtained from ECLIPSE 100 (Black Oil). The aim of this work is to study the mathematical model of non-isothermal oil displacement by polymer/surfactant flooding, and to show the efficiency of the combined method for oil-recovery. 

Polymer, surfactant and salt transport equations can be written as [1] ... 

While fluid-structure interactions clearly create damaging mechanical stresses, surfactant physicochemical interactions are critical to the protection of the airway epithelium. In order to investigate this important interaction during airway reopening, it is necessary to expand the models above by introducing the biophysical properties of surfactant located in the hning fluid. This includes surfactant transport in the bulk phase by convection and diffusion, and the sorption kinetics between the bulk phase and the air-liquid interface. The flow fields in the bulk fluid and sorption kinetics at the interface both contribute to the rate and manner in which surfactant adsorbs to the interface where it locally decreases surface tension. These interactions are depicted in Figure 16.8, and defined below by Equations 16.10 and 16.11. 

Let C(x, t) and a(x, t) be the local surfactant concentrations in the bulk solutiou and in the adsorbed state on the capillary surface. A constant surfactant concentration C(0, t) = Cq< CMC is kept at the capillary inlet. In this case the surfactant transport in the filled portion of the capillary obeys Equation 5.13. 

Numerous investigators have elaborated the theory by including bulk transport and surface diffusion of the surfactant molecules [45], or the effect of shear and dilatation surface viscosity [49], The general conclusion is that the rate of thinning predicted by Reynolds equation is too low (the reader may find the details in ref. [237]). Hopefully our case is different and we do not have to enter into these considerations (see Section 3.2). 

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