Laplace pressure difference

To understand drainage we have to discuss the pressure inside the liquid films. At the contact line between liquid films, a channel is formed. This is called the Plateau border. Due to the small bending radius (rP in Fig. 12.18), there is a significant Laplace pressure difference between the inside of the compartment and the liquid phase. The pressure inside the liquid is significantly smaller than in the gas phase. As a result, liquid is sucked from the planar films into the Plateau s border. This is an important effect for the drainage of foams because the Plateau borders act as channels. Hydrodynamic flow in the planar films is a slow process [574], It is for this reason that viscosity has a drastic influence on the evolution of a foam. Once the liquid has reached a Plateau border the flow becomes much more efficient. The liquid then flows downwards driven by gravitation. 

A possible reason is shown schematically in Figure 15.16. The figure at the left shows the typical pore structure of an SPG membrane the figures at the right-hand side show a schematized version of it. Once the oil emerges from a pore in the membrane, there is a change in Laplace pressure. When the oil is forced into the membrane, one has to overcome the Laplace pressure difference, equal to a/Rp (note that the Laplace pressure in a cylinder is a factor two smaller than in a sphere). The same is true, but reversed, at the downstream side of the membrane. Here, the oil emerges from a pore as a droplet. There is a moment that the droplet has a Laplace pressure difference with the continuous phase that is smaller than that of the pore. At that moment there is a... 

Figure 15.16. Schematic representation of the processes inside an SPG membrane. As soon as an oil droplet emerges, it will have a different Laplace pressure difference with the continuous phase. At a certain droplet size, it will be lower than the Laplace pressure inside the oil phase in the pore. When the pores are highly interconnected, the oil in the pore can be replaced by the continuous phase, leading to spontaneous snap-off.

When will this spontaneous snap-off take place If the droplet has radius Rj (thus giving rise to Laplace pressure difference with the continuous phase of la/Rj), the point at which the oil will start to have a pressure gradient from pore to droplet is ... 

Various direct microfluidic emulsification geometries are discussed in literature, e.g., (straight-through) microcharuiels and T- and Y-junctions (see Eig. 1). Two droplet formation mechanisms can be distinguished one uses Laplace pressure differences for spontaneous droplet generation and the other uses shear to form droplets. 

The other one describes the flow between neck and droplet based on the local Laplace pressure differences and is as follows ... 

Here only a condensed scaling analysis from Aussilous and Quere [151] is given. The front of the bubble may be regarded as spherical with radius r and the Laplace pressure difference across the gas-liquid interface is given by Ap = provided... 

Water collection from fog is a system to extract a discrete phase from a continuous phase. There is another case where the extraction of a discrete phase from a continuous phase has practical application value, which is oil/water separation. The separation device mentioned above was founded on a mesh surface, and the lowest-sized oil droplets that could be collected strongly depended on the size of the mesh in use. Li et al. prepared a separation device that could separate oil droplets much smaller in size than those separated by the mesh separator. The idea is similar to the fog collection mechanism, i.e., the oil droplets are captured and transported by the spine arrays. PDMS arrays on a planar substrate were prepared via puncturing mold replication. As shown in Figure 16.9B, since the oil droplets are carried by the water flow and subjected to the tilted PDMS cone arrays, they collide on the cones, accumulate, and are transported to the cone base via the Laplace pressure difference. Using the separation apparatus shown in Figure 16.9B, the oil droplets can be continuously separated until clean water is obtained. 

Consider now the dynamical deposition of a film caused by a relative motion between the solid and the liquid (Figure 5.11). For low capillary numbers, the thickness of the film results from a compromise similar to the one we have described in the context of a plate. Specifically, the viscous force in the meniscus is balanced out by the capillary force which tends to bring the liquid back into the reservoir. Remarkably, in both geometries considered here, the expression giving the Laplace pressure difference between the film and the reservoir takes on the same form when the thickness is small. In this limit, the film turns out to be in state of overpressure by an amount j/b with respect to the reservoir. 

One can now also investigate other ensembles for the single vesicle adhered to a substrate such as the ensemble in which besides the volume also the surface area A is kept constant. As we discussed previously, this is the ensemble for which Seifert and Lipowsky constructed their phase diagrams of the vesicle unbinding transition. Another ensemble one might wish to consider is the constant pressure ensemble, for which Q is the appropriate free energy, in which instead of the vesicle volume the Laplace pressure difference Ap is... 

This result is obtained by balancing pressures in points M and N of Fig. 8. In equilibrium the pressure in N is equal to the pressure in the outer fluid, Pq, and the pressure in M is pgh + AP + Pq where AP is the Laplace pressure difference. The order of magnitude can be obtained by assuming that the meniscus mean radius of curvature is of the order of the pressure difference is AP = — and h =... 

If instead of two plates a cylindrical tube of diameter d is immersed in the same liquid, the meniscus will take the shape of a spherical cup of radius d/2. So, the Laplace pressure difference across the interface is and the capillary rise inside the... 

Analyzing the Laplace pressure difference in a film sinusoidally perturbed (given by the two principal radii of curvature) it results that the pressure in a valley is larger than in a crest if the wavelength of the perturbation (A.) is larger than Ina, where a is the fiber radius. In this case, the liquid flows from the valley to the crest and the perturbation is increased. A first order instability analysis shows that the fastest growing perturbation has a wavelength... 

In Taylor flow, a thin film is observed near the gas bubble which prohibits the contact between gas and solid. This phenomenon is similar to the role of lubricating viscous fluid in bearing for preventing the solid-solid contact. The spherical region at the front has a constant radius r with the Laplace pressure difference across the interface AP = la jr for small thickness of the film (6 r). The curvature in the axial direction vanishes for the flat film region, and the Laplace... 

Because Ko is proportional to the concentration of the solute in the dispersed phase, we can visualize the effect of the concentration on the evolution of the system just by considering the flux as a function of n, for different values of Ko. When Kq is equal to zero, we have the disappearance of the smaller bubble or droplet due to the Laplace pressure difference. When we take a nonzero value for Ko, we find a zero flux for a nonzero radius n, the equilibrium radius. 

The osmotic pressure difference APosm that drives the flux of the soluble gas opposing the flux driven by the Laplace pressure difference can be written as... 

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