Capillary Pressure Oscillations

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

In this section the influence of the pressure in the capillary and the heat flux fluctuations on the stability of laminar flow in a heated capillary tube is analyzed. All the estimations performed in the framework of the general approach and developed in the previous section are kept also in the present cases. Below we will assume that the single cause for capillary pressure oscillations is fluctuations of the contact angle due to motion of the meniscus, whereas heat flux oscillations are the result of fluid temperature fluctuations only. 

Taking into account that ml = l + l and ui < ml, we anlve at the following relation for capillary pressure oscillations ... 

Chapter 11 consists of following Sect. 11.2 deals with the pattern of capillary flow in a heated micro-channel with phase change at the meniscus. The perturbed equations and conditions on the interface are presented in Sect. 11.3. Section 11.4 contains the results of the investigation on the stability of capillary flow at a very small Peclet number. The effect of capillary pressure and heat flux oscillations on the stability of the flow is considered in Sect. 11.5. Section 11.6 deals with the study of capillary flow at a moderate Peclet number. 

Unlike at adiabatic conditions, the height of the liquid level in a heated capillary tube depends not only on cr, r, pl and 6, but also on the viscosities and thermal conductivities of the two phases, the wall heat flux and the heat loss at the inlet. The latter affects the rate of liquid evaporation and hydraulic resistance of the capillary tube. The process becomes much more complicated when the flow undergoes small perturbations triggering unsteady flow of both phases. The rising velocity, pressure and temperature fluctuations are the cause for oscillations of the position of the meniscus, its shape and, accordingly, the fluctuations of the capillary pressure. Under constant wall temperature, the velocity and temperature fluctuations promote oscillations of the wall heat flux. 

Effect of Capillary Pressure and Heat Flux Oscillations... 

As discussed in more detail below, recent experiments convincingly showed that the flow oscillation in capillary extrusion of LPE is interfacial in nature due to a reversible coil-stretch transition at the melt/die wall boundary. Pressure oscillation phenomenon has also been reported in extrusion of other polymer melts. In particular, there are well-defined oscillations in controlled-rate capillary flow of PB that were found to arise from the same interfacial molecular instability [62]. 

An original method involves quadrupole oscillations of drops K The drop (a) in a host liquid (P) is acoustically levitated. This can be achieved by creating a standing acoustic wave the time-averaged second order effect of this wave gives rise to an acoustic radiation force. This drives the drop up or down in p, depending on the compressibilities of the two fluids, till gravity and acoustic forces balance. From then onwards the free droplet is, also acoustically, driven into quadrupole shape oscillations that are opposed by the capillary pressure. From the resonance frequency the interfacial tension can be computed. The authors describe the instrumentation and present some results for a number of oil-water interfaces. 

Harmonic interfacial disturbances can be induced by bringing an air bubble at the tip of a capillary in oscillation by a piezoelectric excitation system. In more modem Instmments pressure variation in the bubble is directly monitored by pressure transducers. Such a set-up allows the determination of Interfacial dynamic moduli. 

There is an extra oscillation module, based on direct measurements of the capillary pressure, which operates from 1 to 150 Hz. There is also an additional accessory for the PAT1 for low-frequency oscillations. The range of surface and interfacial tension is 1 to 1000 mN/m with a resolution of 0.1 mN/m. The instrument allows for transient relaxation measurements, using perturbations such as ramp, square pulse, or trapezoidal area changes. 

The first problem can be solved easily if the system volume, which is connected with the bubble, is big enough in comparison with the volume of the bubble separating from the capillary. In this case the system pressure is equal to the maximum bubble pressure. On the contrary, the use of an electric pressure transducer for measuring the bubble formation frequency presumes that pressure oscillations in the measuring system are distinct enough. This condition is fulfilled in systems with comparatively small volumes only. As shown by Mysels... 

In recent years, several theoretical and experimental attempts have been performed to develop methods based on oscillations of supported drops or bubbles. For example, Tian et al. used quadrupole shape oscillations in order to estimate the equilibrium surface tension, Gibbs elasticity, and surface dilational viscosity [203]. Pratt and Thoraval [204] used a pulsed drop rheometer for measurements of the interfacial tension relaxation process of some oil soluble surfactants. The pulsed drop rheometer is based on an instantaneous expansion of a pendant water drop formed at the tip of a capillary in oil. After perturbation an interfacial relaxation sets in. The interfacial pressure decay is followed as a function of time. The oscillating bubble system uses oscillations of a bubble formed at the tip of a capillary. The amplitudes of the bubble area and pressure oscillations are measured to determine the dilational elasticity while the frequency dependence of the phase shift yields the exchange of matter mechanism at the bubble surface [205,206]. 

The phenomena of uniform drop formation from a stream of liquid issuing from on orifice were noted as early as 1833 by Savart [2] and described mathematically by Lord Raleigh [3,4] and Weber [5]. In this type of system that is based on their observations, fluid under pressure issues from an orifice, typically 40-80 pm in diameter, and breaks up into uniform drops by the amplification of capillary waves induced onto the jet, usually by an electromechanical device that causes the pressure oscillations to propagate through the fluid. The drops break off from the jet in the presence of an electrostatic field called the charging filed, and thus acquire an electrostatic charge. The charged drops are directed to their... 

The interfacial tension, y, in the Gibbs adsorption equation is used for equilibrium conditions as bitumen components are adsorbed. Measurement techniques available are extensive. Some of these methods are duNouy ring, maximum bubble pressure, drop volume, Wilmhelmy plate, sessile drop, spinning drop, pendant drop, capillary rise, oscillating jet, and capillary ripples. These and many others are referenced extensively by Malhotra and Wasan (153). These authors also showed that there is no correlation between emulsion stability and interfacial tension. The nature of the film dominates stability. Some relationships between interfacial tensions and crude oil properties follow. 

Fast motions of a bubble surface produce sound waves. Small (but non-zero) compressibility of the liquid is responsible for a finite velocity of acoustic signals propagation and leads to appearance of additional kind of the energy losses, called acoustic dissipation. When the bubble oscillates in a sound field, the acoustic losses entail an additional phase shift between the pressure in the incident wave and the interface motion. Since the bubbles are much more compressible than the surrounding liquid, the monopole sound scattering makes a major contribution to acoustic dissipation. The action of an incident wave on a bubble may be considered as spherically-symmetric for sound wavelengths in the liquid lr >Ro-When the spherical bubble with radius is at rest in the liquid at ambient pressure, pg), the internal pressure, p, differs from p by the value of capillary pressure, that is... 

Since the neighbour slugs oscillate in phase opposition under shock load, pressure oscillations in the liquid plug should be restrained. Really the wave here turns out to be monotonous and its amplitude coincides practically with pressixre in the incident wave (Fig. 3). The applicability of Bq. (1) to the wave analysis is confirmed by experimental results ["l0. Fig.4 shows the comparison of these experiments conducted in a capillary horizontal tube ( 5 mm diam.) with air-water medium of slug structure, with the data of calculation by (1). 

Identify a reputable immiscible, two-phase flow simulator for use in this problem, and select a validated problem set (with available solutions) where consistent relative permeability and capillary pressure curves have been successfully tested against field data. Re-mn selected data sets. How do your solutions change as the absolute magnitude of capillary pressure change What happens when the capillary pressure vs. saturation curve is replaced by an approximate straight-line function What if the capillary pressure is set identically to zero In all three scenarios, note the position of the saturation discontinuity, its steepness, and the thickness of the front. Do your solutions oscillate in time If so, numerical instability is indicated. 

Fortran implementation. Equation 21-77 is easily programmed in Fortran. Because the implicit scheme is second-order accurate in space, thus rigidly enforcing the diffusive character of the capillary pressure effects assumed in this formulation, we do not obtain the oscillations at saturation shocks or the saturation overshoots having S > 1 often cited. The exact Fortran producing the results shown later is displayed in Figure 21-5 and in several function statements given later. For convenience, the saturation derivatives F (Sy ) and G (Sy ) are denoted FP and GP (P indicates prime for derivatives). 

---