Capillary Flow Pattern

We deal here with the stability of flow in a heated capillary tube when liquid is evaporating on the meniscus. The capillary, as shown in Fig. 11.1, is a straight vertical pipe with diameter d and length 1. The wall heat flux is uniform = const. The thermal conditions on the capillary inlet and outlet are  

Hereafter, the subscripts G and L denote vapor and liquid, respectively, and in and 0 denote inlet and outlet of the capillary tube, respectively. 

These conditions correspond to a certain design of cooling system, namely, a micro-channel with cooling inlet and adiabatic outlet (Yarin et al. 2002). 

The wall heat flux is the cause for the liquid evaporation, and perturbation of equilibrium between the gravity and capillary forces. It leads to the offset of both phases (heated liquid and its vapor) and interface displacement towards the inlet. In this case the stationary state of the system corresponds to an equilibrium between gravity, viscous (liquid and vapor) and capillary forces. Under these conditions the stationary height of the liquid level is less than that in an adiabatic case 

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. 

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Serizawa et al. (2002) studied experimentally, through visualization, the two-phase flow patterns in air-water two-phase flows in round tubes. The test section for air-water experiments consisted of a transparent silica or quartz capillary tube with circular cross-section positioned horizontally. The two-phase flow was realized through a mixer with different designs, as shown in Figs. 5.4 and 5.5. The air was injected into the mixer co-axially while water was introduced peripherally. 

As demonstrated in Fig. 5.7, the result indicates that two-phase flow patterns observed in a 100 pm quartz tube are almost similar to those observed in a 25 pm silica capillary tube with several exceptions. One such exceptions is that in slug flow encountered at low velocities, small liquid droplets in a gas slug stick to the tube wall (Fig. 5.8). This fact is evidence that no liquid film exisfs befween fhe gas slug and the tube wall. 

The liquid alone pattern showed no entrained bubbles or gas-liquid interface in the field of view. The capillary bubbly flow, in the upper part of Fig. 5.14a, is characterized by the appearance of distinct non-spherical bubbles, generally smaller in the streamwise direction than at the base of the triangular channel. This flow pattern was also observed by Triplett et al. (1999a) in the 1.097 mm diameter circular tube, and by Zhao and Bi (2001a) in the triangular channel of hydraulic diameter of 0.866 mm. This flow, referred to by Zhao and Bi (2001a) as capillary bubbly... 

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. 

Another numerical study of free-surface flow patterns in narrow channels was conducted by Yang et al. [185]. They considered the flow of bubbles of different size driven by body forces, for example the rising of bubbles in a narrow capillary due to buoyancy. The lattice Boltzmann method [186] was used as a numerical scheme... 

For the catalytic oxidation of malonic acid by bromate (the Belousov-Zhabotinskii reaction), fimdamental studies on the interplay of flow and reaction were made. By means of capillary-flow investigations, spatio-temporal concentration patterns were monitored which stem from the interaction of a specific complex reaction and transport of reaction species by molecular diffusion [68]. One prominent class of these patterns is propagating reaction fronts. By external electrical stimulus, electromigration of ionic species can be investigated. 

Flow inside the capillary membranes, depicted in the lower half of the plot and indicated by positive velocities, shows a regular pattern. The single capillaries are resolved, and flow inside each capillary possesses almost identical maximum velocities. Flow outside the membranes (upper half, negative velocities) reveals a different pattern. Flere, the different flow characteristics between the SMC and SPAN modules become distinct. Obviously, the capillaries in the SMC module are not packed in a regular manner. Large spaces in between the capillaries cause an irregular flow pattern in the dialysate-side with a maximum velocity of about -15 mm s-1 (Figure 4.6.2(a)), which is comparable to the maximum velocity in... 

As in the case of capillary-tube units, the shear rate (rotational speed) should be variable over wide ranges (10- to 1000-fold) and baffles or other obstructions which could interfere with the laminar-flow pattern must be absent. Since the fluid is sheared for long periods of time in these instruments, temperature control is much more critical, especially in the case of high-consistency materials, for which temperature rises of over 20°C. (W2) have been recorded. Weltmann and Kuhns (W5) subsequently presented an erudite mathematical analysis of the temperature distribution within the layers of sheared fluid. 

Ballman and co-workers have used carbon particles to determine flow patterns for polystyrene melts in plate-cone and capillary viscometers (70). Complex patterns, rather than the simple flow expected, were observed for high molecular weight samples. These may have been caused, however, by differences in viscosity between adjacent layers of pure melt and melt with suspended particles. 

Thulasidas TC, Abraham MA, Cerro RL. Flow patterns in liquid slugs during bubble-train flow inside capillaries. Chem Eng Sci 1997 52 2947-2962. 

P. J. A. Kenis, R. F. Ismagilov, and G. M. Whitesides, Microfabrication inside capillaries using multiphase laminar flow patterning, Science 285, 83—85 (1999). 

Rose examined the flow pattern in a capillary tube where one immiscible liquid displaces another one. In the front end of the displacing liquid the flow pattern is one he termed fountain flow, and in the other reverse fountain flow. In polymer processing the significance of the former was demonstrated in the advancing melt front in mold filling (see Chapter 13). 

Fig. 12.4, the melt is forced into a converging flow pattern and undergoes a large axial acceleration, that is, it stretches. As the flow rate is increased, the axial acceleration also increases, and as a result the polymer melt exhibits stronger elastic response, with the possibility of rupturing, much like silly putty would, when stretched fast. Barring any such instability phenomena, a fully developed velocity profile is reached a few diameters after the geometrical entrance to the capillary. 

Fig. 12.15 The ratio of entrance pressure drop to shear stress at the capillary wall versus Newtonian wall shear rate, T. . PP , PS O, LDPE +, HDPE , 2.5% polyisohutylene (PIB) in mineral oil x, 10% PIB in decalin A, NBS-OB oil. [Reprinted by permission from J. L. White, Critique on Flow Patterns in Polymer Fluids at the Entrance of a Die and Instabilities Leading to Extrudate Distortion, Appl. Polym. Symp., No. 20, 155 (1973).]...

Fig. 12.18 Flow patterns above the oscillating region of HDPE. (a) Microtome cut along the cylinder axis of HDPE solidified inside a capillary die showing that the flow patterns are formed at the die entrance, (b) Microtome cut of the HDPE extrudate resulting under the same conditions as in (a). [Reprinted by permission from N. Bergem, Visualization Studies of Polymer Melt Flow Anomalies in Extruders, Proceedings of the Seventh International Congress on Rheology, Gothenberg, Sweden, 1976, p. 50.]...

The site of the sharkskin distortion is again the die exit, and so is the screw thread pattern. The site of, and the mechanism for the gross extrudate distortion are problems that have no clear answers. The work of White and Ballenger, Oyanagi, den Otter, and Bergem clearly demonstrates that some instability in the entrance flow patterns is involved in HDPE melt fracture. Clear evidence for this can be found in Fig. 12.18. Slip at the capillary wall, to quote den Otter, does not appear to be essential for the instability region, although it may occasionally accompany it. ... 

Estimation of Entrance Pressure-Pressure Losses from the Entrance Flow Field17 Consider the entrance flow pattern observed with polymer melts and solutions in Fig. 12.16(a). The flow can be modeled, for small values of a, as follows for 0 < a/2 the fluid is flowing in simple extensional flow and for a/2 < 0 < rc/2 the flow is that between two coaxial cylinders of which the inner is moving with axial velocity V. The flow in the outer region is a combined drag-pressure flow and, since it is circulatory, the net flow rate is equal to 0. The velocity V can be calculated at any upstream location knowing a and the capillary flow rate. Use this model for the entrance flow field to get an estimate for the entrance pressure drop. 

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