Drops in a general linear flow

Instead, in this section, we consider the deformation of a drop in a general linear flow, lor ( a 1, where the shape remains approximately spherical. We assume that the density of the drop is equal to that of the suspending fluid and that the surface tension is constant on the drop surface - that is, there are no thermal gradients and no surfactants present. The fact that the drop is neutrally buoyant means that it does not translate relative to the surrounding fluid. Thus, at large distances from the drop, we can assume that the fluid undergoes a general linear flow of the form... 

Problem 8-11. The Effect of Surfactant on Drop Deformation in a General Linear Flow. 

Mechanical and chemical stability of novel stationary phases are basic requirements concerning their application. A lack in stability generally causes a loss in resolution and thus reduces column efficiency. In addition, the reproducibility of retention times, being important for qualitative analysis, may be affected. Evaluation of the mechanical stability of polymeric stationary phases is usually accomplished by the determination of the pressure drop across the column, when employing solvents of different polarity within a wide range of flow rates. A stationary phase can be considered as mechanically stable if a linear relationship between applied flow rate and resulting back pressure is obtained. 

Because the volume of the resin required has been established (to handle 103 bed volumes/h), the inside diameter (D) and the length (L) of the bed can be determined in order to be comparable to the residence time of the quinaldic breakthrough study. Two approaches can be followed in length and diameter determinations. One involves duplicating the breakthrough or the bench-scale studies. The second approach involves the practical aspects such as L/D and the pressure drop phenomenon. As a general rule, L/D should be more than or equal to 10. Therefore, a 16-in. X 1.0-in. i.d. column was constructed with the following characteristics surface area = 4.9 cm2 residence time = 0.57 min bed volume/h = 103 flow rate = 347 mL/min linear velocity = 70.8 cm/min L/D = 16. 

When the Ink Is allowed to rest In the Instrument for 15 minutes or more and then steady shear Is Initiated, there Is a significant stress overshoot (Figure 1). Subsequently, the stress level shows a significant time dependence for a period of time that depends on the experimental conditions but Is generally less than 10 seconds. After this Initial period the stress appears to level-off at what will be termed the short term steady flow value. If the steady shear Is maintained for long periods of time, however. It Is found that the stress Is not constant but shows a small and very slow decrease. For the range of conditions tested here, the stress, and therefore the viscosity, drops by about 15% In one hour (Figure 2). The decrease Is approximately linear In a log (n) vs log (time) plot. 

To obtain a solution for the complete class of flows given by (8 59), we again construct a solution of the creeping-motion equations by means of the superposition of vector harmonic functions. The development of a general form for the pressure and velocity fields in the fluid exterior to the drop follows exactly the arguments of the preceding solution for a solid sphere in a linear flow, and the solution therefore takes the same general form [see Eq. (8-39) and (8 11)], that is,... 

At about 10 to 15 percent of the ram travel which corresponds to the yield strain of materials, a drop in the friction force is observed. Thereafter the friction force decreases almost linearly with ram travel because of the continuously decreasing area of sliding contact. The variation of the material flow stress and normal pressure with ram travel, shown in Figs. 6 and 7, is in general similar to that of the extrusion pressure. The interface shear stress increases rapidly when the dead zone is approached. It also results in the increased value of the coefficient of friction. The actual increase in both the cases would be smaller than that obtained by calculation, because some sliding and cross flow in the dead zone is likely to take place in the latter part of the extrusion process. 

---