Extensional flow capillary

A. E. Everage and R. L. Ballman, The extensional flow capillary as a new method for extensional viscosity measurement. Nature 273, 213-215 (1978). 

Consider drops of different sizes in a mixture exposed to a 2D extensional flow. The mode of breakup depends on the drop sizes. Large drops (R > Caa,tal/xcy) are stretched into long threads by the flow and undergo capillary breakup, while smaller drops (R Cacri,oV/vy) experience breakup by necking. As a limit case, we consider necking to result in binary breakup, i.e., two daughter droplets and no satellite droplets are produced on breakup. The drop size of the daughter droplets is then... 

Fig. 20. Radius of drops produced on capillary breakup in hyperbolic extensional flow (Rdrops), radius of the thread at which the disturbance that causes breakup begins to grow (Rent), and the time for growth of the disturbance (fgrow) for different values of the dimensionless parameters p and /xc feao/tr. The time for capillary breakup of the extending thread ((break) can be obtained from these graphs (see Illustration for sample calculations) (Janssen and Meijer, 1993).

Fig. 22. Radius of drops produced by capillary breakup (solid lines) and binary breakup (dotted lines) in a hyperbolic extensional flow for different viscosity ratios (p) and scaled shear rate (p,cylo) (Janssen and Meijer, 1993). The initial amplitude of the surface disturbances is ao = 10 9 m. Note that significantly smaller drops are produced by capillary breakup for high viscosity ratios.

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. 

Figure 9.7 Photographs of droplet shapes in planar extensional flow for various viscosity ratios M of the dispersed to the continuous phase. The droplets are viewed in the plane normal to the velocity gradient direction. The critical capillary numbers Cac and droplet deformation parameters Dc at breakup are also given. The droplet fluids are silicon oils with viscosities ranging from 5 to 60,000 centistokes, while the continuous fluids are oxidized castor oils both phases are Newtonian. (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

The melt viscosity of LCPs is sensitive to thermal and mechanical histories. Quite often, instrumental influences are important in the value of viscosity measured. For example, the viscosity of HBA/HNA copolyesters are dependent on the die diameter in capillary flow (59). LCP melts or solutions are very efficiently oriented in extensional flows, and as a result, the influence of the extensional stresses at the entrance to a capillary influence the shear flow in the capillary to a much greater extent than is usually found with non-LC polymers. 

All flows can be decomposed into shear and exten-sional components. The effectiveness of the flow field is dependent on the deformation rate, the relative values of shear and extension, and the micro structure of the fluid. Extensional flows are more effective in microstructure development, such as droplet breakup and mechanochemical reactions. However, such flows are difficult to generate and to maintain and therefore, in practical applications, capillary entrance/exit flows provide a suitable means of achieving extensional flows where the shear component of the flow field changes with the capillary entry-exit angle. Indeed, OFRs also generate extensional flows, which result in efficient droplet breakup in emulsification. Mixers in which the deformation rate and the relative values of shear/extension rates can be controlled are known as controlled deformation mixers. 

Problem 7-12. Bubble in an Axisymmetric Flow. A gas bubble is immersed in a viscous Newtonian fluid that is undergoing an axisymmetric extensional flow. The fluid is viscous enough that the relevant Reynolds number is small so that the creeping-motion approximation can be applied. The capillary number based on the extension rate, E, and the surface tension, a, is small, i.e.,... 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

Note that in shear for A, = 1, the critical capillary number = 1, whereas for A, > 1, increases with X and becomes infinite for X > 3.8. This means that the breakup of the dispersed phase in pure shear flow becomes impossible for X > 3.8. This limitation does not exist in extensional flows. 

Ouibrahim and Fruman (47) in 1980 found dilational flow in three distinct flow situations, which each involve an extensional component capillary tube flow, orifice flow, and pitot tube flow. They examined extensively hydrolyzed polyacrylamide (HPAA) and found that the dilatant effect was greatly reduced in the presence of excess salt. This finding was attributed to the effect of the salt ions in screening the charges on the polyelectrolytic HPAA and thus causing the contraction of the highly expanded molecule. 

Filament stretching has been studied for obtaining extensional properties of polymeric solutions and melts. A comprehensive review of the flow dynamics of filament stretching is provided by McKinley and Sridhar [9]. Subsequently, liquid filament rheometers or capillary breakup extensional rheometers (CaBER) has been developed for the characterization of the dynamics of complex fluids undergoing extensional flows. 

But according to de Gennes (69) such a transition is unlikely in a weak flow its experimental evidence in strong flows has been claimed by several authors from pressure losses measurements in short capillaries or in slits [Quibrahim (22), Ambari (70)] and from flow birefringence measurements in extensional flows [Pope et al. (71), Cressely et al. (72)]. It must be... 

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