Extensional flow, axisymmetric

In the context of the preceding model, a drop is said to break when it undergoes infinite extension and surface tension forces are unable to balance the viscous stresses. Consider breakup in flows with D mm constant in time (for example, an axisymmetric extensional flow with the drop axis initially coincident with the maximum direction of stretching). Rearranging Eq. (26) and defining a characteristic length Rip113, we obtain the condition, for a drop in equilibrium,... 

We first derive the kinematics of the deformation. The flow situation is shown in Fig. 14.14. Coordinate z is the vertical distance in the center of the axisymmetric bubble with the film emerging from the die at z = 0. The radius of the bubble R and its thickness 8 are a function of z. We chose a coordinate system C, embedded in the inner surface of the bubble. We discussed extensional flows in Section 3.1 where we defined the velocity field of extensional flows as... 

G. A RIGID SPHERE IN AXISYMMETRIC, EXTENSIONAL FLOW I. The Flow Field... 

Figure 7-13. The streamlines for axisymmetric flow in the vicinity of a solid sphere with uniaxial extensional flow at infinity. When the direction of motion is reversed at infinity, the undisturbed flow is known as biaxial extensional flow. The stream-function values are calculated from Eq. (7-185). Contour values are plotted in equal increments equal to 0.5.

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.,... 

Problem 7-13. The Viscosity of a Dilute Suspension of Spherical Bubbles. Consider again the Problem 7 12. This time, assume that the bubble remains spherical. If you have already solved 7 12, the solution for part (b) provides the velocity and pressure fields for a spherical bubble. If you have not previously solved 7 12, you will need to determine the solution for a spherical bubble in an axisymmetric extensional flow, using the boundary conditions... 

In the extensional flow, the principal axis of deformation is x and the shape is axisymmetric about this axis. The ratio of major to minor axis in this case is... 

Sphere in Linear Flows Axisymmetric Extensional Flow and Simple Shear... 

Figure 9-9. A schematic sketch showing the flow patterns for (a) uniaxial extensional flow, (b) biaxial extensional flow, and (c) hyperbolic (or 2D) extensional flow. Each part of the figure shows the flow from two perspectives one along the z axis toward the xy plane and the other along the x axis toward the yz plane. In the first two cases, the flow is axisymmetric, with the x axis being the symmetry axis. In the 2D case, the flow is invariant in the z direction.

Problem 9-20. Mass Transfer From a Spherical Bubble in an Extensional Flow. Consider a spherical gas bubble that is suspended in a liquid that undergoes an axisymmetric extensional flow. The bubble contains a component A that is soluble in the exterior liquid... 

Another axisymmetric extensional flow is biaxial extension. In this deformation, there is a compression along the axis of symmetry that stretches in the radial direction, as shown in Fig. 9. The principal strain rate is defined as... 

Stone and Leal [70] applied the method to a droplet contaminated with an insoluble surfactant in an axisymmetric extensional flow. The effect of the... 

The sink flow analysis, which assumes a purely extensional flow (i.e., no shear component), was presented by Metzner and Metzner (1970) to evaluate the extensional viscosity from orifice Apen measurements. For an axisymmetric contraction, the flow into the orifice is analogous to a point sink for a planar contraction flow, the analogy is with a line sink (Batchelor, 1967). In the case of axisymmetric contraction (Figure 7.8.1), the use of spherical coordinates and continuity gives the velocity components... 

While we will not need the general tensorial form of the superposition principle for the purposes of this book, we will show the result of its use to describe the special case of axisymmetric, uniaxial (tensile) extensional flow ... 

Figure 2.4 Critical Weber number for break-up of drops in various types of flow. Singledrop experiments in two-dimensional simple shear (a — 0), hyperbolic flow (a = l) and intermediate types/-as well as a theoretical result for axisymmetrical extensional flow (ASE)." The hatched area rrfers to apparent We values obtained in a colloid mill" ...

Steady extensional deformations can be created by impinging two liquid streams, creating a stagnation flow. Figure 7.7.1 illustrates both axisymmetric and planar stagnation flow. These flows are not homogeneous. A material element near the central part of the flow... 

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