Fluid particle deformation

Here r(t) is the stress at a fluid particle given by an integral of deformation history along the fluid particle trajectory between a deformed configuration at time f and the current reference time t. 

In Equation (5,14), (77j ) is found by interpolating existing nodal values at the old time step and then transforming the found value to the convccted coordinate system. Calculation of the componenrs of 7 " and (/7y ) depends on the evaluation of first-order derivahves of the transformed coordinates (e.g, as seen in Equation (5.9). This gives the measure of deformation experienced by the fluid between time steps of n and + 1. Using the I line-independent local coordinates of a fluid particle (, ri) we have... 

Bubble and drop breakup is mainly due to shearing in turbulent eddies or in velocity gradients close to the walls. Figure 15.11 shows the breakup of a bubble, and Figure 15.12 shows the breakup of a drop in turbulent flow. The mechanism for breakup in these small surface-tension-dominated fluid particles is initially very similar. They are deformed until the aspect ratio is about 3. The turbulent fluctuations in the flow affect the particles, and at some point one end becomes... 

Shear stresses develop in fluids when adjacent particles have different velocities. This causes the fluid to deform and undergo turbulent mixing. 

Bubbles and drops tend to deform when subject to external fluid fields until normal and shear stresses balance at the fluid-fluid interface. When compared with the infinite number of shapes possible for solid particles, fluid particles at steady state are severely limited in the number of possibilities since such features as sharp corners or protuberances are precluded by the interfacial force balance. 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

All the work discussed in the preceding sections is subject to the assumptions that the fluid particles remain perfectly spherical and that surfactants play a negligible role. Deformation from a spherical shape tends to increase the drag on a bubble or drop (see Chapter 7). Likewise, any retardation at the interface leads to an increase in drag as discussed in Chapter 3. Hence the theories presented above provide lower limits for the drag and upper limits for the internal circulation of fluid particles at intermediate and high Re, just as the Hadamard-Rybzcynski solution does at low Re. 

As noted in Chapters 2 and 3, deformation of fluid particles is due to inertia effects. For low Re and small deformations, Taylor and Acrivos (T3) used a matched asymptotic expansion to obtain, to terms of order We /Re,... 

Drops and bubbles in highly purified systems are significantly more deformed than corresponding fluid particles in contaminated systems. Increased flattening of fluid particles in pure systems results from increased inertia forces related to the increased terminal velocities discussed above. Some experimental results for drops and bubbles in water (low M systems) are shown in Fig. 7.9. The... 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

It is convenient to divide the discussion of wall effects for bubbles and drops into two parts. Section A covers cases where the diameter ratio, 2 = d /D, is less than about 0.6. At low 2, the walls cause little deformation beyond that which may be present for the fluid particle in an infinite medium, so that the discussion of wall effects for rigid particles forms a good starting point. Section B treats the case of slug flow (2 > 0.6) where the container walls have a dominant effect on the shape of the bubble or drop. 

We recall from Chapters 2 and 3 that fluid particles at low Re in infinite media tend to be spherical and that the interface is usually stagnant due to surface-active contaminants or large values of /c = If 2 is less than about 0.3, deformation due to the container walls tends to be minor and the corrections given above for rigid spheres at low Re may be used. 

Theoretical predictions relating to the orientation and deformation of fluid particles in shear and hyperbolic flow fields are restricted to low Reynolds numbers and small deformations (B7, C8, T3, TIO). The fluid particle may be considered initially spherical with radius ciq. If the surrounding fluid is initially at rest, but at time t = 0, the fluid is impulsively given a constant velocity gradient G, the particle undergoes damped shape oscillations, finally deforming into an ellipsoid (C8, TIO) with axes in the ratio where... 

Gas Bubbles Fluid particles, unlike rigid solid particles, may undergo deformation and internal circulation. Figure 6-59 shows rise velocity data for air bubbles in stagnant water. In the figure, Eo = Eotvos number, g(pL - pG)dJa, where pL = liquid density pG = gas density, de = bubble diameter, a = surface tension, and the equivalent diameter de is the diameter of a sphere with volume equal to that of... 

Fig. 13.15 Schematic representation of the flow pattern in the central portion of the advancing front between two parallel plates. The coordinate system moves in the x direction with the front velocity. Black rectangles denote the stretching deformation the fluid particles experience. [Reprinted by permission from Z. Tadmor, Molecular Orientation in Injection Molding, J. Appl. Polym. Sci., 18, 1753 (1974).]...

To illustrate this, we will discuss the example shown in Fig. 6.6, which presents one deformable fluid particle moving along a streamline. We can describe this system taking into account inertia, resistive (viscous) force and weight force. The magnitude of the inertia force along the streamline can be written as ... 

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