Planar elongational flow

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

If the duration and intensity of the elongational stress are sufficiently large, the macromolecule can be stretched to a greater extent. Keller was able to prove this using a cross-slot device producing a planar elongational flow field with the aid of flow diffraction. 

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

Fig. 5 Schematic diagram of planar elongational flow using cross-slot. Streamlines are marked red and the stagnation point is marked as X. The velocity in the z direction is zero...

D Avino, G., Maffettone, P.L., Hulsen, M.A., Peters, G.M.W., Numerical simulation of planar elongational flow of concentrated rigid particle suspensions in a viscoelastic fluid, J. Non-Newtonian Fluid Mech. 150 (2008) 65. 

Baig, C Edwards, B.J., Keflfer, D.J., and Cochran, H.D. (2005) A proper approach for nonequilibrium molecular dynamics simulations of planar elongational flow. 

Rheological and structural studies of linear polyethylene melts under planar elongational flow using nonequflibrium molecular dynamics simulations. 

Kim, J.M., Kefifer, D.J., Kroger, M., and Edwards, B.J. (2008) Rheological and entanglement characteristics of linear-chain polyethylene liquids in planar Couette and planar elongational flows. 

If each of the three epsilons are zero we recover the boundary conditions of planar Couette (shear) flow. For y = 0, there are three basic elongation flow fields defined as follows if one of the epsilons is also zero, the flow field is referred to as planar elongation flow if one of the epsilons is negative while the other two are positive and equal, the flow field is referred to as biaxial stretching flow and if one of the epsilons is positive while the other two are negative and equal, the flow field is referred to as uniaxial stretching flow. 

The above expression can be applied to idealized systems with three types of deformation planar (plel), uniaxial (unel) extensional, and simple shear (ss) mixers. For a mixing device dominated by planar elongational flow, the distances along the X axis are related to strain or strain rate by... 

We can now make an assessment of the efficiency of the three types of flows to mix materials. Suppose we need to prepare a polymer blend with striation thickness of the minor component in the final product not more than 3 xm. The pellets of the minor component fed into the flowing major component have a characteristic length of 3 mm. If we allow the components to stay in the flow field no more than 10 s, the ratio of the specific power for simple shear flow to that of the uniaxial elongational flow and to that of the planar elongational flow is... 

We consider next the planar elongational flow. The experimental setup is a four-roll apparatus (see Fig. 6.18c) used first by Taylor (1934). Cox (1969) developed a theory similar to that for simple shear flow ... 

Thus, for Newtonian fluids simple shear and planar elongational flows require the same power to produce the same deformation while uniaxial elongation requires 33% more power. 

B.4 Lineal Stretch Efficiency of Planar Elongational Flow. Prove that the time average lineal stretch efficiency, z, of a two-dimensional stagnation flow (planar elongational flow Fig. 6.35) is... 

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