Axisymmetric contraction

Two types of flows were investigated. At CEMEF, experiments on a plane geometry were carried out. Birefringence measurements were performed, the results of which are presented in Section III-l. Comparison with computed results will be presented in paragraph 6.2. At LRMP, flows in axisymmetric contractions were studied. We present in the next pages the experimental results obtained for presstire drops, entrance pressure losses and die swell. 

In equation (42), the quantities j denote the vailues of the function / or those of its derivative / jj and / z. The details of the basis functions, the quantities j and the nodes of an element have been presented elsewhere [26], The first and second derivatives of the mapping function / can be obtained from derivatives of the basis fimctions. To compute the pressiure p and tensor components, the four nodal values at points A, B, C and D of each element are chosen as unknowns. Their derivatives are evaluated by using finite-difference formulae. In Fig 14 we present computed streamlines for 4/1 and 8/1 axisymmetric contractions, and the computed shear stress along the computed streamlines, for a K-BKZ fluid of the form given by Papanastasiou et al [19]. 

Figure 14. Computed streamlines and corresponding shear stresses along the streamhnes, when considering the peripheral stream tube, for a 4/1 and a 8/1 axisymmetric contractions (K-BKZ memory-integral fluid).

Flow experiments were carried out at LRMP for axisymmetric contractions and at CEMEF for plane geometries. Numerical simulations were performed at Laboratoire de Rheologie, with Wagner memory-integral constitutive equations, with the stream-tube method (sub-section 5.1) and at CEMEF, where the finite-... 

The characteristic curve of extrudate flow including adherence to the walls, and hence representative of shghtly to moderately entangled polymer flow in sudden two-dimensional or axisymmetrical contractions [7, 32], is represented in Fig 2. It shows a slope discontinuity above a certain pressm-e level, which depends on the pol3uner-die pair considered. With low flow rates, the flow is stable. Indeed, for these regimes, allowing for entrance effects, the flow curve is in fact representative of the shear rheometry of the polymer imder consideration, at low shear rates [34]. The slope discontinuity of the head loss curve indicates a modification in the structure of flow. It will be seen that this corresponds to the triggering of a hydrodynamic instability upstream of the contraction. 

For an axisymmetric contraction, U2 — ffui (from continuity), which implies... 

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... 

Flow fields of a Boger fluid at various flow rates in an axisymmetric contraction with a contraction ratio of 14.375 (A) Q = 0.08 mL/s, (B)... 

Figure 4.11 Flow patterns of LDPE and HDPE melts flowing through a 20 1 axisymmetric contraction before unstable conditions set in. Note the big vortex for LDPE and the absence of any vortex activity for HDPE [55],...

Flow through a convergent channel, either two-dimensional or axisymmetric, are probably the most investigated geometries (due to their connection with the technical die entry flow problem). Most of the other convergent flows could be derived from the abrupt contraction flow which is depicted schematically in Fig. 25. 

The burner in the test facility, shown in Fig. 18.1, is an axisymmetric nozzle, which is concentrically placed into the circular suction collar. To achieve a top-hat velocity profile with laminar boundary layer at the nozzle exit, a fourth order polynomial with a large contraction ratio of 31.6 1 and an exit diameter of D = 10.16 mm is used in the design. The suction collar assembly is connected to a vacuum pump through a series of solenoid valves so that a counterflow, which is in the opposite direction of the fuel-air mixture flow, can be established... 

This paper reports experiments and numerical simulations related to a linear low-density polye ylene (LLDPE) and a low-density polyethylene (LDPE), in a significant number of axisymmetric and planar mixed flows. Converging and abrupt contraction geometries involving short and long dies were considered as well as extrudate swell flows occurring at the exit of the ducts under investigation. 

Kim M. E., R. A. Brown and R. C Armstrong," The roles of inertia and shearthinning in flow of an inelastic liquid through an axisymmetric sudden contraction," J. Non-Newtonian Fluid Mech., 13 (1983) 341-364. 

For an axisymmetric pipe or channel, in a contracting flow, the kinetic energy change is derived as follows. Letjfl = be the area ratio (d2 < dj for contraction), and use V = Vq [1 - rjRi) ] for the inlet and outlet. Then ... 

Experiments were conducted in the newly built High Temperature Supersonic Jet Facility at the Fluid Mechanics Research Laboratory of the Florida State University in Tallahassee. A schematic of the facility can be seen in Fig. 3.2. In the present experiments, a converging axisymmetric nozzle having an exit diameter of 50.8 mm Wcis used. The nozzle profile was designed using a fifth-order polynomial with a contraction ratio of approximately 2.25. The stagnation pressure and temperature were held constant to within 0.5% of its nominal value during the experiment. 

The effects of the Reynolds number on the extrusion of Newtonian fluid from square and rectangular dies has been considered. As with planar and axisymmetric jets, extrudates from three-dimensional dies swell at low Reynolds numbers but contract at high ones. Depending on its aspect ratio, limiting die swell from the rectangle varies that of the square (0.7255) and that of two-dimensional planar case (0.8333). Wall slip reduces die swell and in the cases of perfect slip, completely eliminates it. 

Figure 5.53 A gas bubble surrounded by a liquid inside a hydrophilic axisymmetric chaimel with contraction...

Figure 5.53 shows a hydrophilic axisymmetric microchannel. The microchannel radius changes from to through a contraction with position-dependent channel radius r x). The local tapering 6f x) of the channel can be expressed as... 

Stable flow with strong vortex growth [31]. Obviously, any steady-state, two-dimensional, axisymmetric simulations are incapable of reproducing three-dimensional, time-dependent effects, such as the pulsating flow patterns exhibited by some polymer solutions in contractions. 

For absorbing particles, the integral in (1.107) is negative. Consequently, the equality in (1.110) transforms into an inequahty which is equivalent to the contractivity of the matrix elements of an axisymmetric particle provided that the z-axis of the particle coordinate system is directed along the axis of symmetry. 

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