Extrusion of Polymer Melts and Melt Flow Instabilities. III. Theoretical Analysis of Extrusion through a Slit Die

1969 ◽  
Vol 42 (3) ◽  
pp. 691-699 ◽  
Author(s):  
James L. White
Keyword(s):  
Reynolds Number ◽  
Stress Field ◽  
Viscoelastic Fluid ◽  
Polymer Melts ◽  
Second Order ◽  
Secondary Flows ◽  
Newtonian Fluids ◽  
Creeping Flow ◽  
Flow Instabilities ◽  
Field Problem

Abstract In the previous sections of this paper we have discussed our own and other experimental studies of flow instabilities in the extrusion of polymer melts as well as various theories of the mechanism of initiation of the instability. It is our belief that one of the keys to a deeper understanding of this phenomenon is a fuller analytical understanding of the stress and velocity fields in the composite reservoir, capillary, and extrudate system. It is to this problem that we turn our attention here. The velocity, stress-field problem in the entrance region of a conduit being fed from a reservoir has received considerable attention for Newtonian fluids. Most authors have followed Schlichting and Goldstein in using boundary-layer theory to analyze this problem. While there are a number of such solutions for viscous non-Newtonian and viscoelastic fluids, they are of little interest for polymer melts. This is not only because they represent a high Reynolds number, inertia-dominated asymptote but because they neglect all phenomena occurring in the reservoir feeding the conduit. Of more interest are the low Reynolds number creeping flow solutions for Newtonian fluids which are based upon the work of Sampson (see also Roscoe and Weissberg). A decade ago Tomita published a pioneering analysis of the creeping flow of a viscous non-Newtonian (power-law) fluid into a sharp edge entrance of a capillary. More recently LaNieve and Bogue have analyzed the creeping flow of a Coleman-Noll second order into the capillary entrance. A recent study of the entry problem has been made by Metzner, Uebler, and Chan Man Fong. The related problem of creeping flow of a viscoelastic fluid in a converging channel or cone has been analyzed by Adams, Whitehead, and Bogue and Kaloni. While the former authors computed the stress field for a second-order fluid and an integral constitutive equation in a presumed velocity field, Kaloni actually evaluated velocity profiles and predicted the formation of secondary flows. A more intuitive, but far less rigorous approach to extrusion of a viscoelastic fluid has been taken by Dexter and Dienes and Smith. These authors presume a virgin material to enter a capillary die in fully developed flow and utilize the theory of linear viscoelasticity to evaluate the stress field.

A three-dimensional computation of the force and torque on an ellipsoid settling slowly through a viscoelastic fluid

1995 ◽  
Vol 283 ◽  
pp. 1-16 ◽  
Author(s):  
J. Feng Feng ◽  
D. D. Joseph ◽  
R. Glowinski ◽  
T. W. Pan
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Tilt Angle ◽  
Three Dimensional ◽  
Major Axis ◽  
Weissenberg Number ◽  
Second Order ◽  
The Body ◽  
Fall Velocity ◽  
Elasticity Number

The orientation of an ellipsoid falling in a viscoelastic fluid is studied by methods of perturbation theory. For small fall velocity, the fluid's rheology is described by a second-order fluid model. The solution of the problem can be expressed by a dual expansion in two small parameters: the Reynolds number representing the inertial effect and the Weissenberg number representing the effect of the non-Newtonian stress. Then the original problem is split into three canonical problems: the zeroth-order Stokes problem for a translating ellipsoid and two first-order problems, one for inertia and one for second-order rheology. A Stokes operator is inverted in each of the three cases. The problems are solved numerically on a three-dimensional domain by a finite element method with fictitious domains, and the force and torque on the body are evaluated. The results show that the signs of the perturbation pressure and velocity around the particle for inertia are reversed by viscoelasticity. The torques are also of opposite sign: inertia turns the major axis of the ellipsoid perpendicular to the fall direction; normal stresses turn the major axis parallel to the fall. The competition of these two effects gives rise to an equilibrium tilt angle between 0° and 90° which the settling ellipsoid would eventually assume. The equilibrium tilt angle is a function of the elasticity number, which is the ratio of the Weissenberg number and the Reynolds number. Since this ratio is independent of the fall velocity, the perturbation results do not explain the sudden turning of a long body which occurs when a critical fall velocity is exceeded. This is not surprising because the theory is valid only for slow sedimentation. However, the results do seem to agree qualitatively with ‘shape tilting’ observed at low fall velocities.

Flow of a PTT Fluid Through Planar Contractions: Vortex Inhibition Using Rounded Corners

2010 ◽  
Author(s):  
M. Khodadadi Yazdi ◽  
A. Ramazani S. A. ◽  
H. Hosseini Amoli ◽  
A. Behrang ◽  
A. Kamyabi
Keyword(s):  
Fluid Flow ◽  
Constitutive Equation ◽  
Viscoelastic Fluid ◽  
Constitutive Equations ◽  
Isothermal Condition ◽  
Critical Value ◽  
Newtonian Fluids ◽  
Creeping Flow ◽  
Contraction Flow ◽  
Ptt Fluid

Contraction flow is one of important geometries in fluid flow both in Newtonian and non-Newtonian fluids. In this study, flow of a viscoelastic fluid through a planar 4:1 contraction with rounded corners was investigated. Six different rounding ratios (RR = 0, 0.125, 0.25, 0.375, 0.438, 0.475, 0.488) was examined using the linear PTT constitutive equation at creeping flow and isothermal condition. Then the resulting PDE set including continuity, momentum, and PTT constitutive equations were implemented to the OpenFOAM software. The results clearly show vortex deterioration with increasing rounding diameter, so that when rounding corner exceeds a critical value, the vortex disappears completely. This phenomenon was also observed at different upstream widths. Furthermore, by increasing rounding diameter, the diminishing vortex approaches to the re-entrant corner.

Small Reynolds Number Electro-Hydrodynamic Flow Around Drops and the Resulting Deformation

1979 ◽  
Vol 46 (3) ◽  
pp. 510-512 ◽  
Author(s):  
M. B. Stewart ◽  
F. A. Morrison
Keyword(s):  
Reynolds Number ◽  
Flow Field ◽  
Order Approximation ◽  
Creeping Flow ◽  
Small Reynolds Number ◽  
Zeroth Order ◽  
Regular Perturbation ◽  
Low Reynolds Number Flow ◽  
Reynolds Number Flow ◽  
Number Flow

Low Reynolds number flow in and about a droplet is generated by an electric field. Because the creeping flow solution is a uniformly valid zeroth-order approximation, a regular perturbation in Reynolds number is used to account for the effects of convective acceleration. The flow field and resulting deformation are predicted.

Non-Isothermal Peristaltic Flow of Newtonian Fluids in a Circular Tube

2009 ◽  
Author(s):  
M. S. Yun ◽  
B. P. Huynh
Keyword(s):  
Reynolds Number ◽  
Flow Pattern ◽  
Circular Tube ◽  
Pressure Change ◽  
Peristaltic Flow ◽  
Isothermal Flow ◽  
Newtonian Fluids ◽  
Fluid Viscosity ◽  
Fluid Properties ◽  
Isothermal Case

Non-isothermal peristaltic flow of Newtonian fluids in a circular tube is investigated numerically, using a commercial Computational Fluid Dynamics (CFD) software package. Simulation is performed over a range of Reynolds-number values, up to 1000. Temperature affects the flow field via fluid viscosity, which is assumed to decrease exponentially with temperature. Other fluid properties are assumed to be constant, and are similar to those of an oil. Allowing for temperature effects alters significantly the flow pattern and reduces pressure change. In the crest region, recirculation appears in non-isothermal flow at a much smaller Reynolds number Re than in isothermal flow. Influence of the Reynolds number itself is also reduced significantly, such that the flow pattern changes very little with increasing Re, in contrast to the isothermal case. Similarly, in non-isothermal flow, flow pattern is unchanged at different flow rate. This is also in contrast to the isothermal situation.

Heat Transfer Enhancement in a Rectangular (AR = 3:1) Channel With V-Shaped Dimples

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
C. Neil Jordan ◽  
Lesley M. Wright
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Nusselt Number ◽  
Liquid Crystal ◽  
Pressure Drop ◽  
Heat Transfer Enhancement ◽  
Thermal Performance ◽  
Secondary Flows ◽  
Transfer Enhancement ◽  
Reduced Pressure

An alternative to ribs for internal heat transfer enhancement of gas turbine airfoils is dimpled depressions. Relative to ribs, dimples incur a reduced pressure drop, which can increase the overall thermal performance of the channel. This experimental investigation measures detailed Nusselt number ratio distributions obtained from an array of V-shaped dimples (δ/D = 0.30). Although the V-shaped dimple array is derived from a traditional hemispherical dimple array, the V-shaped dimples are arranged in an in-line pattern. The resulting spacing of the V-shaped dimples is 3.2D in both the streamwise and spanwise directions. A single wide wall of a rectangular channel (AR = 3:1) is lined with V-shaped dimples. The channel Reynolds number ranges from 10,000–40,000. Detailed Nusselt number ratios are obtained using both a transient liquid crystal technique and a newly developed transient temperature sensitive paint (TSP) technique. Therefore, the TSP technique is not only validated against a baseline geometry (smooth channel), but it is also validated against a more established technique. Measurements indicate that the proposed V-shaped dimple design is a promising alternative to traditional ribs or hemispherical dimples. At lower Reynolds numbers, the V-shaped dimples display heat transfer and friction behavior similar to traditional dimples. However, as the Reynolds number increases to 30,000 and 40,000, secondary flows developed in the V-shaped concavities further enhance the heat transfer from the dimpled surface (similar to angled and V-shaped rib induced secondary flows). This additional enhancement is obtained with only a marginal increase in the pressure drop. Therefore, as the Reynolds number within the channel increases, the thermal performance also increases. While this trend has been confirmed with both the transient TSP and liquid crystal techniques, TSP is shown to have limited capabilities when acquiring highly resolved detailed heat transfer coefficient distributions.

Heat Transfer and Flowfield Measurements in the Leading Edge Region of a Stator Vane Endwall

1999 ◽  
Vol 121 (3) ◽  
pp. 558-568 ◽  
Author(s):  
M. B. Kang ◽  
A. Kohli ◽  
K. A. Thole
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Secondary Flows ◽  
Edge Region ◽  
Leading Edge Vortex ◽  
Stator Vane ◽  
Edge Vortex ◽  
The Mean

The leading edge region of a first-stage stator vane experiences high heat transfer rates, especially near the endwall, making it very important to get a better understanding of the formation of the leading edge vortex. In order to improve numerical predictions of the complex endwall flow, benchmark quality experimental data are required. To this purpose, this study documents the endwall heat transfer and static pressure coefficient distribution of a modern stator vane for two different exit Reynolds numbers (Reex = 6 × 105 and 1.2 × 106). In addition, laser-Doppler velocimeter measurements of all three components of the mean and fluctuating velocities are presented for a plane in the leading edge region. Results indicate that the endwall heat transfer, pressure distribution, and flowfield characteristics change with Reynolds number. The endwall pressure distributions show that lower pressure coefficients occur at higher Reynolds numbers due to secondary flows. The stronger secondary flows cause enhanced heat transfer near the trailing edge of the vane at the higher Reynolds number. On the other hand, the mean velocity, turbulent kinetic energy, and vorticity results indicate that leading edge vortex is stronger and more turbulent at the lower Reynolds number. The Reynolds number also has an effect on the location of the separation point, which moves closer to the stator vane at lower Reynolds numbers.

Settling Behavior of Particles in Bubble Containing Newtonian Fluids: Experimental Study and Model Development

2021 ◽  
Author(s):  
Silin Jing ◽  
Xianzhi Song ◽  
Zhaopeng Zhu ◽  
Buwen Yu ◽  
Shiming Duan
Keyword(s):  
Reynolds Number ◽  
Drag Coefficient ◽  
Volume Concentration ◽  
Settling Velocity ◽  
Particle Density ◽  
Particle Diameter ◽  
Newtonian Fluids ◽  
Spherical Particles ◽  
Bubble Volume ◽  
Particle Reynolds Number

Abstract Accurate description of cuttings slippage in the gas-liquid phase is of great significance for wellbore cleaning and the control accuracy of bottom hole pressure during MPD. In this study, the wellbore bubble flow environment was simulated by a constant pressure air pump and the transparent wellbore, and the settling characteristics of spherical particles under different gas volume concentrations were recorded and analyzed by highspeed photography. A total of 225 tests were conducted to analyze the influence of particle diameter (1–12mm), particle density (2700–7860kg/m^3), liquid viscosity and bubble volume concentration on particle settling velocity. Gas drag force is defined to quantitatively evaluate the bubble’s resistance to particle slippage. The relationship between bubble drag coefficient and particle Reynolds number is obtained by fitting the experimental results. An explicit settling velocity equation is established by introducing Archimedes number. This explicit equation with an average relative error of only 8.09% can directly predict the terminal settling velocity of the sphere in bubble containing Newtonian fluids. The models for predicting bubble drag coefficient and the terminal settling velocity are valid with particle Reynolds number ranging from 0.05 to 167 and bubble volume concentration ranging from 3.0% to 20.0%. Besides, a trial-and-error procedure and an illustrative example are presented to show how to calculate bubble drag coefficient and settling velocity in bubble containing fluids. The results of this study will provide the theoretical basis for wellbore cleaning and accurate downhole pressure to further improve the performance of MPD in treating gas influx.

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
J. Granata ◽  
L. Xu ◽  
Z. Rusak ◽  
S. Wang
Keyword(s):  
Numerical Simulation ◽  
Reynolds Number ◽  
Difference Scheme ◽  
Second Order ◽  
Swirling Flows ◽  
Inviscid Limit ◽  
Simulation Algorithm ◽  
Spatial Integration ◽  
Order Difference ◽  
Breakdown Process

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

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