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1.
A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The first part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability.In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid. 相似文献
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The stability of the Couette flow of the liquid with the power law viscosity in a wide annular gap has been investigated theoretically in this work with the aid of the method of small disturbances. The Taylor number, being a criterion of the stability, has been defined using the mean apparent viscosity value in the main flow. In the whole range of the radius ratio, R
i
/R
o
and the flow index, n, considered (R
i
/R
o
0.5, n = 0.25–1.75 ), the critical value of the Taylor number Ta
c
is an increasing function of the flow index, i.e., shear thinning has destabilizing influence on the rotational flow, and dilatancy exhibits an opposite tendency.In the wide ranges of the flow index, n > 0.5, and the radius ratio, R
i
/R
o
> 0.5, the wide-gap effect on the stability limit is predicted to be almost the same for non-Newtonian fluids as for Newtonian ones. The ratio on the critical Taylor numbers for non-Newtonian and Newtonian fluids: Ta
c
(n) and Ta
c
(n = 1) obey a generalized functional dependence: Ta
c
(n)/Ta
c
(n = 1) = g(n), where g(n) is a function corresponding to the solution for the narrow gap approximation.Theoretical predictions have been compared with experimental results for pseudoplastic liquids. In the range of the radius ratio R
i
/R
o
> 0.6 the theoretical stability limit is in good agreement with the experiments, however, for R
i
/R
o
< 0.6, the critical Taylor number is considerably lower than predicted by theory. 相似文献
6.
Y. L. Yeow 《Flow, Turbulence and Combustion》1983,40(2):121-133
A simple analysis of the periodic extensional flow generated by a pulsating sphere in an infinite sea of viscoelastic fluid has been carried out. The general procedure is illustrated by two specific constitutive equations: the corotational Jeffreys fluid and the Oldroyd fluid model B. The response of these fluids is reflected in the temporal variation of the pressure on the surface of the sphere, with Reynolds and Deborah numbers and parameters of the constitutive equations as independent variables. For the case of pulsation with infinitesimal amplitude the fluid response is summarised in the form of pressure amplitude and phase lag versus Deborah number plots. The role of the pulsating flow in the characterisation of viscoelastic fluids and the extension of the procedure to other constitutive equations are briefly discussed. 相似文献
7.
Summary Earlier parts of this series have described a technique based on the collapse of single bubbles in the fluids for studying the elongational rheology of viscoelastic solutions and melts of moderate viscosities (
0 > 102p) at relatively high strain rates
. The present paper describes the modelling of bubble collapse with both rate and integral type constitutive relations using a body coordinate system. Predictions of the stress at the bubble wall as a function of time during collapse from a BKZ model and a modified corotational Maxwell model compared favorably with experimental data for two polymer solutions, 1% polyacrylamide in water/glycerine and 2% hydroxypropyl cellulose in water.
Nomenclature a material constant - b material constant - g metric tensor, space coordinates - m material constant - n material constant - p pressure - P G pressure within bubble - P R pressure outside bubble at the wall - P pressure far away from the bubble - R bubble radius - dR/dt - R 0 initial bubble radius - t time - u velocity - U potential function - Y R/R 0 Greek symbols covariant body metric tensor - surface tension - rate of deformation matrix, II -second invariant of - strain rate - 0 zero shear rate viscosity - e elongational viscosity - ef effective viscosity - 1, 2, 3 coordinates in body system - 1 1/R 0 3 - body stress tensor - density - space stress tensor - relaxation time - ef effective relaxation time - bubble pressure function, defined in eq. [19] - vorticity tensor With 11 figures and 1 table 相似文献
Zusammenfassung In vorangehenden Veröffentlichungen dieser Reihe wurde eine Methode beschrieben, mit Hilfe derer man aus dem Zerfall von einzelnen Blasen in einer Flüssigkeit auf die Dehn-Rheologie viskoelastischer Lösungen und Schmelzen mittlerer Viskosität ( 0 > 102 P) bei relativ hohen Dehngeschwindigkeiten schließen kann. Die vorliegende Untersuchung beschreibt Modelle des Blasenzerfalls mit Hilfe von Stoffgleichungen sowohl vom rate- als auch vom Integral-Typ, wobei ein körperfestes Koordinatensystem benutzt wird. Die Voraussagen der Spannung an der Blasenwand als Funktion der Zeit während des Zerfalls bei Verwendung eines BKZ- und eines modifizierten korotatorischen Maxwell-Modells zeigen eine recht gute Übereinstimmung mit experimentellen Werten, die an zwei Polymerlösungen, nämlich einer 1%igen Polyacrylamid-Lösung in einer Wasser-Glycerin-Mischung und einer 2%igen wäßrigen Hydropropylcellulose, erhalten worden sind.
Nomenclature a material constant - b material constant - g metric tensor, space coordinates - m material constant - n material constant - p pressure - P G pressure within bubble - P R pressure outside bubble at the wall - P pressure far away from the bubble - R bubble radius - dR/dt - R 0 initial bubble radius - t time - u velocity - U potential function - Y R/R 0 Greek symbols covariant body metric tensor - surface tension - rate of deformation matrix, II -second invariant of - strain rate - 0 zero shear rate viscosity - e elongational viscosity - ef effective viscosity - 1, 2, 3 coordinates in body system - 1 1/R 0 3 - body stress tensor - density - space stress tensor - relaxation time - ef effective relaxation time - bubble pressure function, defined in eq. [19] - vorticity tensor With 11 figures and 1 table 相似文献
8.
Gianbattista Scarpi 《Meccanica》1973,8(2):112-115
Summary After brief reference to energetical principles in order to find suitable constitutive equations for non-linear viscoelastic fluids, the behaviour of stress components is examined, for a simple shear motion with sinusoidal time time varying displacements of a liquid with an oscillating fading memory.
Research supported by C.N.R., Gruppo nazionale per la fisica matematica. 相似文献
Sommario Dopo aver richiamato brevemente un possibile procedimento per la determinazione della equazione costitutiva per liquidi viscoelastici nel caso di grandi deformazioni, si esamina il comportamento delle componenti del tensore degli sforzi nel caso di un moto periodico a filetti rettilinei e paralleli, supposto che il liquido abbia una memoria debole e oscillante nel tempo.
Research supported by C.N.R., Gruppo nazionale per la fisica matematica. 相似文献
9.
A natural generalization of cylindrical Couette flow is the flow of a viscous incompressible liquid between two concentric spheres rotating about the same axis with different angular velocities. As has often been noted, spherical Couette flow is, despite the apparent similarity, considerably more complex than cylindrical flow. It consists of differential rotation about the axis and one- or two-eddy circulation (depending on the ratio between the angular velocities of the two spheres = 2/1) in the meridional plane and depends significantly on the Reynolds number Re = 1r
1
2
and the relative thickness of the layer = (r2-r1)/r1 (1, 2 and r1, r2 are the angular velocities and radii of the inner and outer spheres, respectively. The investigation of spherical Gouette flow and its stability has begun relatively recently (within the last 10 years) and has evidently been stimulated by applied problems associated with astro- and geophysics. Because of the great computational difficulties encountered in investigating such flow theoretically, experimental investigations have yielded more extensive and interesting results [1–8], although all the published results refer to the case of rotation of one inner sphere ( = 0).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–15, March–April, 1978.It remains to thank S. A. Shcherbakov for help in organizing automatic input of the signals to the BÉSM-6 computer. 相似文献
10.
R. Steller 《Rheologica Acta》1985,24(6):547-550
A constitutive equation presented earlier has been generalized to describe the transient shear flow behaviour of polymer liquids. The mathematical structure and properties of such a generalization have been discussed. The practical expressions for the generalized relaxation modulus have been proposed. 相似文献
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This paper reports numerical simulation of the flow past a heated/cooled sphere. A Galerkin finite element method is used to solve the 3D incompressible Boussinesq equations in primitive variable form. Numerical simulations of flow around the sphere for a range of Grashof numbers and moderate Reynolds numbers, were conducted. The drag coefficient for adiabatic flow shows good agreement with standard correlations over the range of the Reynolds numbers investigated. It is shown that the drag can vary considerably with heating of the sphere and that computational fluid dynamics methods can be used to derive constitutive laws for macroscopic momentum and heat exchange in multiphase flow. © 1998 John Wiley & Sons, Ltd. 相似文献
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In this paper visualisation and direct velocity profile measurement experiments for a branched polypropylene melt in a 10:1 axisymmetric contraction demonstrate the onset of wall slip. Video processing of the flow shows the formation of vortices and their diminution with increasing flow rate. Numerical simulations using a multimode K-BKZ viscoelastic and a purely viscous (Cross) model—both of them incorporating a nonlinear slip law—were used to predict the flow kinematics and dynamics as well as to deduce the slip velocity function by performing fitting to the velocity profiles. It was found that the numerical predictions agree well with the experimental results for the velocity profiles, and vortex formation, growth and reduction. It is suggested that such experiments (visualisation of entrance flow and direct velocity profile measurement) can be useful in evaluating the validity of constitutive equations and slip laws in the flow of polymer melts through processing equipment. 相似文献
15.
The unsteady Couette flow between two infinite horizontal plates induced by the non-torsional oscillations of one of the plates in a rotating system under the boundary layer approximations is investigated. An exact solution of the governing equations has been obtained by using Laplace transform technique. It is shown that when the oscillating plate situated at an infinite distance from stationary plate then the problem reduces to the unsteady boundary layer problem in a rotating system with non-torsional oscillations of the free-stream velocity. 相似文献
16.
Andrei Potanin 《ournal of non Newtonian Fluid Mechanics》2010,165(5-6):299-312
Models of the vane-cup and Couette rheometers are compared using computational fluid dynamics as well as approximate solutions. Thixotropy of the fluid is incorporated by means of a model based on experimental data for various toothpastes. Parameters of the model are calculated by fitting the results of step-shear tests in Couette geometry and are subsequently used to predict torque for similar tests in vane-cup geometry. Calculations for the thixotropic model were compared to the calculations for a time-independent equilibrium model with the same steady-shear rheology. There are two important findings in this work. First, at constant angular velocity, torque in the vane-cup rheometer for the thixotropic model turns out greater than for the equilibrium model, because the structure of the thixotropic fluid has not had enough time to reach equilibrium. This has implications for rheometry and the modeling of mixing operations. In both cases, torques may be underestimated if the standard equilibrium model is used in calculations. The second finding is relevant to rheometry. As is well known, equilibrium model predicts a lower torque value for a vane-cup geometry than for an equivalent Couette geometry. We found that taking thixotropy into account either makes the difference less pronounced or in some cases actually makes torque for vane-cup higher than for Couette. End effects are also analyzed. 相似文献
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18.
G. A. Danilin 《Fluid Dynamics》1969,4(3):144-147
We consider equilibrium flow of a multicomponent ionized gas between two catalytic plates of infinite length, one of which moves parallel to the other with constant velocity. The results of [1] are generalized for ionized gaseous mixtures which are in local thermodynamic equilibrium. Formulas are presented for calculating the thermal flux and the effective thermal conductivity for ambipolar diffusion.Then a special ionization case is discussed.Notation Ai
chemical symbol of the i-th component
- Wi
projection of the molar diffusive flux vector of the i-th component on the y-axis
- xi
molar concentration
- Hi
enthalpy
- mi
molecular weight
- Qs
heat of the s-th reaction
- Kps(T)
equilibrium constant of the s-th reaction
- Wi
mass formation rate of the i-th component per unit volume
- Zi
charge number
- e
unit charge (electron charge)
- E
electric field intensity
-
distance between the plates
- N
number of components
-
v
sl
stoichiometric coefficients
-
density
- T
temperature
- p
pressure
- u
projection of average velocity on y-axis
-
viscosity
-
thermal conductivity
- Dij
binary diffusion coefficient
- R
universal gas constant
- k
Boltzmann constant
In conclusion, the author wishes to thank G. A. Tirskii for proposing the study and for suggestions made in the course of the investigation. 相似文献
19.
Dominik Krug Beat Lüthi Hansjörg Seybold Markus Holzner Arkady Tsinober 《Experiments in fluids》2012,52(5):1349-1360
Genuine plane Couette flow is hard to realize experimentally, and no applications of modern spatially resolving measurement
techniques have been reported for this flow so far. In order to resolve this shortcoming, we designed and built a new experimental
facility and present our first results here. Our setup enables us to access the flow via 3D particle tracking velocimetry
and therefore to obtain truly three-dimensional flow fields for the first time experimentally in plane Couette flow. Results
are analyzed in terms of basic flow properties, and a clear distinction of flow regimes (laminar for Re < 320, transitional for 320 < Re < 400, and turbulent when Re > 400) could be made. Comparison with DNS data shows good agreement in the turbulent regime and builds trust in our data.
Furthermore, vortical coherent structures are studied in detail with the additional help of kalliroscope imaging, and the
typical vortex spacing is determined to be roughly one gap width. As a noteworthy result, we find that the onset of the turbulent
regime coincides with the range of Reynolds numbers at which a distance of 100 wall units is comparable to the gap width. 相似文献
20.
Direct numerical simulations have proven of inestimable help to our understanding of the transition to turbulence in wall-bounded
flows. While the dynamics of the transition from laminar flow to turbulence via localised spots can be investigated with reasonable
computing resources in the domains of limited extent, the study of the decay of turbulence in conditions approaching those
in the laboratory requires the consideration of domains so wide as to exclude the recourse to fully resolved simulations.
Using Gibson’s C++ code Channel-Flow, we scrutinise the effects of a controlled lowering of the numerical resolution on the decay of turbulence in-plane Couette
flow at a quantitative level. We show that the number of Chebyshev polynomials describing the cross-stream dependence can
be drastically decreased while preserving all the qualitative features of the solution. In particular, the oblique turbulent
band regime experimentally observed in the upper part of the transitional range is extremely robust. In terms of Reynolds
numbers, the resolution lowering is seen to yield a regular downward shift of the upper and lower thresholds R
t
and R
g
where the bands appear and break down. The study is illustrated with the results of two preliminary experiments. 相似文献