Hyperbolicity and change of type in the flow viscoelastic fluids through pipes

We consider the steady flow of an upper convected Maxwell fluid through a pipe with wavy walls. The analysis is an extension to round pipes of the methods introduced by Yoo and Joseph [1] to study the same problem in plane channels. As in the channel problem, the vorticity in a small cylinder at the center of the pipe becomes hyperbolic when the centerline velocity is larger than the speed of shear waves into rest. The region of hyperbolicity is smaller, but the decay of vorticity is less, when the elasticity parameter is larger.     

Hyperbolicity and change of type in the flow of viscoelastic fluids

The equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators. Propagation of singularities is discussed and conditions for a change of type are investigated. The vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied. This leads to a partitioning of the field of flow into subcritical and supercritical regions, as in the problem of transonic flow.Dedicated to Walter Noll on the Occasion of his 60^th Birthday     

On the flow resistance of viscoelastic fluids through packed beds

Analytical solutions are obtained for the free surface cell model of packed beds using a third order fluid. Second order perturbed results indicate a substantial increase in resistance to the flow of a viscoelastic fluid through a packed bed. This predicted increase is in good agreement with experimental findings.     

A finite element analysis of laminar unsteady flow of viscoelastic fluids through channels with non-uniform cross-sections

The effects of non-Newtonian behaviour of a fluid and unsteadiness on flow in a channel with non-uniform cross-section have been investigated. The rheological behaviour of the fluid is assumed to be described by the constitutive equation of a viscoelastic fluid obeying the Oldroyd-B model. The finite element method is used to analyse the flow. The novel features of the present method are the adoption of the velocity correction technique for the momentum equations and of the two-step explicit scheme for the extra stress equations. This approach makes the computational scheme simple in algorithmic structure, which therefore implies that the present technique is capable of handling large-scale problems. The scheme is completed by the introduction of balancing tensor diffusivity (wherever necessary) in the momentum equations. It is important to mention that the proper boundary condition for pressure (at the outlet) has been developed to solve the pressure Poisson equation, and then the results for velocity, pressure and extra stress fields have been computed for different values of the Weissenberg number, viscosity due to elasticity, etc. Finally, it is pertinent to point out that the present numerical scheme, along with the proper boundary condition for pressure developed here, demonstrates its versatility and suitability for analysing the unsteady flow of viscoelastic fluid through a channel with non-uniform cross-section.     

The inlet layer in the flow of viscoelastic fluids

In this paper we use the method of matched asymptotic expansions to study the effect of a small elasticity on the stress inlet boundary conditions in the flow of Maxwell-like fluids. Renardy's conditions on the normal stresses are discussed in this context and conclusions about some other boundary conditions and models are made.     

Instabilities in multi-hole converging flow of viscoelastic fluids

Studies of the onset of instabilities were conducted on single hole and multi-hole contractions using laser speckle visualization. A well characterized elastic fluid was used with constant viscosity of 13.1 Pa · s and elasticity characterized by a longest relaxation time constant of 2.233 s. The onset of instabilities was characterized in terms of the Deborah number and the contraction ratio. Three types of instabilities were observed: pulsing vortices, azimuthally rotating vortices, and swirling vortices. For the single hole contractions the critical Deborah number for instability increased from 4.4 to 5.07 to 5.25 as the contraction ratio increased from 4: 1 to 8: 1 to 12: 1. The magnitude of the instabilities was much greater for the 4: 1 contraction than for the other two contraction ratios. For the multi-hole contraction a square array of nine holes was used and the ratio of the hole diameter to hole spacing was varied. The height of the vortices is very similar for the single hole and multi-hole contractions at low Deborah numbers. At high Deborah numbers the effect of adjacent holes is to reduce the height of the vortices by a factor of three. For the 4: 1 spacing no secondary vortex was observed below a Deborah number of De = 3.7. Secondary vortices occurred for the 8:1 and 10:1 spacing at all Deborah numbers. Unstable pulsing vortices appeared for all spacings at a critical Deborah number around 5.5. Adjacent holes decreased the strength of the unsteady vortex motions. The centerline velocities were measured for the multi-hole contraction at shear rates of 5, 30, and 300 s^–1. The elongational strain rates are similar at a low shear rate of 5 s^–1. As shear rate is increased the onset of stretching occurs closer to the plane of the contraction for the smaller contraction ratios.     

Laminar shear flow of plastic viscoelastic fluids

Summary The theory of plastic viscoelastic fluids was developed by the author to represent the rheological behavior of polymer melts and solutions with high loading of small particles. The present paper develops an asymptotic formulation of the general theory which applies to laminar shear flows. The formulation is analogous to Criminale, Ericksen and Filbey's theory for viscoelastic fluids. We apply this to study plane Poiseuille and Couette flow.With 2 tables     

Entry flow behaviour of viscoelastic fluids in an annulus

Developing and fully developed velocity profiles in the entrance region of an abrupt 2-to-1 annular contraction were measured for a number of visco-elastic polymer solutions. Experimental results were obtained for Reynolds number and flow behaviour index in the range 9.8 ? Re ? 355 and 0.372 ? n ? 0.55 respectively. A momentum-energy integral technique was employed in the boundary layer analysis. The deviation from inelastic behaviour was indicated by the ratio of elastic to inertial forces, Ws/Re. Within the limits of confidence of the experimental results, good agreement with theoretical predictions was obtained and very little deviation from inelastic behaviour was observed for Ws/Re < 0.08. For the test fluids investigated, the entrance length was found to be longer than that predicted for the corresponding inelastic fluids of the same n.     

Inertia effect in conical converging flow of viscoelastic fluids

The convergent flow of viscoelastic fluids in conical nozzles has been examined. The experimentally determined streamlines agreed with those obtained from calculations with an approximation up to \(\dot V^2 \) Because of the elasticity of the fluid a rotationally symmetric eddy arises. It clings to the wall whereas the output flow proceeds in the middle. When $$\frac{\varrho }{{\eta _0 }}\left( {\frac{{\dot V^2 }}{{t_0 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} > 40$$ > 40 another eddy can arise in front of the first one. This second eddy is situated in the middle and causes the output to proceed as a flow near the wall. The higher the afore-mentioned dimensionless value is, the higher is the inclination of the flow to instability.     

Flow of viscoelastic fluids through an abrupt contraction

Summary Developing and fully developed velocity profiles were measured for viscoelastic fluids flowing through an abrupt 2 to 1 glass-contraction. An R 16Weissenberg-Rheogoniometer was used to measure the rheological properties of the viscoelastic fluids in the shear rates range of interest in the contraction. The measured entry lengths for the viscoelastic fluids were significantly less than predictions and experimental values for inelastic fluids with the same power-law parameters. Deviations from inelastic entry behaviour ranged from 11.6–100%, were independent ofReynolds number, but were strongly dependent on the ratio of the friction velocity to the shear wave velocity. Increasing the friction velocity relative to the shear wave velocity resulted in an increased deviation from inelastic behaviour. When the friction velocity was of the same order as the shear wave velocity a zero entry length and a fully developed entry velocity profile were observed. Further increase in the friction velocity relative to the shear wave velocity resulted in anomalous entry behaviour accompanied by unusual flow patterns upstream of the contraction.

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Zusammenfassung Es wurden sich bildende sowie voll ausgebildete Geschwindigkeitsprofile viskoelastischer Flüssigkeiten in einer scharfkantigen Rohrverengung von 2 zu 1 gemessen. Ein Weissenbergsches Rheogoniometer R 16 diente zur Charakterisierung der viskoelastischen Flüssigkeiten im betreffenden Deformationsgeschwindigkeitsbereich.Meßergebnisse für die Einlauflänge viskoelastischer Flüssigkeiten weichen bedeutend von den Voraussagen sowie von Meßergebnissen für unelastische Flüssigkeiten ab, die, nach demOstwald- de Waeleschen Modell berechnet, die gleichen Kenngrößen aufzeigen.Die Abweichung vom viskosen Einlaufverhalten beträgt 11,6 bis 100%. Sie ist unabhängig von der Reynoldschen Zahl, hängt aber sehr stark ab von dem Verhältnis zwischen zwei Geschwindigkeiten u^*=Schubspannungsgeschwindigkeit undu=Scherwellengeschwindigkeit.Eine Erhöhung vonu ^* gegenüberu verursacht eine erhöhte Abweichung vom unelastischen Verhalten. Wenn die zwei Geschwindigkeitenu ^* undu von der gleichen Größenordnung sind, verschwindet die Einlaufsentwicklung und ein vollausgebildetes Geschwindigkeitsprofil tritt schon am Eingang auf. Ein weiteres Erhöhen vonu ^* überu verursacht anomales Einlaufverhalten mit ungewöhnlichem Strömungsbild oberhalb der Verengung.

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On Sabbatical Leave: Dept. of Chemical Engineering, University of Toronto, Toronto 181, Ontario, Canada.     

Structural instability in the sedimentation of particulate suspensions through viscoelastic fluids

A theory is developed to describe a structural instability that has been observed during the sedimentation of particulate suspensions through viscoelastic fluids. The theory is based on the assumption that the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. In keeping with the experimental observations, it predicts that the suspension structure will stratify into vertical columns when a dimensionless stability parameter exceeds a critical value. The column-to-column separation, measured in particle radii, is predicted to be proportional to the square root of the ratio of the dimensionless dispersion coefficient to the product of the particle volume fraction and the Deborah number. The time for the formation of the columns is predicted to scale with the inverse of the average volume fraction. These predictions are in agreement with experimental data reported in the literature.     

A hybrid method for computing the flow of viscoelastic fluids

A hybrid method for computing the flow of viscoelastic and second-order fluids is presented. It combines the features of the finite difference technique and the shooting method. The method is accurate because it uses central differences. Its convergence is at least superlinear. The method is applied to obtain the solutions to three problems of flow of Walters' B' fluid: (a) flow near a stagnation point, (b) flow over a stretching sheet and (c) flow near a rotating disk. Numerical results reveal some new characteristics of flows which are not easy to demonstrate using the perturbation technique.     

Anomalous heat transfer and drag in laminar flow of viscoelastic fluids

A boundary-layer approach is used to derive an expression for the heat transfer coefficient when a viscoelastic fluid flows past a cylinder. The heat-transfer coefficient becomes independent of velocity at large values of the latter, thus explaining the experimental results of James and Acosta. The agreement with the observations of these authors is, however, only qualitative, since their experiments were carried out at relatively low Reynolds numbers. By changing the exponent of the Reynolds number from the boundary-layer approximation value to one which is valid in the range used by James and Acosta, a correlation is suggested which is in satisfactory agreement with their data. Similar considerations are used to explain the additional experimental observation of James and Acosta, i.e. that the drag coefficient for the flow past a cylinder also becomes independent of velocity at large values of the latter quantity.     

Finite element simulation of viscoelastic fluids of the integral type

In the present paper, we develop an algorithm for calculating the steady flow of viscoelastic fluids of the integral type. The calculation is based upon a simple integration of the strains along the streamlines, and the method remains valid for arbitrary elements. The technique is applied to the flow of a Maxwell fluid through a wedge and to the calculation of the hole-pressure error for the Doi-Edwards fluid.     

Stability of plane Poiseuille flow of slightly viscoelastic fluids

Summary The title subject has been examined by the author in a series of papers (Cousins, 1970, 1972a, b), and the assumptions and principal results of those papers are discussed here. The work is motivated by the phenomenon evinced in fluid flow situations, of turbulent drag reduction by certain polymer additives. From a survey of experimental work it is clear that molecular elongation plays an important role in reducing drag by suppressing transverse motions. This effect may be interpreted as a normal stress effect in a continuum theory. A second-order fluid, which is a simple model exhibiting such a property, is used in a linear analysis of disturbances to planePoiseuille flow. Unlike theNewtonin case Squire's theorem is not valid (Lockett, 1969a) and a three-dimensional analysis is required. The viscoelastic terms are in general destabilising. Under certain conditions the first growing disturbance will propagate at an angle to the basic flow, giving a longitudinal vortex structure close to the channel boundaries not present at the onset of instability in aNewtonian fluid. The analysis is extended to finite-amplitude disturbances by introducing a time-dependent amplitude, but calculations are here confined to the simpler two-dimensional case. Disturbances which would decay under linear theory may in fact grow provided the initial amplitude is sufficiently large. A threshold amplitude for instability is found as a function ofReynolds number. The viscoelastic terms are again found to be destabilising. Finally, a further viscoelastic property, that of stress relaxation, is introduced through an integral representation of the stress. A linear analysis is developed and stress relaxation is also shown to be a destabilising influence.With 6 figures     

Oscillatory flow of linear viscoelastic fluids in thin-walled elastico-viscous tubes

Rheologica Acta - The oscillatory flow of a linear visco-elastic fluid in a thin-walled elastico-viscous tube is investigated for the case of long waves. It is found that the elasticity of the...     

Computation of flow of viscoelastic fluids by parameter differentiation

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three-dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first-order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.