Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

Mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder

The steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder in a stream flowing vertically upwards is numerically studied for both cases of heated and cooled cylinders. The governing partial differential equations are transformed into dimensionless forms using an appropriate transformation and then solved numerically using the Keller-box method. The comparison between the solutions obtained and those for a Newtonian fluid is found to be very good. Effects of the mixed convection and elasticity parameters on the skin friction and heat transfer coefficients for a fluid having the Prandtl number equal to one are also discussed. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point. However, for a sufficiently cold cylinder there will not be a boundary layer.     

Analysing flow and heat transfer of a viscoelastic fluid over a semi-infinite horizontal moving flat plate

This paper presents a numerical study of the flow and heat transfer of an incompressible homogeneous second grade type fluid above a flat plate moving with constant velocity U. Such a viscoelastic fluid is at rest and the motion is created by the sheet. The effects of the non-Newtonian nature of the fluid are governed by the local Deborah number K (the ratio between the relaxation time of the fluid and the characteristic time of the flow). When , a new analytical solution for this flow is presented and the effects of fluid's elasticity on flow characteristics, dimensionless stream function and its derivatives are analysed in a wide domain of K. A novel result of the analysis is that a change in the flow solution's behaviour occurs when the dimensionless stream function at the edge of the boundary layer, f_∞, equals 1.0. It is found that velocity at a point decreases with increase in the elasticity of the fluid and, as expected, the amount of fluid entrained diminishes when the effects of fluid's elasticity are augmented. In our heat transfer analyses we assume that the surface temperature has a power-law variation. Two cases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed heat flux (PHF case). Local similarity heat-transfer solutions are given for PST case when s=2 (the wall temperature parameter) whereas when a similarity solution takes place in the case of prescribed wall heat flux. The numerical results obtained are fairly in good agreement with the aforementioned analytical ones.     

Numerical predictions of bubble growth in viscoelastic stretching filaments

In this study, we investigate the growth of bubbles within predominately extensional-deformation flows of thin film stretching form. This involves more than one free-surface to the flow (multiple surfaces), typically as inner (bubble) and outer (filament) boundaries that introduces fluid–gas interfacial treatment. Various bubble initial states and locations may be considered. The problem is discretised in space–time through a hybrid-finite element/volume pressure-correction formulation, coupled with an arbitrary Lagrangian–Eulerian (ALE) coupled with VOF scheme to track domain-mesh and free-surface movement. We contrast these results against the results from a complete ALE algorithm. Various fluid-filament materials have been considered, covering such properties as constant viscosity fluids (Newtonian), low-polymeric/high-solvent viscosity Boger-type (Oldroyd-B) fluids and high-polymeric/low-solvent viscosity elastic-type fluids (Oldroyd-B and Phan-Thien/Tanner). Numerical solutions are presented in terms of comparison between algorithms (ALE versus hybrid ALE/VOF), shapes (bubble shapes, filament shapes), contours of extra-stress (magnitude and location), mid-filament radius and extensional viscosity.     

粘弹性地基上粘弹性输流管道的稳定性分析

从Winkler假设和单轴线性粘弹性本构方程出发,推导了Kelvin-Voigt粘弹性地基上三参量固体模型输流管道的运动微分方程,采用改进的有限差分法,分析了管道和地基的粘弹性参数对输流管道无量纲复频率和无量纲流速之间的变化关系的影响。     

Pressure-oscillation defoaming for viscoelastic fluid

The enhancement of bubble rising velocity was experimentally investigated by mechanically applying an oscillating pressure to a single small air bubble (e.g., 1 mm³) in a viscoelastic fluid. For shear-thinning fluids, the cyclic change in bubble diameter induced by the oscillating pressure generates a continuous strong local flow near the bubble surface. Consequently, the apparent liquid viscosity is reduced and the bubble rising velocity increases by 400 times or more compared to the case without oscillating pressure. However, for a Newtonian fluid, almost no effect was observed with oscillating pressure. Time-series data of the longitudinal and horizontal bubble diameters were obtained experimentally using a stroboscope and a video system, and these data were used to estimate the local shear rate and the local shear viscosity. The increase in bubble rising velocity estimated from the shear viscosity behavior agreed well with the experimental data. Additionally, a periodic change in the bubble shape from a sphere at the maximum bubble size to a cusped shape at the minimum bubble size was observed under strong oscillatory pressure.     

Natural convection of a viscoelastic fluid

The influence of elasticity and shear thinning viscosity on the temperature distribution and heat transfer in natural thermal convection is discussed. The numerical investigations are based on a four-parameter Oldroyd constitutive equation, which represents the typical fluid response of dilute solutions and melts. It was found that especially the second normal-stress difference affects the heat transfer mechanism.     

Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments

The surface tension driven breakup of viscoelastic filaments into droplets is qualitatively different from that of Newtonian liquid filaments. Disturbances on filaments of dilute polymer solutions often grow to a configuration consisting of nascent droplets connected by a thin ligament; the breakup time for this configuration is much longer than that predicted by extensions of Rayleigh's linear stability theory. We present here a nonlinear analysis of surface tension driven breakup of viscoelastic filaments using two complementary approaches that given equivalent results: a transient finite element solution and a one-dimensional thin filament approximation. We show that significant nonlinear effects lead to the experimentally-observed nascent droplet-ligament configuration, and we predict the entire evolution of the filament profile. Agreement with available experimental data for profile evolution and breakup of jets of Newtonian fluids and dilute polymer solutions is excellent.     

Entropy effects in viscoelastic fluid theory

A derivation of viscoelastic fluid theory is given which includes the effect of the entropy dependence upon deformation history. The theory is then specialized to model second-order effects. The resulting theory predicts the observed effects of shear thinning and stress overshoot. Comparison is made with experimental data.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

On the stability of viscoelastic fluid flow

Summary In continuation of two foregoing investigations where stability of rectilinear flows of viscoelastic fluids between two plane parallel plates was analysed, the present paper discusses the analogous problem for flow in a cylindrical tube (Poiseuille flow) and a cylindrical annulus. The linearized disturbance equations for steady instabilities of cellular type are derived taking into account viscoelasticity according to second-order theory as well as inertial forces. ForPoiseuille flow analytical solutions are obtained, whereas for flow in an annulus a technique of numerical integration is utilized. It is found that curves of neutral stability are similar to the ones for plane channel flow giving a hint in this ease also that possibly two different types of instabilities may occur simultaneously.

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Zusammenfassung Im Anschluß an zwei vorangegangene Untersuchungen, in denen die Stabilität von geradlinigen Strömungen viskoelastischer Flüssigkeiten zwischen zwei ebenen, parallelen Platten behandelt worden war, wird hier das analoge Problem für die Strömung in einem Kreiszylinder (Poiseuille-Strömung) und in einem Ringspalt analysiert. Die Störungsgleichungen für stationäre Störströmungen vom Zellulartyp werden unter Zugrundelegung einer Theorie zweiter Ordnung für die Viskoelastizität und Einbeziehung der Trägheit in linearisierter. Form abgeleitet. Für diePoiseuille-Strömung lassen sich analytische Lösungen finden, wohingegen für die Ringspalt-Strömungen die Lösungen durch numerische Integration gewonnen werden. Die Indifferenzkurven sind von ähnlicher Gestalt wie bei der Strömung durch den ebenen Kanal und weisen ebenfalls auf die Möglichkeit eines gleichzeitigen Vorhandenseins zweier verschiedener Störströmungen hin.

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The main results of this paper were presented as the second part of a talk entitled On stability of rectilinear flows of viscoelastic fluids at the annual meeting of German Rheologists held in Bad Münster a. Stein from 28–30 May 1969.     

The squeeze-film flow of a viscoelastic fluid

We report on the use of a numerical method to solve the (inertia-less) squeeze-film flow problem for a viscoelastic fluid. The method is based on a Boundary Element formulation and relies on a time-marching scheme. The viscoelastic fluid is modelled by a constitutive law that allows for a shear stress overshoot mechanism in a suddenly started shear flow. The results show convincingly that the load enhancement sometimes observed experimentally is due to stress overshoot. A simple explanation for the enhancement is suggested; the stress overshoot appears quickly and takes a long time to die away, so that steady-state viscous behaviour is not very relevant.     

On the stability of viscoelastic fluid flow

Summary This paper supplements the earlier ones of this series, which dealt withstationary instabilities of cellular type and generalizes the case of planeCouette flow foroscillatory cellular disturbances, usually termed as over-stability. Using the well-known scheme of linearization the disturbance equations are derived and the analytical solutions have been given. By numerical evaluation of these, the curves of neutral stability are calculated explicitely for certain typical cases. It is found that, in general, the overstability mode is higher than the stationary mode but that both can come close to each other if certain conditions are satisfied.

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Zusammenfassung Die vorliegende Veröffentlichung ergänzt die vorangehenden dieser Serie, welchestationäre Instabilitäten vom Zellulartyp zum Gegenstand haben, indem sie für den Fall der ebenenCouette-Strömungoszillierende Zellularströmungen (overstability) in die Untersuchung einbezieht. Unter Verwendung des wohlbekannten Linearisierungsverfahrens werden die Störungsgleichungen abgeleitet und analytisch gelöst. Aus diesen Lösungen werden für einige typische Fälle die Indifferenzkurven numerisch bestimmt. Man findet ganz allgemein, daß die Indifferenzkurven der oszillierenden Störungen höher liegen als diejenigen der stationären Störungen, daß beide jedoch unter gewissen Bedingungen sehr nahe beieinander liegen können.

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The main results of this paper were presented as the first part of a talk entitled Stationäre und periodische Instabilitäten in Strömungen viskoelastischer Flüssigkeiten vomCouette- undPoiseuille-Typ at the annual meeting of German Rheologists held in Berlin from May 11–13, 1970.     

On the stability of viscoelastic fluid flow

Summary The stability of a viscoelastic fluid flowing through a plane parallel channel, in the presence of constant pressure gradient is analyzed. The linearized disturbance equations for steady instabilities of cellular type are derived (taking inertia fully into account but viscoelasticity as in second-order theory) and solved using an approximation method as well as direct integration procedure. It is found that, as also in the case of planeCouette flow, such types of instabilities should be expected only for materials for which second normal-stress difference is positive. The curves of neutral stability possess more complicated nature than those forCouette flow, and may give a hint to the simultaneous existence of two types of disturbances with different cell widths, provided the parameter representing the ratio of inertial to elastic forces lies in a certain range.

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Zusammenfassung Es wird die Stabilität der Strömung einer viskoelastischen Flüssigkeit durch einen ebenen, parallelen Kanal bei Anwesenheit eines konstanten Druckgradienten untersucht. Bei Erfassung der Viskoelastizität durch eine Theorie zweiter Ordnung und vollständiger Berücksichtigung der Trägheit werden die linearisierten Störungsgleichungen abgeleitet und sowohl mittels einer Näherungsmethode als auch durch ein direktes Integrationsverfahren gelöst. Man findet, daß genauso wie im Fall der ebenenCouette-Strömung Instabilitäten eines derartigen Typs nur dann zu erwarten sind, wenn die zweite Normalspannungsdifferenz des Materials positiv ist. Die Indifferenzkurven sind von komplizierterer Form als bei derCouette-Strömung und deuten darauf hin, daß u. U. zwei Arten von Störungen mit verschiedener Zellgröße gleichzeitig auftreten können, wenn der Parameter, welcher das Verhältnis von Trägheits- und elastischen Kräften bezeichnet, in einem gewissen Wertebereich liegt.

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Herrn Professor Dr. K.Hansen zum 60. Geburtstag gewidmet

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The main results of this paper were presented as the first part of a talk entitled On stability of rectilinear flows of viscoelastic fluids at the annual meeting of German Rheologists held in Bad Münster a. Stein from 28–30 May 1969.     

Unsteady motion of a non-linear viscoelastic fluid

In this paper, we study the unsteady flow of a generalized second grade fluid. Specifically, we solve numerically the linear momentum equations for the flow of this viscoelastic shear-thinning (shear-thickening) fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, especially in oil-drilling applications.     

Unsteady flows of a third-order viscoelastic fluid

Summary Unsteady third-order viscoelastic flow is examined for the purpose of determining a method by which certain material constants in theRivlin-Ericksen theory can be measured. The concept of an intrinsic pressure hole error is used, and an expression for this error is derived for third-order flow. It is also shown that the velocity field for unsteady creeping motion of a third-order fluid is identical at any time to a corresponding steady third-order flow. The pressure field, however, depends upon the rate of change of the flow with time.

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Zusammenfassung Es wird die instationäre Strömung dritter Ordnung einer viskoelastischen Flüssigkeit untersucht zu dem Zweck, eine Methode anzugeben, mit deren Hilfe gewisse Stoffkonstanten in der Theorie vonRivlin undEricksen gemessen werden können. Dazu wird das Konzept des intrinsic pressure hole error verwendet, und es wird für diesen Fehler im Rahmen der Theorie dritter Ordnung ein Ausdruck abgeleitet. Weiter wird gezeigt, daß das Geschwindigkeitsfeld für die instationäre schleichende Strömung einer Flüssigkeit dritter Ordnung zu jeder Zeit mit der zugeordneten stationären Strömung dritter Ordnung identisch ist. Im Gegensatz dazu hängt das Druckfeld jedoch von der zeitlichen Änderung der Strömungsgeschwindigkeit ab.

     

Interaction of spheres in a viscoelastic fluid

Summary The interaction between two identical spheres of radiusa in a second-order fluid is studied, if the undisturbed flow is a general homogeneous flow. WithR the (instantaneous) distance between the sphere centers only the situationa/R 1 is considered. It turns out that it is not sufficient to know thea/R-term of the perturbation velocity, since certain contributions of the (a/R)²-terms are also needed. For two spheres sedimenting in a quiescent fluid a change of the relative position vector is predicted: the distance decreases and so does the orientation, i.e. the spheres tend to fall along their line of centers. If the motion of the individual sphere is restrained via a rigid connection (rigid dumbbell) this change of orientation implies that the dumbbell rotates until its axis is parallel to the direction of the applied force (stable orientation). In simple shear the first-order dumbbell (a/R-terms due to interaction) ultimately ends up in the plane normal to the gradient direction, independent of the rate of shear. This contrasts the behavior of a second-order dumbbell: if the symmetry axis lies in the plane of flow it will rotate around the vorticity axis at small rates of shear. Increasing the shear rate this dumbbell reaches a spinfree terminal state in which the angle between the symmetry axis and the flow direction is non-zero (although it is small). It is conjectured that for arbitrary initial orientations (not in the flow plane) the axis of the second-order dumbbell will not rotate in the Jeffrey orbits but rather show a systematic drift to become oriented parallel to the vorticity axis.

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Zusammenfassung Die Wechselwirkung zweier identischer Kugeln vom Radiusa in einer beliebigen homogenen Strömung einer Flüssigkeit zweiter Ordnung wird untersucht. MitR dem (augenblicklichen) Abstand der Kugelzentren beschränken wir uns auf die Situationa/R 1. Es zeigt sich, daß es nicht genügt, diea/R-Glieder der Störungsgeschwindigkeit zu kennen, da einige Beiträge der (a/R)²-Terme ebenso benötigt werden. Für zwei in einer ruhenden Flüssigkeit sedimentierende Kugeln wird eine Änderung der relativen Position vorausgesagt: der Abstand verkleinert sich, und das gleiche gilt für die Orientierung, d. h. die Kugeln streben die Situation, hintereinander zu fallen, an. Schränkt man die Bewegung der individuellen Kugeln durch eine starre Verbindung ein (starre Hantel), so zieht diese Orientierungsänderung eine Rotation nach sich, die die Hantelachse parallel zur Richtung der angreifenden Kraft ausrichtet (stabile Orientierung). Bei einer einfachen Scherung wandert unabhängig von der Schergeschwindigkeit die Achse einer Hantel erster Ordnung (die nur die Wechselwirkungsgliedera/R enthält) in der Ebene, deren Normale in Gradientenrichtung zeigt. Damit verhält sie sich völlig anders als eine Hantel zweiter Ordnung: Liegt bei letzterer die Symmetrieachse in der Strömungsebene, so rotiert diese bei kleinen Schergeschwindigkeiten um eine Achse senkrecht zur Strömungsebene. Bei einer Vergrößerung der Schergeschwindigkeit wird dagegen eine rotationsfreie Lage erreicht, bei der die Hantelachse unter einem kleinen Winkel zur Strömungsrichtung steht. Bei einer beliebigen Anfangsorientierung (außer in der Strömungsebene) schließen wir auf eine Wanderung der Achse einer Hantel zweiter Ordnung, bis diese parallel zur indifferenten Richtung steht.

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With 5 figures