Flow Of Viscoelastic Liquids Research Articles

Role of slip in the stability of viscoelastic liquid flow through a channel

For a horizontal channel flow of a weakly viscoelastic liquid that follows the constitutive model of Walters’ liquid B′′, a linear stability analysis is performed when the channel walls are slippery. In this case, the viscoelastic channel flow is driven by the streamwise pressure gradient. We explore the primary instability for a symmetric slip flow with the same slip length of the channel walls, an asymmetric slip flow with different slip lengths of the channel walls, and an analogy to the Poiseuille–Couette flow of a viscoelastic liquid with a slip effect on the lower wall but no slip effect on the upper wall. We observe that the viscoelastic parameter shows a destabilizing impact, but the slip length shows a stabilizing impact on the most unstable shear mode in all three flow configurations. Moreover, the Poiseuille and Poiseuille–Couette flows for the viscoelastic liquid are linearly more unstable to infinitesimal disturbances than those of the Newtonian liquid. As a result, wall velocity needs to be higher than in the Newtonian liquid to stabilize the viscoelastic Poiseuille–Couette flow. Using the asymptotic analysis, we have determined the critical value of the slip length, or equivalently, the critical value of the lower wall velocity, above which the asymmetric slip flow and the Poiseuille–Couette flow of the viscoelastic liquid are permanently stable to infinitesimal disturbances.

Two layered immiscible flow of viscoelastic liquid in a vertical porous channel with Hall current, thermal radiation and chemical reaction

This investigation deals with the combined effects of Hall current, thermal radiation, heat source and chemical reaction on heat and mass characteristics in a vertical porous channel filled with a two layered viscoelastic liquid under the influence of oscillatory pressure gradient. The exact solution of the governing equations for the fluid velocity, temperature, and concentration are obtained by applying the regular perturbation method with appropriate boundary and interface conditions. The numerical values of the shear stress at the plates are shown in tabular form for various values of the relevant parameters, whereas those of the fluid velocity, temperature, concentration, and rate of heat transfer are displayed graphically. Our analysis indicates that with the raise of first-order chemical reaction and ratio of mass diffusion parameters, the temperature and concentration of the liquid raises. With the raise of Hall parameter the skin friction decreases at the walls of the channel. To validate the analytical results, a comparative study of the numerical evaluation of the closed-form results is performed with the results obtained by shooting technique along with the RK-fourth order method. Graphical results demonstrate that the imposed boundary and interface conditions are satisfied by the flow variables.

Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime

We report the temporal and spatiotemporal stability analyses of antisymmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We). The resulting fourth-order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations that are numerically integrated starting from the rescaled far-field conditions, i.e., via Compound Matrix Method. The temporal stability analysis indicates that with increasing We, (a) the entire range of the most unstable mode is shifted toward longer waves (i.e., the entire region of temporal instability is gradually concentrated near zero wavenumber), (b) the vorticity structure contours are dilated, and (c) the residual Reynolds stresses are diminished. All these analogous observations previously reported in the inertial limit [J. Azaiez and G. M. Homsy, “Linear stability of free shear flow of viscoelastic liquids,” J. Fluid Mech. 268, 37–69 (1994).] suggest a viscoelastic destabilization mechanism operating at low and moderate Re. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities, as well as evanescent modes. The main result is that the free shear flow of dilute polymeric liquids is either (absolutely/convectively) unstable for all Re or the transition to instability occurs at comparatively low Re, a finding attributed to the fact that viscoelasticity aggravates instabilities via shear-induced anisotropy and the slow relaxation effects.

Mixed convective radiative flow of vicoelastic liquid subject to space dependent internal heat source and chemical reaction

Present study addresses Soret and Dufour effects in mixed convection MHD flow of viscoelastic liquid with chemical reaction. Flow induced by an exponential stretching sheet is addressed in the presence of magnetic field. Energy expression is modelled by exponential space dependent internal heat source, thermal radiation, and convective condition. Relevant problems are modelled by employing boundary-layer concept. The partial differential systems are reduced to ordinary differential systems, and problem is solved by homotopic technique. Physical insight of results is arranged by graphs and tables.

Numerical computations of fractional nonlinear Hartmann flow with revised heat flux model

In this communication we have examined the magneto-hydrodynamic flow of viscoelastic liquid with revise version of thermal flux. Thermal conductivity of the fluid is variable. Effects of body forces such as gravity are also examined in terms of convection. Non-integer time derivatives are used for the better understanding of viscoelastic flow characteristics. In literature, these derivatives have been proved suitable for the predictions of hereditary systems such as viscoelastic flows. Heat transfer is analyzed in the flow field via modified fractional thermal gradient. Temperature gradient exists between the flow boundaries and increases with the passage of time. Flow is generated by the variable movement of flow boundary. All the segments of numerical modeling are completely addressed in this paper i.e. physical system, mathematical model, simulation, prediction, validation and verification. Unsteady motion of in-compressible fluid is directed by differential equations with fractional partial derivatives, for the solutions of these equations numerical techniques are necessary to use. Governing equations with appropriate flow conditions have been discretized by finite difference along with finite element algorithms. Theoretical error appraisals are hypothesized and created numerical plan has been approved by doing error investigations numerically regarding spatial directions. We have shown the estimate error and plotted the error curves in logarithmic scales of L2(Ω) and H1(Ω) norms by taking absolute approximations of the space discretization. Local Nusselt number and coefficients of skin friction are determined for non-integer viscoelastic model. Flow field is simulated for various values of physical parameters. The obtained results showed that fractional exponent α has opposite effects on the velocity, temperature profiles for time t=2. It is also noted that temperature gradient increases by the increase of fractional number α and thermal relaxation parameter γ. Viscoelastic flows in polymer and plastics industries can be addressed by similar technique.

Linear stability of a viscoelastic liquid flow on an oscillating plane

Linear stability of a viscoelastic liquid on an oscillating plane is studied for disturbances of arbitrary wavenumbers. The main aim is to extend the earlier study of Dandapat & Gupta (J. Fluid Mech., vol. 72, 1975, pp. 425–432) to the finite wavenumber regime, which has not been attempted so far in the literature. The Orr–Sommerfeld boundary value problem is formulated for an unsteady base flow, and it is resolved numerically based on the Chebyshev spectral collocation method along with the Floquet theory. The analytical solution predicts that U-shaped unstable regions appear in the separated bandwidths of the imposed frequency, and the dominant mode of the long-wave instability intensifies in the presence of the viscoelastic parameter. The numerical solution shows that oblique neutral curves come out from the branch points of the U-shaped neutral curves at finite wavenumber and continue with the imposed frequency until the curves cross the next U-shaped neutral curve. As a consequence, in the finite wavenumber regime, no stable bandwidth of the imposed frequency is predicted by the long-wavelength analysis. Further, in some frequency ranges, the finite wavenumber instability is more dangerous than the long-wave instability.

Lid-driven cavity flow of viscoelastic liquids

The lid-driven cavity flow is a well-known benchmark problem for the validation of new numerical methods and techniques. In experimental and numerical studies with viscoelastic fluids in such lid-driven flows, purely-elastic instabilities have been shown to appear even at very low Reynolds numbers. A finite-volume viscoelastic code, using the log-conformation formulation, is used in this work to probe the effect of viscoelasticity on the appearance of such instabilities in two-dimensional lid-driven cavities for a wide range of aspect ratios (0.125≤Λ=height/length≤4.0), at different Deborah numbers under creeping-flow conditions and to understand the effects of regularization of the lid velocity. The effect of the viscoelasticity on the steady-state results and on the critical conditions for the onset of the elastic instabilities are described and compared to experimental results.

The oscillating drop method for measuring the deformation retardation time of viscoelastic liquids

The use of the oscillating drop method for measuring the deformation retardation time λ2 of viscoelastic liquids is proposed. For small oscillation amplitudes, λ2 may be determined from the characteristic equation of the drop derived from linear theory. In the experiment an acoustically levitated individual drop is excited to shape oscillations of the fundamental mode m=2 by ultrasound modulation. Once the excitation is terminated, the drop performs damped oscillations, and the angular frequency and damping rate are measured from drop shapes recorded by a high-speed camera. A numerical method is used for determining a pair of liquid properties from the characteristic equation – the liquid zero-shear dynamic viscosity η0 and the deformation retardation time λ2. The spherical Bessel functions involved in the equation produce a manifold of solutions, from which the correct one must be identified. Comparison of the computed value of η0 with the result from a rheometric liquid characterisation is found to be the right criterion for the identification. The values of λ2 obtained by this measurement are found to depend weakly on uncertainties of the experiment. They deviate strongly from the values typically used in simulations of viscoelastic liquid flow.

Single particle motion in a sheared colloidal dispersion

The motion of a spherical Brownian “probe” particle addressed by an external force immersed in shear flow of a colloidal dispersion of spherical neutrally buoyant “bath” particles is quantified. The steady-state nonequilibrium microstructure of bath particles around the probe—induced by the applied force and ambient shear—is calculated to first order in the volume fraction of bath particles, ϕ. The distortion to the equilibrium microstructure caused by the moving probe is characterized by a Péclet number Pef (a dimensionless pulling force), and the distortion due to the shear flow is represented by another Péclet number Pes (a dimensionless shear rate). Matched asymptotic expansions are employed to quantify the microstructure at small Péclet numbers; specifically, within the distinguished limits Pes3/2≪Pef≪Pes1/2≪1. The nonequilibrium microstructure is subsequently utilized to compute the average rectilinear velocity of the probe through O(ϕPes3/2Us), for an arbitrary orientation of the external force to the shear flow. Here, Us is the Stokes velocity of the probe in a pure Newtonian fluid. It is also shown that to O(ϕPes3/2Us) the torque-free probe simply rotates with the ambient shear; a modification to the angular velocity of the probe is at most O(ϕPesPefUs). In particular, a probe forced along the flow axis of shear is demonstrated to experience a cross-streamline drift velocity of O(ϕPesUs), to leading order, which acts to propel the particle to streamlines of the ambient shear that move in the same direction as the external force. A mathematical connection between this result and cross-streamline drift of a particle in a Newtonian fluid at small, but nonzero, Reynolds numbers is drawn. The magnitude of the cross-streamline drift velocity is found to be sensitive to the degree of hydrodynamic interactions between the probe and bath particles, which are tuned via an excluded-annulus model. It is also demonstrated that a probe forced along the vorticity axis of the shear experiences a shear-driven enhancement in rectilinear velocity of O(ϕPes3/2Us), to leading order: This nonanalytic dependence originates from the microstructural deformation in the shear dominated (outer) region far from the probe. A connection of this finding to recent work on particle sedimentation in orthogonal shear flow of viscoelastic liquids is discussed.

A New Stabilization of Adaptive Step Trapezoid Rule Based on Finite Difference Interrupts

The adaptive step trapezoid rule (TR) is a generally effective numerical integrator, but it is prone to ringing instability and solution stall. We introduce a new stabilization algorithm based on finite difference interrupts (FDI) that reduces ringing and prevents stall. Unlike previously reported stabilization schemes, our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion. TR with FDI is at least as stable and accurate as existing methods when solving the prototypical scalar problem. We demonstrate that it has better stability, accuracy, and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids. Though we demonstrate TR with FDI in the context of a finite element method-of-lines, it is applicable to any TR-based algorithm.

Modeling Polymer Melt Flow at the Outlet from an Extruder Molding Tool

Use of a Phan-Thien–Tanner model for simulating viscoelastic liquid flow at the outlet from an extruder molding tool is proposed. A dependence for free extruded surface profile of cylindrical extruded objects on Deborah number is determined. Modeling results may be used for selecting extrusion processing regimes and in designing new extrusion heads with the aim of improving object quality characteristics.

Suppression of purely elastic instabilities in the torsional flow of viscoelastic fluid past a soft solid

Experiments are performed to explore the role of a soft, deformable solid layer on the purely elastic instability in the torsional flow of polymer solutions between two circular discs. The gel layer is placed on the stationary bottom plate of a rheometer, and the polymer solution is placed between the gel and the rotating top disc. The observed variation of viscosity with shear rate (or shear stress) is correlated with the presence or absence of purely elastic instability in the viscometric flow. Earlier work has shown that with increase in shear rate, the torsional flow of a polymer solution between rigid discs undergoes transition from the simple viscometric flow state to elastic turbulence via a sequence of instability modes. We combine rheological observations and flow visualization to show that the deformable solid has a profound effect on the stability of the torsional flow. In marked contrast to flow between rigid plates (where the fluid shows apparent shear-thickening at the onset of instability), the apparent viscosity continues to decrease up to a much larger value of shear rate with the presence of a soft gel. At a fixed shear rate, for flow past a soft gel, the measured stress does not exhibit marked temporal fluctuations that would otherwise be present without the soft gel. Using flow visualization, we show that secondary flow patterns that form after the instability for a rigid surface disappear for flow on soft gel surfaces. In the case of rigid surfaces, the instability is sub-critical and exhibits hysteresis behavior, which again is absent when the flow occurs past a soft solid layer. Our results show that the role of the soft deformable solid is to suppress the purely elastic instability in torsional flows of polymeric liquids for intermediate shear rates. While it is known that soft deformable solids destabilize the flow of Newtonian liquids in the absence of inertial effects, our study shows that the effect of deformability can be opposite in the torsional flow of viscoelastic liquids.

Boundary conditions for hyperbolic systems of partial differentials equations

An easy-to-apply algorithm is proposed to determine the correct set(s) of boundary conditions for hyperbolic systems of partial differential equations. The proposed approach is based on the idea of the incoming/outgoing characteristics and is validated by considering two problems. The first one is the well-known Euler system of equations in gas dynamics and it proved to yield set(s) of boundary conditions consistent with the literature. The second test case corresponds to the system of equations governing the flow of viscoelastic liquids.

A mechanism for oscillatory instability in viscoelastic cross-slot flow

Interior stagnation-point flows of viscoelastic liquids arise in a wide variety of applications including extensional viscometry, polymer processing and microfluidics. Experimentally, these flows have long been known to exhibit instabilities, but the mechanisms underlying them have not previously been elucidated. We computationally demonstrate the existence of a supercritical oscillatory instability of low-Reynolds-number viscoelastic flow in a two-dimensional cross-slot geometry. The fluctuations are closely associated with the ‘birefringent strand’ of highly stretched polymer chains associated with the outflow from the stagnation point at high Weissenberg number. Additionally, we describe the mechanism of instability, which arises from the coupling of flow with extensional stresses and their steep gradients in the stagnation-point region.

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink

This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat source, where the flow is generated due to a linear stretching of the sheet and influenced by uniform magnetic field applied vertically. Here an analysis has been carried out to study the effect of magnetic field on the visco-elastic liquid flow and heat transfer over a stretching sheet with non-uniform heat source. The non-linear boundary layer equation for momentum is converted into ordinary differential equation by means of similarity transformation and is solved exactly. Heat transfer differential equation is also solved analytically. The effect of magnetic field on velocity, skin friction and temperature profiles are presented graphically and discussed.

Instability suppression in viscoelastic film flows down an inclined plane lined with a deformable solid layer

The linear stability of viscoelastic (Oldroyd-B) film flow down an inclined plane lined with a deformable (neo-Hookean) solid layer is analyzed using low-wave-number asymptotic analysis and the Chebyshev-Tau spectral numerical method. The free surface of film flows of viscoelastic liquids, unlike that of their Newtonian counterparts, becomes unstable in flow down a rigid inclined surface even in the absence of fluid inertia, due to the elastic nature of the liquids. For film flow past a deformable solid, our low-wave-number asymptotic analysis reveals that the solid deformability has a stabilizing effect on the free-surface instability, and, remarkably, this prediction is insensitive to rheology of the liquid film, be it viscoelastic or Newtonian. Using the spectral numerical method, we demonstrate that the free-surface instability can be completely suppressed at all wave numbers when the solid becomes sufficiently deformable. For the case of pure polymeric liquids without any solvent, when the solid layer is made further deformable, both the free surface and the liquid-solid interface are destabilized at finite wave numbers. We also demonstrate a type of mode exchange phenomenon between the modes corresponding to the two interfaces. Importantly, our numerical results show that there is a sufficient range of shear modulus of the solid where both the modes are stable at all wave numbers. For polymer solutions described by the Oldroyd-B model, while the free-surface instability is suppressed by the deformable solid, a host of new unstable modes appear at finite Reynolds number and wave number because of the coupling between liquid flow and free shear waves in the solid. Our study thus demonstrates that the elastohydrodynamic coupling between liquid flow and solid deformation can be exploited either to suppress the free-surface instability (present otherwise in rigid inclines) in viscoelastic film flows, or to induce new instabilities that are absent in flow adjacent to rigid surfaces.

Instability of coextruded elastic liquids at high Weissenberg number

We reconsider the linear stability of pressure-driven flow of two Oldroyd-B or UCM fluids symmetrically placed in a channel or a pipe (a “coextrusion flow”). The fluids have matched viscosities but different relaxation times. Inertia and surface tension are neglected. We focus on the high Weissenberg number limit, where the distance travelled by a fluid particle in a relaxation time is large compared to the width of the channel. Given a prescribed location of the unperturbed interface, the growth rate at sufficiently high wavenumber can be deduced from previous analysis of Couette flow [Y. Renardy, Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newtonian Fluid Mech. 28 (1988) 99–115; K. Chen, D.D. Joseph, Elastic short-wave instability in extrusion flows of viscoelastic liquids, J. Non-Newtonian Fluid Mech. 42 (1992) 189–211; J.C. Miller, J.M. Rallison, Interfacial instability between sheared elastic liquids in a channel, J. Non-Newtonian Fluid Mech. 143 (2007) 71–87; however, we find that the structure of the perturbation flow differs from the Couette case. A new regime with different stability properties was first observed in a companion paper [J.C. Miller, J.M. Rallison, Interfacial instability between sheared elastic liquids in a channel, J. Non-Newtonian Fluid Mech. 143 (2007) 71–87]. In this narrow-core regime the fraction of the channel occupied by the inner fluid is small and the wavenumber is large. We examine the corresponding distinguished limit and demonstrate that instability can occur at intermediate wavenumbers in flow geometries for which the long-wave and short-wave limits are both stable, contradicting some literature claims [P. Laure, H.L. Meur, Y. Demay, J.C. Saut, S. Scotto, Linear stability of multilayer plane Poiseuille flows of Oldroyd-B fluids, J. Non-Newtonian Fluid Mech. 71 (1997) 1–23; S. Scotto, P. Laure, Linear stability of three-layer Poiseuille flow for Oldroyd-B fluids, J. Non-Newtonian Fluid Mech. 83 (1999) 71–92]. The wavespeed of this mode can be substantially higher than the speed of the base flow. The perturbation grows on the time-scale of the shorter relaxation time of the two fluids. Although most of the analysis is performed for asymptotically large Weissenberg numbers, the instability is found to persist for values which should be experimentally accessible. For core-annular pipe flow the curvature of the unperturbed interface plays a role, and the stability characteristics of channel and pipe flow are found to be qualitatively different. In addition, we find some new instabilities in channel flow which fall outside the categories previously identified. We demonstrate finally that on symmetry grounds analogous narrow-core instabilities must arise for a wider class of elastic constitutive properties than Oldroyd-B.

Plane sudden expansion flows of viscoelastic liquids

We report a systematic numerical investigation of the creeping flow of three different viscoelastic models, the UCM, Oldroyd-B and the linear form of the PTT model, through a 1:3 planar sudden expansion. Although the effect of elasticity is to reduce both the length and intensity of the recirculation region downstream of the expansion, we show that this reduction is much lower than previous studies have suggested and that, at high Deborah number, a significant region of recirculation still exists for all of the models studied.

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case. Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].

Numerical modelling of viscoelastic cavity driven flow using finite difference simulations

A study of the unsteady flow of viscoelastic liquids contained in an cavity driven by a moving wall has been undertaken as a first step toward understanding heat and mass transport in polymer processing equipment. The material was represented by the constitutive equation for a upper convected Criminale–Ericson–Filbey (CEF) model. Solution to the vorticity equation for various moving walls and different aspect ratios were obtained numerically. The solutions were found to be stable and convergent for limited value of Weissenberg numbers. Despite this fact, some new results, which are governed by inertia, elasticity and aspect ratio, were obtained and this has been documented first time.