Oldroyd-B Constitutive Equation Research Articles

Convection instability of linear Oldroyd-B fluids in a vertical channel with non-Fourier heat flux model

Owing to the importance of non-Fourier heat flux model in several natural and engineering processes, the convection of binary viscoelastic fluid in a vertical channel with non-Fourier heat flux model is investigated. The linear Oldroyd-B constitutive equation is used to model viscoelasticity. The presence of the basic flow in the vertical y-direction makes the problem challenging compared with the case in Rayleigh–Bénard convection. We use the Chebyshev collocation method to explore the instability characteristics of the linear Oldroyd-B fluid under a wide variety of physical parameters. Results show that the non-Fourier effect and relaxation time contribute to destabilize the system for oscillatory convection. The retardation time can inhibit the instability of the convective system. In the absence of the non-Fourier effect, the vertical fluid layer cannot support oscillatory motions. Oscillatory motion is possible, and the neutral stability curve branches when the non-Fourier effect is taken into account in the fluid. In addition, a new interesting phenomenon can be found: under the coupling action of viscoelastic fluids and the non-Fourier effect, the neutral stability curve would change from single to two branches and then to a single branch with the increase in relaxation time.

Benchmarking the immersed boundary method for viscoelastic flows

We present and analyze a series of benchmark tests regarding the application of the immersed boundary (IB) method to viscoelastic flows through and around non-trivial, stationary geometries. The IB method is widely used to simulate biological fluid dynamics and other modeling scenarios in which a structure is immersed in a fluid. Although the IB method has been most commonly used to model systems involving viscous incompressible fluids, it also can be applied to visoelastic fluids and has enabled the study of a wide variety of dynamical problems including the settling of vesicles and the swimming of elastic filaments in fluids modeled by the Oldroyd-B constitutive equation. In the viscoelastic context, however, relatively little work has explored the accuracy or convergence properties of this numerical scheme. Herein, we present benchmarking results for an IB solver applied to viscoelastic flows in and around non-trivial geometries using either the idealized Oldroyd-B constitutive model or the more physically realistic, polymer-entanglement-based Rolie-Poly constitutive equations. We use two-dimensional numerical test cases along with results from rheology experiments to benchmark the IB method and compare it to more complex finite element and finite volume viscoelastic flow solvers. Additionally, we analyze different choices of regularized delta function and relative Lagrangian grid spacings which allow us to identify and recommend the key choices of these numerical parameters depending on the present flow regime.

A new elastic instability in gravity-driven viscoelastic film flow

We examine the linear stability of the gravity-driven flow of a viscoelastic fluid film down an inclined plane. The viscoelastic fluid is modeled using the Oldroyd-B constitutive equation and, therefore, exhibits a constant shear viscosity and a positive first normal stress difference in viscometric shearing flows; the latter class of flows includes the aforesaid film-flow configuration. We show that the film-flow configuration is susceptible to two distinct purely elastic instabilities in the inertialess limit. The first instability owes its origin entirely to the existence of a free surface and has been examined earlier [Shaqfeh et al., “The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number,” J. Non-Newtonian Fluid Mech. 31, 87–113 (1989)]. The second one is the analog of the centermode instability recently discovered in plane Poiseuille flow [Khalid et al., “Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow,” Phys. Rev. Lett. 127, 134502 (2021)] and owes its origin to the base-state shear; it is an example of a purely elastic instability of shearing flows with rectilinear streamlines. One may draw an analogy of the aforesaid pair of unstable elastic modes with the inertial free-surface and shear-driven instabilities known for the analogous flow configuration of a Newtonian fluid. While surface tension has the expected stabilizing effect on the Newtonian and elastic free-surface modes, its effect on the corresponding shear modes is, surprisingly, more complicated. For both the Newtonian shear mode and the elastic centermode, surface tension plays a dual role, with there being parameter regimes where it acts as a stabilizing and destabilizing influence. While the Newtonian shear mode remains unstable in the limit of vanishing surface tension, the elastic centermode becomes unstable only when the appropriate non-dimensional surface tension parameter exceeds a threshold. In the limit of surface tension being infinitely dominant, the free-surface boundary conditions for the film-flow configuration reduce to the centerline symmetry conditions satisfied by the elastic centermode in plane Poiseuille flow. As a result, the regime of instability of the film-flow centermode becomes identical to that of the original channel-flow centermode. At intermediate values of the surface tension parameter, however, there exist regimes where the film-flow centermode is unstable even when its channel-flow counterpart is stable. We end with a discussion of the added role of inertia on the aforementioned elastic instabilities.

Viscoelastic effects on the deformation and breakup of a droplet on a solid wall in Couette flow

The deformation, movement and breakup of a wall-attached droplet subject to Couette flow are systematically investigated using an enhanced lattice Boltzmann colour-gradient model, which accounts for not only the viscoelasticity (described by the Oldroyd-B constitutive equation) of either droplet (V/N) or matrix fluid (N/V) but also the surface wettability. We first focus on the steady-state deformation of a sliding droplet for varying values of capillary number ( $Ca$ ), Weissenberg number ( $Wi$ ) and solvent viscosity ratio ( $\beta$ ). Results show that the relative wetting area $A_r$ in the N/V system is increased by either increasing $Ca$ , or by increasing $Wi$ or decreasing $\beta$ , where the former is attributed to the increased viscous force and the latter to the enhanced elastic effects. In the V/N system, however, $A_r$ is restrained by the droplet elasticity, especially at higher $Wi$ or lower $\beta$ , and the inhibiting effect strengthens with an increase of $Ca$ . Decreasing $\beta$ always reduces droplet deformation when either fluid is viscoelastic. The steady-state droplet motion is quantified by the contact-line capillary number $Ca_{cl}$ , and a force balance is established to successfully predict the variations of $Ca_{cl}/Ca$ with $\beta$ for each two-phase viscosity ratio in both N/V and V/N systems. The droplet breakup is then studied for varying $Wi$ . The critical capillary number of droplet breakup monotonically increases with $Wi$ in the N/V system, while it first increases, then decreases and finally reaches a plateau in the V/N system.

Heat and mass transfer analysis of constitutive model with autocatalytic chemical reaction within the Jeffery–Hamel flow perspective

This article examines the heat and mass transfer capabilities of a constitutive model in a thermally evolving steady laminar Jeffery–Hamel flow through a convergent-plate channel, including streamwise conduction with step changes in uniform wall temperature. A Jeffery–Hamel problem with a simple shear flow is used to undertake a comparative computational analysis of the thermal behavior of a viscoelastic fluid subjected to autocatalytic processes. The flow is tracked in a purely radial orientation with the deployment of coupled stresses in momentum conservation. The computational solutions for the flow, temperature and concentration distribution, and heat and mass transfer coefficient of a viscoelastic fluid obeying the complex Oldroyd-B constitutive equation in laminar converging channel flows are established. The analysis of the impacts of the thermal radiation, the heat source, and the chemical reaction as an autocatalytic process is included in the model, which is valid for fully developed thermal and hydrodynamic flow conditions with a constant heat and mass flux imposed at the wall. In the diverging part of the channel, where vortex compression is the predominant flow topology, there exist patches of local flow compression. On the flow field, the modified relaxation and retardation parameters show an opposing behavior. An Oldroyd-B fluid exhibits higher interactions with nearby vortices in the divergent channel, allowing a complex flow structure. The viscoelastic characteristics are anticipated to change the homogeneous–heterogeneous reaction transport processes, offering tremendous potential for applications in associated sectors. The deceleration flow in the diverging channel and the acceleration flow in the converging channel augment the average Nusselt numbers.

Numerical simulation of electroosmotic mixing of non-Newtonian fluids in a micromixer with zeta potential heterogeneity

Reagent mixing for microchemical analysis is critical. Most biological fluids and polymer solutions used in the biomedical field belong to the category of non-Newtonian fluids. In this study, the mixing of polyacrylamide (PAA) solutions driven by an electric field in a micromixer is numerically investigated. The micromixer is equipped with wall-mounted obstacles and surface potential heterogeneity. The simulation is performed by solving the Laplace equation, Poisson-Boltzmann equation, Navier–Stokes equations, Oldroyd-B constitutive equation, and species transport equation. The mixing efficiency increases from 63.9% to 97.6% when the obstacle surface zeta potential is increased from 20mV to 80mV. However, when the obstacle surface zeta potential is increased from 80mV to 120mV, the mixing efficiency decreases. When the EDLs overlap in the channel (κ = 0.5), the heterogeneous potential has little effect on the mixing efficiency, and the mixing efficiency can be close to 99%. When Wi increases from 0 to 0.3320, the mixing efficiency changes from stable to unstable, and the average mixing efficiency increases.

Adjoint-based sensitivity analysis of viscoelastic fluids at a low deborah number

This work performs an adjoint-based sensitivity analysis of a viscoelastic fluid flow around a cylinder in a confined channel. The viscoelastic behavior of the fluid is modeled by the Oldroyd-B constitutive equation and the quantity of interest of the flow is set to the resistance of the cylinder. A set of continuous adjoint governing equations is formulated to quantify the local sensitivities of rheological model parameters on the drag of the cylinder. The adjoint equations are solved together with the original resistance prediction problem, at the Deborah number 0.5. Final outcomes are the field resistance-sensitivity maps for the Oldroyd-B model parameters that imply the local relative importance of different underlying physical mechanisms reflected in the model parameters. In particular, the sensitivity map of relaxation parameter distinguishes two opposing effects of fluid elasticity to resistance of the cylinder depending on the location. While the fluid elasticity near the top and bottom surfaces reduces the resistance, it works adversely near the front and back surfaces. The example illustrates how adjoint-based sensitivity analysis can be utilized for developing effective flow control strategies in many applications working with viscoelastic fluids.

Numerical study of electric plume in viscoelastic fluids within a blade-plate structure

A two-dimensional numerical study on the behavior of electric plumes in viscoelastic fluids within a blade-plate structure is conducted herein. The Oldroyd-B constitutive equation is used to model the viscoelastic fluid. The effects of elasticity on the development processes of electric plumes and the heat transfer characteristics are mainly investigated. Moreover, the flow structure of the electric plume with self-similar characteristics which is similar to the thermal plume has been successfully characterized using the evolution of the charge distribution. Additionally, the results show that the elastic effect can promote electric plume development. The stronger the elastic effect is, the faster the electric plume rises. Furthermore, the influence of other dimensionless parameters on the electric plume is analyzed. When considering the impingement cooling effect of the electric plume, the elastic effect inhibits the heat transfer of the system after the plume fully develops.

Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method

An efficient immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for fully resolved simulations of suspended solid particles in viscoelastic flows. Stress LBM based on Giesekus and Oldroyd-B constitutive equation are used to model the viscoelastic stress tensor. A boundary thickening-based direct forcing IB method is adopted to solve the particle–fluid interactions with high accuracy for non-slip boundary conditions. A universal law is proposed to determine the diffusivity constant in a viscoelastic LBM model to balance the numerical accuracy and stability over a wide range of computational parameters. An asynchronous calculation strategy is adopted to further improve the computing efficiency. The method was firstly applicated to the simulation of sedimentation of a single particle and a pair of particles after good validations in cases of the flow past a fixed cylinder and particle migration in a Couette flow against FEM and FVM methods. The determination of the asynchronous calculation strategy and the effect of viscoelastic stress distribution on the settling behaviors of one and two particles are revealed. Subsequently, 504 particles settling in a closed cavity was simulated and the phenomenon that the viscoelastic stress stabilizing the Rayleigh–Taylor instabilities was observed. At last, simulations of a dense flow involving 11001 particles, the largest number of particles to date, were performed to investigate the instability behavior induced by elastic effect under hydrodynamic interactions in a viscoelastic fluid. The elasticity-induced ordering of the particle structures and fluid bubble structures in this dense flow is revealed for the first time. These simulations demonstrate the capability and prospects of the present method for aid in understanding the complex behaviors of viscoelastic particle suspensions.

Steady sphere translation in weakly viscoelastic UCM/Oldroyd-B fluids with perfect slip on the sphere

We study analytically the influence of perfect slip boundary conditions to the steady translation of a spherical particle in a viscoelastic fluid which consists of a Newtonian solvent and a polymer solute. The viscoelastic fluid is modeled with the Oldroyd-B constitutive equation with the ratio of polymer zero shear rate viscosity to the total viscosity, η, considered as a fluid parameter and varying from zero (Newtonian limit) to 1 (Upper Convected Maxwell (UCM) limit). The solution for all the dependent flow variables is expanded as asymptotic power series with the small parameter being the Weissenberg number, Wi=λU/R, where λ is the single relaxation time of the fluid, R is the radius of the particle and U is its steady translational velocity. The resulting sequence of equations is solved analytically up to eight-order in Wi. Hence, the current work is a substantial expansion of the previous work by Gkormpatsis et al., (2020) which has examined the most general case of Navier slip, but the perturbation series has been found only up to fourth order in Wi. The current work confirms the results obtained in that study in the limit of perfect slip conditions. Moreover, the higher-order analytical solution allows for the better investigation of the convergence of the results with techniques that accelerate the convergence of series. Under perfect slip conditions, it is shown that the relative drag force on the sphere increases monotonically with viscoelasticity, as opposed to the cases of no-slip or partial slip where the drag can also decrease or attain a local minimum. This difference in behavior is attributed to the higher dominance of extensional deformations close to the stagnation points of the flow, while the shear deformations prevail away from these points. Finally, evidence is provided that the UCM/Oldroyd-B models become singular at a finite Weissenberg number, Wiu, which depends weakly on η. Although the exact value of Wiu could not be calculated, a sequence of approximations which results from the eight-order perturbation solution reveals that 0.5<Wiu<1. Wiu appears to decrease as η increases, with its minimum value being observed for the UCM model.

On the effect of boundary vibration on mucus mobilization

In this study, we investigate the mobilization of mucus in a cylindrical tube of constant cross-section subjected to a Small Amplitude Oscillatory Shear (SAOS) assuming the viscoelastic behavior of mucus is governed by the Oldroyd-B constitutive equation. The Laplace transform method was adopted to derive expressions for the velocity profile, average velocity, instantaneous flowrate and mean flowrate (i.e., mobilization) of the mucus within the tube. Additionally, a 2D finite element model (FEM) was developed using COMSOL Multiphysics software to verify the accuracy of the derived analytical solution considering both a Newtonian and an Oldroyd-B fluid. Furthermore, sensitivity studies were performed to evaluate the influence of the vibration frequency, vibration amplitude, mean relaxation time, and zero-shear viscosity on mucus mobilization. Results prove that the analytical and numerical results are within acceptable error tolerance and thus in excellent agreement. Besides, the parametric studies indicate a 48%, 57%, and 343% improvement in mucus mobilization when the mean relaxation time, vibration amplitude, and vibration frequency, respectively were each increased by a factor of 6 from the assumed nominal values. Conversely, the mucus mobilization decreased by 25% when the zero-shear viscosity was increased by a factor of 6 from the assumed reference value. In general, this study confirms that mucus mobilization in a tube can be improved by increasing the magnitude of vibration amplitude and vibration frequency. Similarly, the larger the magnitude of mucus mean relaxation time the better the mucus mobilization when the tube is subjected to boundary vibrations. Finally, mucus mobilization decreases as the magnitude of mucus viscosity increases.

Onset of Thermal Instabilities in the Plane Poiseuille Flow of Weakly Elastic Fluids: Viscous Dissipation Effects

The onset of thermal instabilities in the plane Poiseuille flow of weakly elastic fluids is examined through a linear stability analysis by taking into account the effects of viscous dissipation. The destabilizing thermal gradients may come from the different temperatures imposed on the external boundaries and/or from the volumetric heating induced by viscous dissipation. The rheological properties of the viscoelastic fluid are modeled using the Oldroyd-B constitutive equation. As in the Newtonian fluid case, the most unstable structures are found to be stationary longitudinal rolls (modes with axes aligned along the streamwise direction). For such structures, it is shown that the viscoelastic contribution to viscous dissipation may be reduced to one unique parameter: γ=λ1(1−Γ), where λ1 and Γ represent the relaxation time and the viscosity ratio of the viscoelastic fluid, respectively. It is found that the influence of the elasticity parameter γ on the linear stability characteristics is non-monotonic. The fluid elasticity stabilizes (destabilizes) the basic Poiseuille flow if γ&lt;γ* (γ&gt;γ*) where γ* is a particular value of γ that we have determined. It is also shown that when the temperature gradient imposed on the external boundaries is zero, the critical Reynolds number for the onset of such viscous dissipation/viscoelastic-induced instability may be well below the one needed to trigger the pure hydrodynamic instability in weakly elastic solutions.

Dynamics of a viscoelastic thread surrounded by a Newtonian viscous fluid inside a cylindrical tube

A viscoelastic thread in vacuum is known to evolve into a beads-on-a-string structure at large deformations. If the thread is immersed in another fluid, the surrounding medium may influence the topological structure of it, which remains unexplored. To get some insights into the nonlinear behaviour of a viscoelastic thread in a two-phase flow system, a one-dimensional model is developed under the slender body approximation, in which the thread of a highly viscoelastic fluid described by the Oldroyd-B or Giesekus constitutive equation is immersed in a Newtonian viscous fluid of much smaller density and viscosity inside a cylindrical tube. The effect of the outer viscous fluid layer and the confinement of the tube is examined. It is found that the outer fluid may change substantially the beads-on-a-string structure of the viscoelastic thread. Particularly, it may induce the formation of secondary droplets on the filament between adjacent primary droplets, even for large wavenumbers. The outer fluid exerts a resistance force on the extensional flow in the filament, but the necking of the thread is slightly accelerated, due to the redistribution of capillary and elastic forces along the filament accompanied by the formation of secondary droplets. Decreasing the tube radius leads to an increase in secondary droplet size but affects little the morphology of the thread. The non-uniformity of the filament between droplets is more pronounced for a Giesekus viscoelastic thread, and pinch-off of a Giesekus thread always occurs in the neck region connecting the filament to the primary droplet in the presence of the outer viscous fluid.

Numerical simulations of small amplitude oscillatory shear flow of suspensions of rigid particles in non-Newtonian liquids at finite inertia

We perform immersed-boundary-method numerical simulations of small amplitude oscillatory shear flow of suspensions of monodisperse noncolloidal rigid spherical particles in non-Newtonian liquids from the dilute to the concentrated regime. We study the influence of suspending liquid inertia and rheology and particle concentration on the computationally measured storage and loss moduli of the suspensions. In particular, the rheology of the suspending liquid is modeled through the inelastic shear-thinning Carreau–Yasuda constitutive equation and the viscoelastic Giesekus and Oldroyd-B constitutive equations. The role of inertia is quantified by the Stokes number, St, whereas the relevance of the non-Newtonian effects of the suspension matrix is measured through the Carreau number, Cu, for the Carreau–Yasuda liquid and the Deborah number, De, for the viscoelastic liquids. In suspensions with a Carreau–Yasuda matrix, both the storage and the loss modulus increase with St and decrease with Cu, yet the order of magnitude of Cu has to be greater than unity for these effects to be visible. In suspensions with a viscoelastic matrix, both the moduli increase with St and have a nonmonotonic trend with De, showing a maximum with no quantitative differences between the results pertaining suspensions with Giesekus and Oldroyd-B constitutive equations.

Triple diffusive convection in a viscoelastic Oldroyd-B fluid layer

The stability of a triply diffusive viscoelastic fluid layer in which the fluid density depends on three stratifying agencies possessing different diffusivities is investigated. The viscoelastic fluid is modeled by means of the Oldroyd-B constitutive equation. Analytical expressions are obtained for steady and oscillatory onset by carrying out the linear instability analysis and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameters plane. The occurrence of disconnected closed oscillatory neutral curve lying well below the stationary neutral curve is established for some values of governing parameters indicating the requirement of three critical values of thermal Rayleigh number to specify the linear instability criteria. However, the possibility of quasiperiodic bifurcation from the motionless basic state is not perceived and this is in contradiction to the case of inelastic couple stress and Newtonian fluids. The corresponding weakly nonlinear stability of stationary and oscillatory modes has been carried out using a perturbation method. The cubic Landau equations are derived and the stability of bifurcating solution is discussed. The viscoelastic parameters influence the stability of stationary bifurcation despite their effect is not felt on the stationary onset. The stationary and oscillatory finite amplitude solution is found to bifurcate either subcritical or supercritical depending on the choice of governing parameters. The effect of Prandtl number and viscoelastic parameters on stationary and oscillatory convection modes of heat and mass transfer is analyzed.

Lattice Boltzmann Modeling of the Apparent Viscosity of Thinning–Elastic Fluids in Porous Media

Many non-Newtonian fluids, including polymers, exhibit both shear-thinning and viscoelastic rheological properties. A lattice Boltzmann (LB) model is developed for simulation of the flow of thinning–elastic fluids through porous media. This model applies the Oldroyd-B constitutive equation and the Carreau model, respectively, to account for the viscoelastic and shear-thinning behaviors of the thinning–elastic fluid in porous media. Both rheological features are captured well by this model and are verified against analytical solutions. The thinning-then-thickening viscosity curve of the thinning–elastic fluid observed in experiments is reproduced by the present pore-scale simulations. In addition to the traditional extensional theory, we propose other important mechanisms for the increase in apparent viscosity of viscoelastic fluids at higher shear rates. The mechanisms proposed include the reduction in conductivity due to stagnant fluid, the compressed effective flow region, and larger energy dissipations caused by the viscoelastic instability. We find that the viscoelastic thickening effect is more prominent in porous geometries with a large pore–throat ratio.

Second-Order Finite Difference Approximations of the Upper-Convected Time Derivative

In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of second-order in time for solving a simplified constitutive equation in one and two dimensions. The theoretical analysis of the truncation errors of the methods takes into account the linear and quadratic interpolation operators based on a Lagrangian framework. Numerical experiments illustrating the theoretical results for the model equation defined in one and two dimensions are included. Finally, the finite difference approximations of second-order in time are also applied for solving a two-dimensional Oldroyd-B constitutive equation subjected to a prescribed velocity field at different Weissenberg numbers.

Linear stability analysis of a non-Newtonian liquid jet under AC electric fields

When AC electric fields are applied to a non-Newtonian liquid jet, the parametric resonance occurs in the interface. In this paper, effects of the fluid viscoelasticity are mainly studied by Floquet stability analysis. The viscoelasticity is modeled using the Oldroyd-B constitutive equation. The fluid elasticity has weak effects on the linear capillary instability as in the DC cases, but strongly affects the parametric instability. When the oscillation period is comparable to the relaxation time, the elasticity increases the maximum growth rate to a great extent. The critical frequency, beyond which the resonance decays, also increases greatly with the increasing fluid elasticity. An increase in the time constant ratio (i.e. the deformation retardation time) leads to a smaller maximum growth rate, but the reduction of the parametric mode is much more distinct. In addition, the dominate wavenumber is insensitive to the fluid elasticity and the time constant ratio. Moreover, the competition between the capillary and the parametric instabilities reveals that strengthening the electric field, improving the fluid elasticity and lowering the frequency are helpful to observe the resonant instability experimentally.

Predicting biofilm deformation with a viscoelastic phase-field model: Modeling and experimental studies.

Biofilms commonly develop in flowing aqueous environments, where the flow causes the biofilm to deform. Because biofilm deformation affects the flow regime, and because biofilms behave as complex heterogeneous viscoelastic materials, few models are able to predict biofilm deformation. In this study, a phase-field (PF) continuum model coupled with the Oldroyd-B constitutive equation was developed and used to simulate biofilm deformation. The accuracy of the model was evaluated using two types of biofilms: a synthetic biofilm, made from alginate mixed with bacterial cells, and a Pseudomonas aeruginosa biofilm. Shear rheometry was used to experimentally determine the mechanical parameters for each biofilm, used as inputs for the model. Biofilm deformation under fluid flow was monitored experimentally using optical coherence tomography. The comparison between the experimental and modeling geometries, for selected horizontal cross sections, after fluid-driven deformation was good. The relative errors ranged from 3.2 to 21.1% for the synthetic biofilm and from 9.1 to 11.1% for the P. aeruginosa biofilm. This is the first demonstration of the effectiveness of a viscoelastic PF biofilm model. This model provides an important tool for predicting biofilm viscoelastic deformation. It also can benefit the design and control of biofilms in engineering systems.

An Oldroyd-B solver for vanishingly small values of the viscosity ratio: Application to unsteady free surface flows

This work is concerned with time-dependent axisymmetric free surface flows of Oldroyd-B fluids for any value of ▪, the ratio of solvent to total viscosities. The Oldroyd-B constitutive equation is dealt with by employing a novel technique to calculate the conformation tensor while an EVSS transformation allows the solution of the momentum equations coupled with the free surface stress conditions: this avoids numerical instabilities that can arise when using small values of ▪. The convergence of this new methodology is verified on pipe flow and also by comparing results from the literature for the time-dependent impacting drop problem. This approach is then used to predict the time-dependent free surface flow after a viscoelastic drop impacts a solid surface for ▪ values in the range [0, 1]. The impacting drop problem is investigated for polymer solutions containing a small solvent contribution (▪) or without any solvent viscosity (▪). In addition, a study of the bouncing drop problem for different values of ▪, Weissenberg and Reynolds numbers is undertaken.