Oldroyd-B Model Research Articles

On the validity of the Jeffreys (Oldroyd-B) model to describe the oscillations of a viscoelastic pendant drop

We study both theoretically and experimentally the forced and free oscillations of a viscoelastic pendant drop in the linear regime. The Jeffreys model is numerically solved using the zero-shear viscosity of the steady shear flow in a rotational rheometer, and the stress relaxation time measured with a filament stretching elongational rheometer. The damping factor characterizing the first oscillation mode is one order of magnitude larger than that measured in the experiments. Three possible conclusions can be drawn from this study: (i) the Jeffreys model severely underestimates the viscosity decrease due to elasticity; (ii) the zero-shear viscosity measured in the rotational rheometer is much larger than the effective viscosity responsible for energy dissipation in the pendant droplet oscillations; and (iii) the relaxation time measured with the filament stretching elongational rheometer is much smaller than that characterizing those oscillations. The failure of the Jeffreys model necessarily entails that of all nonlinear constitutive relationships which the Jeffreys model can be derived from, like the commonly used Oldroyd-B model.

Geometric scaling of elastic instabilities in the Taylor–Couette geometry: A theoretical, experimental and numerical study

We investigate the curvature-dependence of the visco-elastic Taylor–Couette instability. The radius of curvature is changed over almost a decade and the critical Weissenberg numbers of the first linear instability are determined. Experiments are performed with a variety of polymer solutions and the scaling of the critical Weissenberg number with the curvature against the prediction of the Pakdel–McKinley criterion is assessed. We revisit the linear stability analysis based on the Oldroyd-B model and find, surprisingly, that the experimentally observed scaling is not as clearly recovered. We extend the constitutive equation to a two-mode model by incorporating the PTT model into our analysis to reproduce the rheological behaviour of our fluid, but still find no agreement between the linear stability analysis and experiments. We also demonstrate that that conclusion is not altered by the presence of inertia or viscous heating. The Pakdel–McKinley criterion, on the other hand, shows a very good agreement with the data.

The Mechanism of Size-Based Particle Separation by Dielectrophoresis in the Viscoelastic Flows

Viscoelastic solution is encountered extensively in microfluidics. In this work, the particle movement of the viscoelastic flow in the contraction–expansion channel is demonstrated. The fluid is described by the Oldroyd-B model, and the particle is driven by dielectrophoretic (DEP) forces induced by the applied electric field. A time-dependent multiphysics numerical model with the thin electric double layer (EDL) assumption was developed, in which the Oldroyd-B viscoelastic fluid flow field, the electric field, and the movement of finite-size particles are solved simultaneously by an arbitrary Lagrangian–Eulerian (ALE) numerical method. By the numerically validated ALE method, the trajectories of particle with different sizes were obtained for the fluid with the Weissenberg number (Wi) of 1 and 0, which can be regarded as the Newtonian fluid. The trajectory in the Oldroyd-B flow with Wi = 1 is compared with that in the Newtonian fluid. Also, trajectories for different particles with different particle sizes moving in the flow with Wi = 1 are compared, which proves that the contraction–expansion channel can also be used for particle separation in the viscoelastic flow. The above results for this work provide the physical insight into the particle movement in the flow of viscous and elastic features.

Spatial instability of double-layer viscoelastic liquid sheets in a viscous gas medium

The spatial instability of double-layer viscoelastic liquid sheets in a viscous gas medium is investigated. A linear stability analysis is used to simplify the governing equations and boundary conditions combined with the Oldroyd-B model for describing the viscoelastic liquid. The growth rate and the cut-off wavenumber have been studied by analyzing the dispersion curve which is based on the spectral collocation method. Moreover, we use the decomposition of growth rate to interpret some special cases. We conclude that the thickness ratio, the liquid density ratio, the liquid viscosity ratio, and the gas-to-liquid viscosity ratio stabilize the liquid sheets, while the gas-to-liquid density ratio, the Reynolds number, and the Weber number destabilize the liquid sheets. The dispersion curves are nearly-identical when the surface tension ratio is smaller than one, while the surface tension ratio is a stabilizing factor when the ratio is larger than one, but the second maximum and the counterintuitive destabilization exist in the large wavenumber region. As specific parameters concerning the viscoelasticity, the elasticity number transforms from the destabilizing factor into the stabilizing factor with the increased Reynolds number in the region nearby the cut-off wavenumber, because the relationship between the change rates of viscous dissipation terms and elasticity terms in the decomposition of the growth rate is varying. Besides, it is discovered that the trend of the change of the growth rate under the varying stress relaxation time ratio is non-monotonic and double-layer liquid sheets are most stable when the stress relaxation time ratio is near 3 (not 1), because the difference between the change rate of the viscous dissipation and the deformation retardation dissipation determines the most stable region (not the elasticity difference), which can also be interpreted with the variation of the energy terms.

Global well-posedness and decay rates for the three dimensional compressible Oldroyd-B model

In this paper, we are concerned with the global well-posedness and decay rates of strong solutions for the three-dimensional compressible Oldroyd-B model. We prove that this set of equations admits a unique global strong solution provided the initial data are close to the constant equilibrium state in H2-framework. Moreover, if the initial data belong to L1, the convergence rate of the solutions in Lp-norm with 2⩽p⩽6 and convergence rates of their spatial derivatives in L2-norm are obtained. It is noteworthy that the smallness restriction on the coupling constant ω is not necessary in this paper.

On the Rayleigh–Taylor instability in compressible viscoelastic fluids

We mathematically investigate the Rayleigh–Taylor (abbr. RT) instability in the compressible viscoelastic fluid in the presence of a uniform gravitational field in a bounded domain based on Oldroyd-B model. We first analyze the linearized equations around the viscoelastic RT equilibrium solution, and obtain an instability condition. Then we construct solutions of the linearized viscoelastic RT problem that grow in time in the Sobolev space H3 under an instability condition, thus leading to the linear instability. Finally, with the help of the constructed unstable solutions of the linearized viscoelastic RT problem and a local well-posedness result of smooth solutions to the nonlinear viscoelastic RT problem, we mathematically prove the instability of viscoelastic RT (abbr. VRT) problem in the sense of Hadamard.

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids Incorporating Internal and Cosserat Rotation Rates

This paper considers conservation and balance laws for non-classical fluent continua in the presence of internal rotation rates due to the velocity gradient tensor and the rotation rates due to Cosserat rotations. In these balance laws, the internal rotation rates are completely defined as functions of the velocity gradient tensor, but the Cosserat rotation rates are additional three degrees of freedom at each material point. When these rotation rates are resisted by the deforming continua, conjugate moments are created. For thermoviscoelastic fluent continua, these result in additional dissipation mechanism as well as rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such fluids based on the entropy inequality in conjunction with the representation theorem using integrity, i.e., complete basis. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as presence of the balance of moments of moments as a balance law and are compared with the classical continuum theories. Retardation moduli corresponding to the Cauchy stress tensor, both symmetric and antisymmetric, as well as the symmetric and antisymmetric Cauchy moment tensors are derived. The constitutive theories presented in this paper are ordered rate theories. These incorporate convected time derivatives of the strain tensor up to order [Formula: see text] and the convected time derivatives of the stress and moment tensors up to certain orders. Simplified forms of the constitutive theories are also derived. It is shown and concluded that the constitutive theories for thermoviscoelastic fluids based on classical continuum theory such as Maxwell model, Oldroyd-B model, Giesekus model, etc., are all a subset of the constitutive theories presented in this paper for non-classical physics. Both compressible and incompressible thermoviscoelastic fluids are considered.

Analytical study on motion and shape of creeping Boger drops falling through viscoelastic media

In the present study, an analytical solution based on a double perturbation method is obtained to simulate the creeping motion of a Boger drop descending in an immiscible viscoelastic quiescent medium. The perturbation parameters are considered to be the Deborah number defined based on the relaxation times of interior and exterior media and the Capillary number. The Oldroyd-B model is implemented as a constitutive equation for both interior and exterior phases. The analytical solutions are obtained up to second order of perturbation expansion for both phases. The motion of the droplet in the surrounding medium is considered to be creeping. Effects of different parameters including, Deborah numbers, viscosity ratios, elasto-capillary number, and viscosity ratio between the drop and external flows are investigated on the shape, motion, and stream function in detail. In contrast to the generally spherical shape of creeping drops in Newtonian regimes, it is shown that the effect of elasticity in a non-Newtonian scenario can feature prolate or oblate shapes. It is shown that an increment in elastic properties of interior fluid is followed by decrement in the terminal velocity of the droplet, while an increment in elasticity of exterior fluid has a contrary effect and increases the terminal velocity. The results have shown suitable agreements with previously reported experimental results.

Elastic modifications of an inertial instability in a 3D cross-slot

We numerically investigate inertial flows of viscoelastic fluids within a three-dimensional cross-slot geometry of square cross-section. Our study focuses on the inertial instability that occurs above a critical Reynolds number (Re) resulting in a transition to a steady flow asymmetry. We investigate numerically the effects of elasticity upon its characteristics by employing the upper-convected Maxwell (UCM), the Oldroyd-B and the modified Chilcott–Rallison finitely extensible nonlinear elastic (FENE–MCR) models. In so doing, we show that the UCM and the Oldroyd-B model results are restricted to very low nominal Weissenberg numbers at non-negligible Reynolds numbers, due to a significant increase in the strain rate at the stagnation point caused by inertia. The resulting steady-asymmetric flow at these critical conditions gives rise to the formation of a single axially-aligned spiral vortex, which is formed along the outlet channels of the geometry in good agreement with experimental observations. Below these critical conditions the flow remains steady-symmetric, varying from a nearly two-dimensional flow at very low Re to a more complex three-dimensional flow at higher Re. In a recent publication [Burshtein et al. [Phys. Rev. X, 7, 041039, (2017)]], we demonstrated experimentally, accompanied by limited complementary numerical simulations, the impact of elasticity upon the critical conditions for which the instability develops and the behaviour of the subsequent growth of vorticity of the single spiral vortex. Our results here show how different viscoelastic models influence the instability in a 3D cross-slot and demonstrate an interesting behaviour of the first normal-stress difference, providing an additional insight on the potential mechanisms which are responsible for the suppression of the spiral-vortex. We also elucidate the role played by solvent-to-total viscosity ratio and the extensibility parameter in the FENE–MCR model on the instability.

Nanoparticles and nonlinear thermal radiation properties in the rheology of polymeric material

The present analysis is related to the dynamics of polymeric liquids (Oldroyd-B model) with the presence of nanoparticles. The rheological system is considered under the application of nonlinear thermal radiations. Energy and concentration equations are presented when thermophoresis and Brownian motion effects are present. Bidirectional form of stretching is considered to interpret the three-dimensional flow dynamics of polymeric liquid. Making use of the similarity transformations, problem is reduced into ordinary differential system which is approximated by using HAM. Influence of physical parameters including Deborah number, thermophoresis and Brownian motion on velocity, temperature and mass fraction expressions are plotted and analyzed. Numerical values for local Sherwood and Nusselt numbers are presented and discussed.

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids with Internal Rotation Rates

The paper presents constitutive theories for non-classical thermoviscoelastic fluids with dissipation and memory using a thermodynamic framework based on entirety of velocity gradient tensor. Thus, the conservation and the balance laws used in this work incorporate symmetric as well as antisymmetric part of the velocity gradient tensor. The constitutive theories derived here hold in coand contra-variant bases as well as in Jaumann rates and are derived using convected time derivatives of Green’s and Almansi strain tensors as well as the Cauchy stress tensor and its convected time derivatives in appropriate bases. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as balance law. It is shown that the dissipation mechanism and the fading memory in such fluids are due to stress rates as well as moment rates and their conjugates. The material coefficients are derived for the general forms of the constitutive theories based on integrity. Simplified linear (or quasi-linear) forms of the constitutive theories are also presented. Maxwell, Oldroyd-B and Giesekus constitutive models for non-classical thermoviscoelastic fluids are derived and are compared with those derived based on classical continuum mechanics. Both, compressible and incompressible thermoviscoelastic fluids are considered.

Mathematical study of a single leukocyte in microchannel flow

The recruitment of leukocytes and subsequent rolling, activation, adhesion and transmigration are essential stages of an inflammatory response. Chronic inflammation may entail atherosclerosis, one of the most devastating cardiovascular diseases. Understanding this mechanism is of crucial importance in immunology and in the development of anti-inflammatory drugs. Micropipette aspiration experiments show that leukocytes behave as viscoelastic drops during suction. The flow of non-Newtonian viscoelastic fluids can be described by differential, integral and rate-type constitutive equations. In this study, the rate-type Oldroyd-B model is used to capture the viscoelasticity of the leukocyte which is considered as a drop. Our main goal is to analyze a mathematical model describing the deformation and flow of an individual leukocyte in a microchannel flow. In this model we consider a coupled problem between a simplified Oldroyd-B system and a transport equation which describes the density considered as non constant in the Navier–Stokes equations. First we present the mathematical model and we prove the existence of solution, then we describe its numerical approximation using the level set method. Through the numerical simulations we analyze the hemodynamic effects of three inlet velocity values. We note that the hydrodynamic forces pushing the cell become higher with increasing inlet velocities.

Relative Entropy, Weak-Strong Uniqueness, and Conditional Regularity for a Compressible Oldroyd--B Model

We consider the compressible Oldroyd-B model derived in \cite{Barrett-Lu-Suli}, where the existence of global-in-time finite energy weak solutions was shown in two dimensional setting. In this paper, we first state a local well-posedness result for this compressible Oldroyd-B model. In two dimensional setting, we give a (refined) blow-up criterion involving only the upper bound of the fluid density. We then show that, if the initial fluid density and polymer number density admit a positive lower bound, the weak solution coincides with the strong one as long as the latter exists. Moreover, if the fluid density of a weak solution issued from regular initial data admits a finite upper bound, this weak solution is indeed a strong one; this can be seen as a corollary of the refined blow-up criterion and the weak-strong uniqueness.

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

We present an Eulerian free-surface flow solver for incompressible pseudoplastic and viscoelastic non-Newtonian fluids. The free-surface flow solver is based on the streamfunction flow formulation and the volume-of-fluid method. The streamfunction solver computes the vector potential of a solenoidal velocity field, which ensures by construction the mass conservation of the solution, and removes the pressure unknown. Pseudoplastic liquids are modelled with a Carreau model. The viscoelastic fluids are governed by differential constitutive models reformulated with the log-conformation approach, in order to preserve the positive-definiteness of the conformation tensor, and to circumvent the high Weissenberg number problem. The volume fraction of the fluid is advected with a geometric conservative unsplit scheme that preserves a sharp interface representation. For the sake of comparison, we also implemented an algebraic advection scheme for the liquid volume fraction. The proposed numerical method is tested by simulating the planar extrudate swell with the Carreau, Oldroyd-B and Giesekus constitutive models. The swell ratio of the extrudates are compared with the data available in the literature, as well as with numerical simulations performed with the open-source rheoTool toolbox in OpenFOAM®. While the simulations of the generalized Newtonian fluids achieved mesh independence for all the methods tested, the flow simulations of the viscoelastic fluids are more sensitive to mesh refinement and the choice of numerical scheme. Moreover, the simulations of Oldroyd-B fluid flows above a critical Weissenberg number are prone to artificial surface instabilities. These numerical artifacts are due to discretization errors within the Eulerian surface-capturing method. However, the numerical issues arise from the stress singularity at the die exit corner, and the unphysical predictions of the Oldroyd-B model in the skin layer of the extrudate after the die exit, where large extensional deformations occur.

Saffman–Taylor instability of viscoelastic fluids in anisotropic porous media

In this study, the effect of anisotropy of porous media on Saffman–Taylor instability (viscous fingering instability) of viscoelastic fluids is investigated for the first time. The Oldroyd-B model is used as the constitutive equation of viscoelastic phase. In linear stability analysis, the quasi-steady state approximation and six order shooting method are used to predict the growth rate of the disturbance in the flow field. In addition, the spectral method based on Hartley transforms and fourth-order Adams–Bashforth technique is used to numerically simulate this instability and the results include concentration contours, transversely averaged concentration profiles, mixing length and sweep efficiency. The results show that the flow becomes more stable by increasing the Deborah number, i.e., elasticity property of displacing viscoelastic fluid. Also, increasing permeability of media in longitudinal direction to the transverse leads to more stable flow. On the other hand, when the ratio of transverse to longitudinal dispersion is increased, the intensity of instability decreases.

Finite element approximation of the FENE-P model

We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain $D ⊂ R d , d = 2 or 3$, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics ($d = 2$) or a reduced version, where the tangential component on each simplicial edge ($d = 2$) or face ($d = 3$) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes, based on the backward Euler type time discretiza-tion, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case $d = 2$, as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions.

Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity

We characterise the stress singularity of the Oldroyd-B, Phan-Thien–Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces. By integrating along streamlines, we verify the polymer stress behavior of r−4/11 for PTT and r−5/16 for Giesekus, where r is the radial distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity. A three-region asymptotic structure is again apparent with now a polymer stress singularity of r−4/5. The high Weissenberg boundary layer equations are found to manifest themselves at the stick surface and are of thickness r3/2. At the slip surface, dominant balance between the upper convected stress and rate-of-strain terms gives a slip boundary layer of thickness r2. The solution of the slip boundary layer shows that the polymer stress is now singular along the slip surface. These results are supported through numerical integration along streamlines of the Oldroyd-B equations in a Newtonian velocity field. The Oldroyd-B model thus extends the point singularity at the join of the stick and slip surfaces to the whole of slip surface. As such, it does not have a physically meaningful solution in a Newtonian velocity field. We would expect a similar stress behavior for this model in the true viscoelastic velocity field.

Stability analysis of an electrospinning jet of polymeric fluids

The axisymmetric instability, which has a potential to cause bead formation along the fibers produced by electrospinning technique, is examined with the help of linear stability theory. The stability of a straight jet is analyzed under conditions corresponding to the electrospinning of two types of polymeric fluids, the PIB-based Boger fluid with low electrical conductivity and the highly conductive PEO solution in ethanol/water. For the former, the rheology is described by the Oldroyd-B model, suitable for unentangled polymers under weak elongational flow. On the other hand, the highly conductive polymer solution experiences a strong elongational flow due to very high axial electric force, for which the viscoelasticity is appropriately described using the eXtended Pom-Pom (XPP) model, the nonlinear rheological model for entangled polymeric systems. Contrary to previous studies, which oversimplifies the electrified jet as a cylindrical jet with uniform radius and other jet variables, we analyze the stability of the realistic non-uniform thinning jet as observed in electrospinning experiments. The stability of the thinning jet profile, obtained using the 1D slender body model, is examined by imposing non-periodic axisymmetric disturbances and constructing the spectrum of disturbance growth rate. The thinning jet is found to be relatively less unstable than the uniform jet, which is attributed to the stabilizing role of extensional stresses, in addition to the axial variation in surface charge density and electric field, present in the non-uniform deforming jet, but ignored in the analysis of the uniform jet. For both the reference fluids considered, the polymer addition renders the jet stable and thus, suppresses the bead formation during straight jet path of electrospinning. Also, the enhancement in fluid elasticity, characterized by the flow Deborah number, plays a stabilizing role for the thinning jet of Oldroyd-B fluid. However, for the XPP fluid, the fluid elasticity shows a rich behavior with a stabilizing effect for moderate values of Deborah number, attributed to stretching of polymer chain between the branch points, and a destabilizing effect for highly elastic fluids, due to strain rate softening. Increasing the strain hardening effect in the polymer solution, achieved by increasing number of arms at the branch point in the XPP molecule, tends to stabilize the electrospinning jet against axisymmetric disturbances potentially producing smooth bead-less fibers. While the instability in low conductivity fluid is driven by capillary forces, the instability in highly conductive fluid is an oscillatory conducting mode driven by the coupling of the surface charges and the axial electric field.

Splash singularities for a 2D Oldroyd-B model with nonlinear Piola-Kirchhoff stress

In this paper consider a 2-D free boundary Oldroyd-B model at infinite Weissenberg number, under the assumption that the Piola-Kirchoff tensor, entering in the description of the extra-stress tensor, is given by a quadratic, convex energy functional. Our main goal is to investigate the existence of splash type singularities, namely points of self-intersection of the free boundary. The analysis of this problem requires to map the equations via a conformal transformation, in order to separate the singular points, and then to fix the free boundary via a Lagrangian change of coordinates. The investigation starts by proving local existence and stability results for a family of smooth initial configurations which, by considering a special class of initial data, allow us to show the existence of solutions having a self-intersecting configuration. As a consequence of this fact, we can conclude there exists a configuration, which has a singularity of splash type.

Axisymmetric and non-axisymmetric instability of a charged viscoelastic jet under an axial magnetic field

This paper investigates the axisymmetric and the first non-axisymmetric linear instability of a charged viscoelastic jet under an applied axial magnetic field. The Oldroyd-B model is employed to account for the viscoelastic effect. Viscoelastic fluid is considered to be a leaky dielectric, and the surrounding inviscid gas a perfect dielectric. Results suggest that increasing the axial magnetic field suppresses both the axisymmetric and non-axisymmetric instabilities, and moves the main unstable range of the non-axisymmetric instability to small wavenumber regions. The magnetic field is coupled with the electric field by increasing the critical electric intensity of the dual effects on the axisymmetric mode; it also favors the transition from axisymmetric mode (jetting-dripping motion) to non-axisymmetric mode (jetting-whipping motion). It is also discovered that the magnetic field weakens the effects of fluid elasticity and deformation retardation time, but has little influence on the role played by the ambient gas.