Maxwell Liquids Research Articles

Slippery boundary and radiative transport in unsteady features of Maxwell fluid over stretched cylinder

This article employs the Cattaneo-Christov double diffusion concept to examine thermal and solute energy transfer processes in Maxwell liquid movement. The irregular two-dimensional movement of a Maxwell liquid with changing heat conductivity across an extended cylinder is investigated, together with thermal radioactivity and velocity slip. We develop partial differential equations for heat and mass transmission in Maxwell liquid using the Cattaneo-Christov pattern instead of Fourier's and Fick's law. Numerical shooting solves ordinary differential equations obtained from controlling partial differential equations via similarity transformations. We noticed that unstable factor should be no more than one for optimal outcomes. Greater Maxwell values minimize the movement field and increase liquid energy transfer. Heat and concentration diffusions in Maxwell liquids decline as thermal and concentration relaxation times approach maximum. In addition, a low thermal conductivity characteristic improves the temperature field.

Transient MHD flow of fractional Burger's fluid model through a parallel microchannel with heat transfer and slip boundary condition

The combined influence of magnetic field, electro-osmotic flow (EOF), and pressure gradient on the unsteady magnetohydrodynamic fluid flow through a parallel microchannel is investigated. The Burger's liquid model is used for the fractional partial differential equation, which allows us to study the behavior of viscoelastic liquid velocity profile in the parallel microchannel. The Laplace transform (LT) in concert with the Fourier cosine transform are used to obtain the analytical solution of the velocity profile. Furthermore, using the method of separation of variables, the energy equation, Joule heating, energy dissipation, and electromagnetic effects are calculated to obtain the temperature within the microchannel. The influence of some relevant parameters like heat transfer, temperature distribution, Hartmann number (Ha), and Brinkman number (Br) on fluid flow velocity are presented graphically and discussed. The results obtained show that temperature decreases with Ha and Br of the transverse electric field for slip flow. Close to the middle of the microchannel, fluid flow velocity increases with decrease in the delay time parameter value and increase in the Burger's parameter value, while the opposite trend is found for the velocity near to the middle of the microchannel. In addition, Burger's liquid is quite general, such that Oldroyd-B, Maxwell, and Newtonian liquids are readily obtained as limiting cases. The study is significant in the application fields of chemical analysis and biological analysis, drug delivery, bacteria detection, and some others.

Radiative heat and slip effects on blood-based hydrodynamic film flow of Maxwell/Oldroyd-B cross nanofluids: a bioapplication

Human blood has elastic energy deposited in the distortion of red blood cells. Due to this, blood behaves like a viscoelastic fluid. The Maxwell and Oldroyd-B fluid models are the best models having viscosity and elasticity properties. Due to the numerous biomedical applications of gold and silver nanoparticles, we assumed blood-based gold-silver mixed nanoliquid. This study explores the consequence of radiative heat and velocity slip on the thin film flow of time-dependent hydrodynamic Oldroyd-B and Maxwell cross nanoliquids. A relevant mathematical model is established and the modeled physical problem is resolved numerically using the bvp5c Matlab scheme after applying appropriate similarities. The stimulus of applicable limits on motion and thermic fields is discussed with graphs and tabular illustrations. It originated that the heat diffusion rate is high in Maxwell cross nanoliquid when associated with Oldroyd-B hybrid nanofluid. The increasing film thickness effectively enlarges the heat relocation rate of both fluids. Also, the rheological nature of Oldroyd-B and Maxwell liquids is not uniform. The results are validated and found good agreement.

Stability analysis of the interface between two weak viscoelastic liquids under periodic oscillations

We consider the motion and the linear hydrodynamic instabilities of two immiscible viscoelastic liquids above a horizontal solid surface induced by the periodic oscillations of the horizontal plate along its plane. A planar interface, parallel to the oscillating plate, separates the lower layer from the other viscoelastic fluid that extends vertically to infinity. The two-dimensional motion of these fluids is studied together with the conditions under which the flow becomes unstable, deforming the planar interface and promoting the mixing of both liquids. The study extends the previous work by Isakova et al. [“A model for the linear stability of the interface between aqueous humor and vitreous substitutes after vitreoretinal surgery,” Phys. Fluids 26, 124101 (2014)] by considering non-Newtonian fluids, particularly liquids with weak viscoelasticity (neglecting normal stress differences), which may model more accurately the physical behavior of the aqueous humor and, especially, the vitreous humour substitute in the vitreous chamber of the eye after vitrectomy. A novel approach to the quasi-steady stability analysis of unsteady flows of Maxwell liquids is developed in the present paper. We focus on the effect of the small Deborah numbers on the motion and on the hydrodynamic instability of the two fluids as the other non-dimensional parameters are varied within the range of interest for the biofluiddynamics of the eye. The special case in which the lower layer modelling the aqueous humor is a Newtonian liquid and the upper vitreous substitute is a Maxwell liquid is considered with detail. We find that, even for a very small Deborah number of the vitreous substitute, the dynamics and the hydrodynamic stability of the two fluids can be qualitatively very different to the Newtonian case, especially as the viscosity ratio is varied, showing that weak viscoelasticity may change dramatically the dynamics of the eye. An exhaustive characterization of the influence of the different parameters on the hydrodynamic stability is given, which may be useful also for the study of the dynamics of other systems where two viscoelastic liquids in contact are subjected to periodic oscillations.

Front-Rear Polarization by Mechanical Cues: From Single Cells to Tissues.

Directed cell migration is a complex process that involves front-rear polarization, characterized by cell adhesion and cytoskeleton-based protrusion, retraction, and contraction of either a single cell or a cell collective. Single cell polarization depends on a variety of mechanochemical signals including external adhesive cues, substrate stiffness, and confinement. In cell ensembles, coordinated polarization of migrating tissues results not only from the application of traction forces on the extracellular matrix but also from the transmission of mechanical stress through intercellular junctions. We focus here on the impact of mechanical cues on the establishment and maintenance of front-rear polarization from single cell to collective cell behaviors through local or large-scale mechanisms.

Surface tension driven convection in viscoelastic liquids with thermorheological effect

Oscillatory onset of convection is studied numerically for Rivlin–Ericksen, Maxwell and Jeffreys liquids by considering free–free and rigid–free isothermal/adiabatic boundaries. The effect of variable viscosity parameter is shown to destabilize the system. The problem reveals the stabilizing nature of strain retardation parameter and the destabilizing nature of stress relaxation parameter, on the onset of convection. The Maxwell liquids are found to be more unstable than the one subscribing to the Jeffreys description whereas the Rivlin–Ericksen liquids are comparatively more stable. Rigid-free adiabatic boundary combination is found to give rise to a most stable system, whereas the free isothermal free adiabatic combination gives rise to a most unstable system.

Elastic effects on Rayleigh-Bénard convection in liquids with temperature-dependent viscosity

A linear stability analysis of convection in viscoelastic liquids with temperature-dependent viscosity is studied using normal modes and Galerkin method. Stationary convection is shown to be the preferred mode of instability when the ratio of strain retardation parameter to stress relaxation parameter is greater than unity. When the ratio is less than unity then the possibility of oscillatory convection is shown to arise. Oscillatory convection is studied numerically for Rivlin-Ericksen, Maxwell and Jeffreys liquids by considering free-free, rigid-rigid and rigid-free isothermal/adiabatic boundaries. The effect of variable viscosity parameter is shown to destabilize the system. The problem reveals the stabilizing nature of strain retardation parameter and destabilizing nature of stress relaxation parameter, on the onset of convection. The Maxwell liquids are found to be more unstable than the one subscribing to Jeffreys description whereas the Rivlin-Ericksen liquid is comparatively more stable. Free-free adiabatic boundary combination is found to give rise to a most unstable system, whereas the rigid isothermal rigid adiabatic combination gives rise to a most stable system. The problem has applications in non-isothermal systems having viscoelastic liquids as working media.

Efficient computation of the spectrum of viscoelastic flows

The understanding of viscoelastic flows in many situations requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations. Linear stability analysis leads to a generalized eigenvalue problem (GEVP), whose numerical analysis may be challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to non-physical eigenvalues at infinity. For viscoelastic flows, the difficulties increase due to the presence of continuous spectrum, related to the constitutive equations. The Couette flow of upper convected Maxwell (UCM) liquids has been used as a case study of the stability of viscoelastic flows. The spectrum consists of two discrete eigenvalues and a continuous segment with real part equal to −1/ We ( We is the Weissenberg number). Most of the approximations in the literature were obtained using spectral expansions. The eigenvalues close to the continuous part of the spectrum show very slow convergence. In this work, the linear stability of Couette flow of a UCM liquid is studied using a finite element method. A new procedure to eliminate the eigenvalues at infinity from the GEVP is proposed. The procedure takes advantage of the structure of the matrices involved and avoids the computational overhead of the usual mapping techniques. The GEVP is transformed into a non-degenerate GEVP of dimension five times smaller. The computed eigenfunctions related to the continuous spectrum are in good agreement with the analytic solutions obtained by Graham [M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341].

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.

Interfacial instability between sheared elastic liquids in a channel

We consider the linear stability of the interface between two sheared elastic liquids at large Weissenberg number ( Wi) with negligible inertia. The liquids are of Oldroyd-B or UCM type and have matched viscosity. In UCM liquids, Renardy [Y. Renardy, Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newton. Fluid Mech. 28 (1988) 99–115] found a purely elastic instability for short-waves in the absence of surface tension for which the perturbation flow decays exponentially away from the interface. For UCM liquids at large Wi we show that this instability persists even though the wavelength is larger than the channel width and the disturbance occupies the entire channel. Surprisingly, the growth rate is not affected by the location of the walls, even though the mode structure is altered. This analysis suggests a reappraisal of the appropriateness of the short-wave and long-wave classifications for instabilities of viscoelastic liquids in order to accommodate the additional length scale introduced by fluid velocity and relaxation. The instability persists for Oldroyd-B liquids even as the elastic contribution to viscosity approaches zero. Surprisingly too, the inclusion of surface tension does not affect the asymptotic growth rate at large wavenumber. When more modest values of Wi are considered, we find parameter values for which arbitrarily large surface tension reduces the growth rate but does not stabilize the flow; previously proposed mechanisms based on the interface displacement are therefore inadequate to explain the instability. Because the instability is locally generated, it appears in other high Wi flows with interfaces, both in channels and in pipes.

OSCILLATORY CONVECTION IN VISCOELASTIC, FERROMAGNETIC/DIELECTRIC LIQUIDS

Oscillatory convection in viscoelastic ferromagnetic and dielectric liquids of the Rivlin-Ericksen, Maxwell and Oldroyd types is studied analytically by considering free-free, isothermal boundaries with idealized conditions on the magnetic / electric potential. The linear theory reveals the stabilizing nature of the strain retardation parameter and the destabilizing nature of the stress relaxation and magnetization / dielectric parameters. The Maxwell liquids are found to be more unstable than the one subscribing to the Oldroyd description whereas the Rivlin-Ericksen liquid is comparatively more stable. The results have implications in many non-isothermal applications of ferromagnetic and dielectric liquids especially in energy conversion devices.

Thin film core-annular flow of upper-convected maxwell liquids

Pressure-driven pipe flow of two upper-convected Maxwell liquids in a vertical core-annular arrangement is studied. The annulus and core liquids have different relaxation times and viscosities. Weakly nonlinear evolution equations which describe the interface shape are derived. Lubrication theory is used in the annulus but not in the core. Motions periodic in the streamwise direction are addressed, with the aim of describing short-time behavior driven by capillary forces. Numerical and analytical results for the spatio-temporal dynamics are given for two subcases. In the first case, the liquids have the same viscosity. White noise non-axisymmetric initial data are found to evolve into axisymmetric motion. When axisymmetry is assumed, the evolution equation is a Kuramoto-Sivashinsky equation; the bifurcation parameter depends on the fluid elasticities, interfacial tension, Reynolds number and Weissenberg number. The second case concerns axisymmetric motions with wavelengths in the axial direction that are long compared with the annulus thickness. This asymptotic analysis places a restriction on the size of the Weissenberg number. A jump in the viscosities introduces dispersion, which may be enhanced by fluid elasticity; this can lead to a transition from an unstable regime or a chaotic regime to one in which organized traveling wave pulses move in the axial direction.

Weakly nonlinear behavior of periodic disturbances in two-layer plane channel flow of upper-convected Maxwell liquids

The stability of a plane Couette-Poiseuille flow consisting of two layers of different fluids is analyzed. The fluids have different viscosities, densities and relaxation times and there is surface tension at the interface. The center manifold reduction method is used to derive the final amplitude evolution equation. The nonlinear calculations are carried out with two alternative approaches. One approach is to keep the combined volume flux fixed, and the other is to keep the pressure gradient in the horizontal direction fixed. Numerical results are presented for the one-fluid Poiseuille flow, and some two-layer Poiseuille flows with reference to co-extrusion through a slit die. Situations with subcritical and supercritical bifurcations are described.

Analysis of simple constitutive equations for viscoelastic liquids

The behavior of general Maxwell-like constitutive equations is analyzed. The total set of the constitutive equations includes the evolution equations for the “configuration tensor” c and an expression for the stress tensor through the tensor c. Several problems are considered. (i) It is shown that the general form for these equations derived recently from the Poissonbracket formalism, is more easily obtained by using a local approach of irreversible thermodynamics for both compressible and incompressible cases. Also, general Maxwell-like constitutive equations, with mixed upper and lower convected time derivatives, are derived using this method, (ii) The problem of Hadamard stability for these constitutive equations is then studied by using the method of “frozen coefficients”. The necessary and sufficient conditions for evolution are established. These conditions demonstrated the common results of 3D instability for the Johnson-Segalman and Phan-Thien-Tanner generalized constitutive equations, with exceptions only for equations with upper and lower convected derivatives. Therefore only Maxwell liquids with a “perfect elastic limit” are further studied. For these liquids, the evolution conditions are expressed explicitly as some constraints imposed on possible forms of free energy (elastic potential). It is also demonstrated that, for the latter liquids, the evolution conditions are the same as those for thermodynamic stability. For the incompressible case, with det c = 1, simple sufficient conditions for global evolutionarity are also found. (iii) It is shown that the configuration tensor c is positive definite for any piecewise smooth strain history. Also, simple sufficient conditions are found for the boundedness of the stress tensor at a given strain history. (iv) The results are applied to some viscoelastic constitutive equations proposed in the literature. (v) The importance of det c in the formulations of the constitutive equations and the relations between the instabilities in rheological equations and instabilities observed in flows of polymer liquids are then discussed.

On Rankine—Hugoniot conditions for Maxwell liquids

We discuss the jump conditions which must hold across discontinuities in the solutions of the equations governing the motion of upper and lower convected Maxwell liquids. In two space dimensions, we can also give such jump relations of Johnson—Segalman fluids.

Stability of the interface in two-layer couette flow of upper convected maxwell liquids

The linear stability of a two-layer Couette flow of upper convected Maxwell liquids is considered. The fluids have different densities, viscosities, and elasticities, with surface tension at the interface. At low speeds, the interfacial mode may become unstable, while other modes stay stable. The shortwave asymptotics of the interfacial mode is analyzed. It is found that an elasticity difference can stabilize or destabilize the flow even in the absence of a viscosity difference. As the viscosity difference increases, the range of elasticities for which there is shortwave stability widens. A linearly stable arrangement can be achieved by placing the less viscous fluid in a thin layer to stabilize longwaves and using elasticities to stabilize shortwaves. Such an arrangement can be stable even when the density stratification is adverse.

Tangent simple systems method applied to a precise study of viscoelastic behaviour of human blood

The tangent simple systems (TSS) method, proposed in (1), is applied in order to study the viscoelastic behaviour of human blood in transient flow for a rectangular low shear rate step. The tangent simple systems which were used are Maxwell liquids. These systems allow one to obtain plots of variations of instantaneous values of viscosity coefficient mu, elasticity modulus G and retardation time tau = mu/G of the studied blood samples, as a function of flow duration. Variations of both parameters mu and G versus time are represented by two exponential functions which involve three couples of parameters (mu o, mu infinity), (Go, G infinity) and (tau mu, tau G). These parameters can be considered as the characteristics of each blood sample. Another representation of the results, called the dual rheogram, is also indicated. The dual rheogram enables one to follow the evolution of the blood structure. Several examples of application of the TSS method to normal blood sample and to suspensions of artificially modified red blood cells (RBC) are given.

Flexural perturbations of free jets of maxwell and Doi-Edwards liquids

Flexural perturbations of high-velocity free jets of drop liquids moving in air are reinforced by the fact that the air pressure on the concave sections of the jet surface is greater than on the convex sections. The linear and nonlinear stages of development of flexural perturbations were studied in [1–5] for viscous Newtonian fluids. The effect of elastic stresses in the fluid on the growth of flexural perturbations of jets was first examined in [6], where it was assumed in an analysis of the growth of small disturbances that surface tension was constant along the jet, i.e., the investigators actually studied a tensed string. The studies [7, 8] examined the linear stage of growth of flexural perturbations of jets of Maxwell liquids. Our goal here is to analyze the dynamics of long-wave flexural perturbations of jets of viscoelastic fluids in both the linear and nonlinear stages of development. The rheological behavior of the fluid is described by two models — the phenomenological (Maxwell) model and the physical-molecular (Doi-Edwards) model. It is shown that the disturbances are oscillatory in character in the nonlinear stage of development. Meanwhile, the results of calculations performed with the Maxwell (M) and Doi-Edwards (DE) rheological models in the given problem agree with each other quantitatively as well as qualitatively.

Tentative application of the tangent simple system method to the study of the viscoelastic behaviour of blood.

The method of the tangent simple systems is applied to the study of the viscoelastic behaviour of human blood in unstationary flow for rectangular steps and triangular ramps of shear rate. The tangent systems we utilize, Maxwell liquids, enable us to determine, at every point of the rheograms, apparent instantaneous values of retardation or relaxation time, viscosity coefficient and elasticity modulus of the studied blood samples, and to plot the curves of variation of these parameters as a function of flow duration. A qualitative interpretation of the results is proposed from data on the aggregation-disaggregation kinetics of red blood cells. Examples are given for samples of normal and pathological bloods.

Some results in the theory of melt spinning for model viscoelastic liquids

Calculation of a “spinning viscosity” for a Maxwell liquid shows that this in not a unique function of the strain rate and the material, but depends on the flow history. It is argued therefore that use of this term or (worse) “elongational viscosity” in presenting experimental data is a dangerous oversimplification. The technique of subtracting out the influence of initial stress values is also show to be unjustified and is not recommended. Following these discussions for the simplest Maxwell liquids, more complicated models involving inertia, a retardation time and a spectrum of relaxation times are examined. The main results concern the existence or non-existence of solutions to initial-value and boundary-value problems for various Maxwell models, where generally the boundary-value problems which are better representations of spinning are better behaved mathematically. Limiting solutions of various types are discussed, and some preliminary computational evidence is presented. The emphasis is however on what may be deduced qualitatively without extensive computation.