Upper-convected Derivative Research Articles

On the use of the Astarita flow field for viscoelastic fluids to develop a generalised Newtonian fluid model incorporating flow type (GNFFTy)

The two-dimensional, steady, homogeneous flow field proposed by Astarita (J. Rheol., vol. 35, 1991, pp. 687–689) is studied for a range of viscoelastic constitutive equations of differential form including the models due to Oldroyd (the upper and lower convected Maxwell; UCM/LCM), Phan-Thien and Tanner (simplified, linear form; sPTT) and Giesekus. As the flow is steady and homogeneous, the sPTT model results also give the FENE-P model solutions via a simple transformation of parameters. The flow field has the interesting feature that a scalar parameter may be used to vary the flow ‘type’ continuously from solid-body rotation to simple shearing to planar extension whilst the rate of deformation tensor $\boldsymbol{\mathsf{D}}$ remains constant (i.e. independent of flow type). The response of the models is probed in order to determine how a scalar ‘viscosity’ function may be rigorously constructed which includes flow-type dependence. We show that for most of these models – the Giesekus being the exception – the first and second invariants of the resulting extra stress tensor are linearly related, and for models based on the upper convected derivative, this link is simply via a material property, i.e. half the modulus. By defining a frame-invariant coordinate system with respect to the eigenvectors of $\boldsymbol{\mathsf{D}}$ , we associate a ‘viscosity’ for each of the flows to a deviatoric stress component and show how this quantity varies with the flow-type parameter. For elliptical motions, rate thinning is always observed and all models give essentially the UCM response. For strong flows, i.e. flow types containing at least some extension, thickening occurs and only a small element of extension is required to remove any shear thinning inherent in the model (e.g. as occurs in steady simple shearing for the sPTT/Giesekus models). Finally, a functional form of a viscosity equation which could incorporate flow type, but be otherwise inelastic, the so-called GNFFTy (generalised Newtonian fluid model incorporating flow type, pronounced ‘nifty’), is proposed. In the frame-invariant coordinate system proposed, this model is also capable of capturing normal-stress differences.

Non-hyperbolicity of the inviscid Cattaneo–Christov system for compressible fluid flow in several space dimensions

SummaryWe consider the coupling between the equations of motion of a compressible fluid in two and three space dimensions with Christov’s equation for the heat flux. Christov’s equation is a frame indifferent formulation of the classical model of Cattaneo that allows for the heat flux to be eliminated to obtain a single hyperbolic equation for the temperature. The obtained system is written in quasilinear form for the state variables density, velocity, temperature and heat flux. It is then shown that this system is not of hyperbolic type as a consequence of the presence of the Lie–Oldroyd upper-convected material derivative involved in Christov’s formulation.

An Extended Reynolds Equation for Non-Newtonian Lubrication With the Upper Convected Maxwell Model

Abstract The upper convected Maxwell (UCM) model is one of the ways to describe lubricant’s viscoelasticity. In this paper, a new modified Reynolds equation for the UCM model is derived. This equation may be called extended as compared to its analogs because it incorporates the complete structure of the upper convected derivative without the explicit omitting of any term or using the perturbation technique. Then, a numerical scheme for solving the viscoelastic lubrication problems with the employment of the derived equation is described. A mixed Euler–Lagrange approach is used here: the constitutive rheological equations are resolved by a semi-Lagrangian technique and the extended Reynolds equation is discretized by the finite volume method. A constant surface slope problem is considered as a test case. Excellent agreement is achieved between the numerical solution at low Deborah number and one of the approximate solutions. The results of simulations with different types of time derivative used in Maxwell model both for two- and three-dimensional cases are also discussed.

Creep, recovery and vibration of an incompressible viscoelastic material of the rate type: Simple tension case

We consider three-dimensional nonlinear viscoelastic models that account for both stress relaxation and creep/recovery phenomena. These models are based on different frame indifferent time derivatives: the Oldroyd (or upper-convected) derivative, the Jaumann (or co-rotational) derivative and the Cotter–Rivlin (or lower-convected) derivative. Under a simple tension creep process, these constitutive equations predict the same stress relaxation but lead to different situations. The models based on the Oldroyd and the lower-convected derivative require restrictions on the values of the material parameters as well as on the traction/compression stress. The model based on the Jaumann derivatives does not require any restriction. All the constitutive models examined are used to study the finite amplitude, horizontal oscillatory motion of a mass attached to a rate-type viscoelastic string. In this way we generalize the classical results by Beatty and Zhou (1991).

A theoretical model for studying the nonlinear viscoelastic response of an active fluid undergoing oscillatory shear

In this work, a nonlinear phenomenological model for neutrally buoyant force-free active suspension of nematodes is proposed and tested. Just a few limited studies were found linked with nonlinear viscoelastic response of the active suspension investigated in this paper. The stress is decomposed through Fourier transform into elastic and viscous stress contributions. The stress response at large strain deviates drastically from the harmonic forcing in a nonlinear regime. In this case, the standard linear viscoelastic moduli cannot describe the nonlinear response of the fluid. Lissajous–Bowditch loops are used as rheological fingerprints to examine the behavior of nonlinear response of the investigated active fluid. The results show time-strain separable nonlinearity, therefore providing a new physically meaningful interpretation. When self-propelled particles interact with each other (i.e., a collective effect), they produce stresses that result in dynamic self-organization at spatial and temporal scales much larger than those of single particles. Complex rheological behavior in active matter depends on the interplay between the external forcing and the circulating flow induced by active agents. The active matter examined in this work is based on the nematode Caenorhabditis elegans motion, whose shape is defined by a dynamic balance between elastic, hydrodynamic, and muscular forces. The orientational instabilities of the active suspension of C. elegans observed in recent experiments carried out by the authors are considered in the present theoretical study. A new time evolution equation for the active stress tensor is proposed in terms of an Oldroyd–Maxwell upper convected material derivative for a dilute active suspension in the absence of thermal or active fluctuations. On the other hand, the Gordon–Schowalter material derivative is used in order to modify the model for the case of non-diluted suspensions. The constitutive equations are nondimensional, and the results are addressed on both linear (small amplitude oscillatory shear) and nonlinear (large amplitude oscillatory shear) regimes. We show results of the viscoelastic moduli as a function of strain in the linear region and in the nonlinear region. The associated Lissajous loop curves illustrating the nonlinear response and the transitions of elastic to viscous behavior of the material at high strain are also presented. The dissipated energy over oscillation cycle is associated with the area enclosed by the closed Lissajous loops curves. Lissajous–Bowditch loops are also computed for the first normal stress differences using our theoretical model, and the results are compared with experimental work that was previously published by the authors.

Stability of an oscillatory Taylor–Couette flow in an upper convected Maxwell fluid

The stability of pulsed bi-dimensional flow between two co-oscillating cylinders in a linear Maxwell fluid was studied by Riahi et al. [J. Soc. Rheol. 42, 321–327 (2014)]. In the present paper, we revisit this flow configuration with emphasis on the effect of the non-linear terms in the constitutive equation of the model, measured by the Weissenberg number, on the dynamics of the system. Under these assumptions and using the upper convected Maxwell derivative, we examine this model to large amplitude oscillatory shear giving rise to the appearance, in comparison to the linear Maxwell model, of the azimuthal normal stress in the basic state. Using the spectral method and the Floquet theory for the spatiotemporal resolution of the obtained eigenvalue problem, numerical results exhibit numerous classes of Taylor vortex flows depending on the order of magnitude of the fluid elasticity. The resulting stability diagram consists of several branches intersecting at specific frequencies where two different Taylor vortex flows simultaneously branch off from the basic state. This feature is accompanied by the occurrence of several co-dimension two bifurcation points besides jumps/drops in the corresponding critical wave number. In addition, it turns out that the elasticity produces strong destabilizing and stabilizing effects in the limit of high and low frequency regimes, respectively, attributed solely to the non-linearities considered by the rheological model.

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.

On Hiemenz flow of Maxwell incompressible viscoelastic medium

Steady axisymmetric flow of a viscoelastic incompressible fluid near the critical point of the plane surface is considered. Model of Maxwellian fluid with upper convected derivative in the rheological constitutive law is used. Velocity and extra stresses field are discussed for various Weissenberg numbers. The subclass of solutions in a stationary case is completely described. The asymptotic solutions for small Weissenberg numbers are compared to the solutions of the complete nonlinear problem. It is found that the fluid elasticity decreases the boundary layer thickness.

Impact of homogeneous–heterogeneous reactions and non-Fourier heat flux theory in Oldroyd-B fluid with variable conductivity

This article scrutinizes the influence of chemical reactions on flow of an Oldroyd-B fluid due to stretched cylinder. In vision of non-Fourier heat flux model, the heat transfer phenomenon is scrutinized. This enhanced constitutive model anticipates the time space upper-convected derivative which is recycled to depicting heat conduction mechanism. Additionally, heat transfer scrutiny is considered with the influence of thermal conductivity which is temperature dependent. Apposite conversions are engaged to acquire ODEs which are then deciphered analytically via homotopic approach. To highlight their physical consequences, the graphical portrayal of diverse considerations on velocity, temperature and concentration fields is depicted and conferred. It is scrutinized from this study that all the profiles are higher in the instance of the cylinder as equated to a flat plate. This scrutiny also reported that the thermal relaxation parameter decreases the temperature field while the Schmidt number and homogeneous response parameter display the conflicting performance on concentration field. In addition, an assessment in restrictive instance is also presented in this exploration, which ensures us that our outcomes are more precise.

Change in internal energy of thermal diffusion stagnation point Maxwell nanofluid flow along with solar radiation and thermal conductivity

Abstract This paper concerns the characteristics of heat and mass transfer in upper convected Maxwell fluid flow over a linear stretching sheet with solar radiation, viscous desperation and temperature based viscosity. After boundary layer approximation, the governing equations are achieved (namely Maxwell, upper convected material derivative, thermal and concentration diffusions). By using the self-similarity transformations the governing PDEs are converted into nonlinear ODEs and solved by RK-4 method in combination with Newton Raphson (shooting technique). The effects of developed parameters on velocity, temperature, concentration, fraction factor, heat and mass diffusions are exemplified through graphs and tabular form and are deliberated in detail. Numerical values of fraction factor, heat and mass transfer rates with several parameters are computed and examined. It is noticed that the temperature is more impactable for higher values of radiative heat transport, thermal conductivity and viscous dissipation. The comparison data for some limiting case are acquired and are originated to be in good agreement with previously published articles.

Non-Fourier's and Fick's laws applications in the flow of temperature dependent thermal conductivity generalized Newtonian liquids: A 3D computational study

The current article presents a numerical study for local solutions of Sisko liquid flow by bidirectional stretched surface. Additionally, the analysis of relaxation times for the heat and mass transfer mechanisms are made by utilizing modified heat flux and mass diffusion models. These improved relations are the generalized form of Fourier's and Fick's laws in which the time space upper-convected derivative are employed to portray the heat conduction and mass diffusion mechanisms. Appropriate transformations lead to a strongly nonlinear differential system of equations which is then solved numerically by employing the applications of bvp4c package in Matlab software. Another numerical method namely shooting technique with RK45 Fehlberg and Newton-Raphson method is utilized to authenticate the results. Graphical illustrations demonstrating the impacts of sundry physical parameters with required discussion highlighting their physical effect are also a part of this exploration. It is perceived that the temperature and concentration of Sisko liquid are diminishing functions of relaxation times for the heat and mass transfer mechanisms. It is also fascinating to perceive that the temperature and concentration of Sisko liquid are higher in case of classical form in comparison with an improved constitutive relations.

Cattaneo–Christov intensity of magnetised upper-convected Maxwell nanofluid flow over an inclined stretching sheet: A generalised Fourier and Fick's perspective

The present study exemplifies the systematic analysis of the boundary layer Cattaneo-Christov model of heat and mass transport of the upper-convected Maxwell nanofluid passed by an inclined stretching surface in the presence of magnetic field. The pioneering step i.e. the upper-convected material derivative has been constructed to frame generalised Fourier's and Fick's phenomenon on behalf of the Cattaneo-Christov model. Governing non-linear PDEs are renovated to ODEs and then worked out via RK-4 procedure. Influences of flow factors inside flow regime, namely, fluid relaxation parameter δm, thermal relaxation parameter δe, nanoparticle concentration relaxation parameter δc are deliberated through tables and graphs to predict the double diffusive phenomenon truthfully. Our investigation authenticates that higher temperature distribution profile is noticed for δe=0.0 i.e. so called Fourier's law as compared to Cattaneo-Christov heat flux model. Thus the significance of the current literature is to explore its unique endeavour towards the generalised version of conventional Fourier's law and Fick's law at nanostructure level.

Consequences of non-Fourier’s heat conduction relation and chemical processes for viscoelastic liquid

The current work aims to investigate the features of an improved heat conduction relation and chemical processes on the unsteady second grade fluid over a stretching surface. The features of unsteady Cattaneo-Christov heat relation is incorporated in the energy equation. The unsteady Cattaneo-Christov heat flux relation is the generalization of Fourier’s laws in which the time space upper-convected derivative is utilized to describe the heat conduction phenomena. Additionally, the mass transfer phenomena is conducted by utilizing heterogeneous-heterogeneous processes. Heterogeneous-homogenous phenomena’s are taking place on the surface of the wall and ambient fluid, respectively. Moreover, these processes are given by isothermal cubic auto-catalator and first order kinetics. The self-similarity transformation is used to transfer the governing PDEs into the ODEs. The resulting problem under consideration is solved analytically by using HAM. The effect of non-dimensional pertinent parameters on the temperature and concentration distribution are deliberated by using graphs and tables. Results show that the temperature profile diminishes for augmented values of the thermal relaxation parameter. In addition to this, it has come to the observation that the liquid temperature is higher for classical Fourier’s law when compared to non- Fourier’s law.

Homotopic solutions for unsteady second grade liquid utilizing non-Fourier double diffusion concept

Main purpose of the current work is to investigate the features of unsteady Cattaneo–Christov heat and mass flux models on the second grade fluid over a stretching surface. The characteristics of unsteady Cattaneo–Christov heat and mass flux models are incorporated in the energy and concentration equations. The unsteady Cattaneo–Christov heat and mass flux models are the generalization of Fourier’s and Fick’s laws in which the time space upper-convected derivative are utilized to describe the heat conduction and mass diffusion phenomena. The suitable transformations are used to alter the governing partial differential equations into the ordinary differential equations. The resulting problem under consideration is solved analytically by using the homotopy analysis method (HAM). The effect of non-dimensional pertinent parameters on the temperature and concentration distribution are deliberated by using graphs and tables. Results show that the temperature and concentration profiles diminish for augmented values of the thermal and concentration relaxation parameters. Additionally, it is perceived that the temperature and concentration profiles are higher in case of classical Fourier’s and Fick’s laws as compared to non-Fourier’s and non-Fick’s laws.

Boundary layer heat and mass transfer with Cattaneo–Christov double-diffusion in upper-convected Maxwell nanofluid past a stretching sheet with slip velocity

A systematic study is presented for the boundary layer Cattaneo–Christov double-diffusion model of heat and mass transfer in an upper-convected Maxwell nanofluid over a stretching sheet. The innovative constitutive model, namely Cattaneo–Christov upper-convected material derivative, is first introduced in characterizing the boundary layer slip shear flow, thermal diffusion and nanoparticles concentration diffusion. Viscoelastic relaxation framework system of upper-convected Maxwell nanofluid is constructed uniquely in which heat and mass transfer are both determined by Cattaneo–Christov model uniformly. The effects of Brownian motion & thermophoresis and the dynamic viscosity assumed as the linear function of temperature are taken into account. The highly coupled boundary layer governing equations including momentum, energy & mass conservation equations are transformed to similarity equations via appropriate dimensionless variables. The velocity, temperature & concentration distributions affected by respective relaxation parameters are obtained by homotopy analysis method (HAM) and analyzed thoroughly. Results indicated that this viscoelastic relaxation framework system makes us possible to predict the relaxation times transport characteristics. The internal elastic stress aggregates initially (with big skin friction) and then release (small stress) along with the development of the boundary layer, which is the main reason to generate macroscopic relaxation phenomena. Moreover, the effects of velocity slip on boundary layer transport are also discussed.

Lattice Boltzmann Formulation for Linear Viscoelastic Fluids Using an Abstract Second Stress

The kinetic theory of gases implies an independent evolution equation for the momentum flux tensor that closely resembles an evolution equation for the elastic stress in continuum descriptions of viscoelastic liquids. However, kinetic theory leads to a nonobjective convected derivative for the evolution of the deviatoric stress, and a fixed relation between the stress relaxation rate and the viscosity. We show that simulations of freely decaying shear flow using the standard two-dimensional lattice Boltzmann kinetic model develop a tangential stress consistent with this nonobjective convected derivative, and this fixed relation between parameters. By contrast, viscoelastic liquids are typically modeled by an upper convected derivative, and with two independent parameters for the viscosity and stress relaxation rate. Although we are unable to obtain an upper convected derivative from kinetic theory with a scalar distribution function, we show that introducing a general linear coupling to a second stress tensor yields the linear Jeffreys viscoelastic model with three independent parameters in the incompressible limit. Unlike previous work, we do not attempt to represent the additional stress through moments of additional distribution functions, but treat it only as an abstract tensor that couples to the corresponding tensorial moment of the hydrodynamic distribution functions. This greatly simplifies the derivation, and the implementation of flows driven by body forces. The utility of the approach is demonstrated through simulations of Stokes' second problem for an oscillating boundary, of the four-roller mill, and of three-dimensional Arnold--Beltrami--Childress and Taylor--Green flows.

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.

Semi‐Lagrangian discretization of the upper‐convective derivative in Non‐Newtonian fluid flow

AbstractNon‐Newtonian fluid flow is governed by the Navier‐Stokes equations with an additional source term resulting from Non‐Newtonian stresses. The Non‐Newtonian stresses evolve along particle paths according to an evolution equation. The temporal derivative in this case is the upper convected derivative. We describe a semi‐Lagrangian discretization of the upper convected derivative. Numerical results for the flow over a contraction are given. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the nonlinear continuum mechanics of space and the notion of luminiferous medium

We prove that, when linearized, the governing equations of an incompressible elastic continuum yield Maxwell’s equations as corollaries. Through judicious distinction between the referential and local descriptions, the principle of material invariance is established and shown to be a true covariance principle, unlike the Lorentz covariance, which is valid only for non-deforming frames in rectilinear relative motion. Thus, this paper establishes that electrodynamics can be fully explained if one assumes that it is the manifestation of the internal forces of an underlying elastic material which we term the metacontinuum. The new frame-indifferent formulation of electrodynamics is shown to incorporate the Lorentz force as an integral part of Faraday’s law, rather than as an additional empirical variable. Respectively, if the upper-convected derivative is added in Maxwell’s displacement current it can explain Biot–Savart’s and Oersted–Ampere’s laws. An immediate corollary of the material invariance is the Galilean invariance of the model. The possible detection of the absolute continuum is also discussed. First, the famous experiment of Ives and Stilwell is reexamined with a modified Bohr–Rydberg formula for the emitted frequencies from a moving atom, and it is shown that the results are fully compatible with the presence of an absolute medium. Second, a new interferometry experiment is proposed in which the first-order Doppler effect can be measured, and thus the presence of a medium at rest can be unequivocally established.

On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction

A material-invariant (frame indifferent) version of the Maxwell–Cattaneo law is proposed in which the relaxation rate of the heat flux is given by Oldroyd’s upper-convected derivative. It is shown that the new formulation allows for the elimination of the heat flux, thus yielding a single equation for the temperature field. This feature is to be expected from a truly frame indifferent description.