Viscoelastic Plane Research Articles

Maximum drag enhancement asymptote in spanwise-rotating viscoelastic plane Couette flow of dilute polymeric solutions

The existence of a maximum drag enhancement (MDE) asymptote at high rotation ( $Ro$ ) and Weissenberg ( $Wi$ ) numbers in turbulent viscoelastic spanwise-rotating plane Couette flow has been demonstrated. Specifically, it is shown that above a critical $Wi$ , drag enhancement plateaus and the MDE asymptote is realized in a broad range of $Ro$ . The mean velocity profiles at MDE appear to closely follow a log-law profile that has a nearly identical slope but different intercepts as a function of $Ro$ . Much like the maximum drag reduction (MDR) asymptote, the logarithmic function in MDE is closely followed if the mean velocity is plotted using the traditional inner variable scaling; however, the logarithmic function is not well defined when examined by the indicator function. Hence, in this study, we have used the logarithmic fit as a visual guide for the mean velocity profile. Last and perhaps the most intriguing finding of this study is that MDE occurs in the elasto-inertial turbulence (EIT) flow state; hence, it is mainly sustained by elastic forces much like the MDR flow state. To that end, a universal picture of elastically induced drag modification asymptotes is emerging, namely these asymptotic states are an inherent property of the elastically sustained EIT flow state.

Viscoelasticity-Induced Instability in Plane Couette Flow at Very Low Reynolds Number

Elasto-inertial turbulence (EIT), a new turbulent state found in polymer solutions with viscoelastic properties, is associated with drag-reduced turbulence. However, the relationship between EIT and drag-reduced turbulence is not currently well-understood, and it is important to elucidate the mechanism of the transition to EIT. The instability of viscoelastic fluids has been studied in a canonical wall-bounded shear flow to investigate the transition process of EIT. In this study, we numerically deduced that an instability occurs in the linearly stable viscoelastic plane Couette flow for lower Reynolds numbers, at which a non-linear unstable solution exists. Under instability, the flow structure is elongated in the spanwise direction and regularly arranged in the streamwise direction, which is a characteristic structure of EIT. The regularity of the flow structure depends on the Weissenberg number, which represents the strength of elasticity; the structure becomes disordered under high Weissenberg numbers. In the energy spectrum of velocity fluctuations, a steep decay law of the structure’s scale towards a small scale is observed, and this can be recognized as a ubiquitous feature of EIT. The existence of instability in viscoelastic plane Couette flow supports the idea that the transitional path toward EIT may be mediated by subcritical instability.

Relaminarization of spanwise-rotating viscoelastic plane Couette flow via a transition sequence from a drag-reduced inertial to a drag-enhanced elasto-inertial turbulent flow

Direct numerical simulation of spanwise-rotation-driven flow transitions in viscoelastic plane Couette flow from a drag-reduced inertial to a drag-enhanced elasto-inertial turbulent flow state followed by full relaminarization is reported for the first time. Specifically, this novel flow transition begins with a drag-reduced inertial turbulent flow state at a low rotation number $0\leqslant Ro \leqslant 0.1$ , and then transitions to a rotation/polymer-additive-driven drag-enhanced inertial turbulent regime, $0.1\leqslant Ro \leqslant 0.3$ . In turn, the flow transitions to a drag-enhanced elasto-inertial turbulent state, $0.3\leqslant Ro \leqslant 0.9$ , and eventually relaminarizes at $Ro=1$ . In addition, two novel rotation-dependent drag enhancement mechanisms are proposed and substantiated. (1) The formation of large-scale roll cells results in enhanced convective momentum transport along with significant polymer elongation and stress generated in the extensionally dominated flow between adjacent roll cells at $Ro\leqslant 0.2$ . (2) Coriolis-force-generated turbulent vortices cause strong incoherent transport and homogenization of significant polymer stress in the bulk via their vortical circulations at $Ro=0.5 - 0.9$ .

Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

Continuous Pathway between the Elasto-Inertial and Elastic Turbulent States in Viscoelastic Channel Flow.

Viscoelastic plane Poiseuille flow is shown to become linearly unstable in the absence of inertia, in the limit of high elasticities, for ultradilute polymer solutions. While inertialess elastic instabilities have been predicted for curvilinear shear flows, this is the first ever report of a purely elastic linear instability in a rectilinear shear flow. The novel instability continues up to a Reynolds number (Re) of O(1000), corresponding to the recently identified elasto-inertial turbulent state believed to underlie the maximum-drag-reduced regime. Thus, for highly elastic ultradilute polymer solutions, a single linearly unstable modal branch may underlie transition to elastic turbulence at zero Re and to elasto-inertial turbulence at moderate Re, implying the existence of continuous pathways connecting the turbulent states to each other and to the laminar base state.

Transition Processes in Tippe-Top Dynamics

The simplest model of a tippe-top is a nonhomogeneous dynamically symmetric ball whose center of mass lies on the axis of dynamic symmetry, but does not coincide with the geometric center. Local analysis of this model is presented in [1, 2] and global qualitative analysis is presented in [3–5]. Numerical studies for multicomponent dry friction were carried out in [6]. A comparative analysis of various models was carried out in [7]. A method of generalized Smale diagrams [9, 10] regarding the problem of the motion of a tippe-top top on a viscoelastic plane is presented in [8]. In this paper, approximate equations that describe the dynamics of the top and allow us to supplement the qualitative analysis with quantitative estimates for multicomponent models of dry and viscous friction under a certain class of initial conditions have been presented.

A symplectic analytical singular element for V-notched analyses in elastic and viscoelastic plane problems

The linear elastic and viscoelastic problems with V-shaped notches under in-plane loading are investigated in this paper. We consider two different sets of boundary conditions on the radial edges: Free-Free case and Clamped-Clamped case. A novel singular finite element (SASE) with arbitrary high-order accuracy for the notched problems is proposed. Firstly, via Laplace transform, the original viscoelastic problem is transformed into a corresponding elastic one. Then we construct the SASE by using elastic symplectic eigen solutions with higher order expanding terms. The SASE can depict the characteristics of displacement fields and singular stress fields in the vicinity of the notch vertex having arbitrary opening angle. By taking advantage of the symplectic eigen solutions in the SASE, Mode I and/or Mode II notch stress intensity factors can be determined directly without any post-processing. Fine finite element meshes are not required. Numerical examples are provided to illustrate the validity and accuracy of the present method.

Enriched homogenized model for viscoelastic plane wave propagation in periodic layered composites

An enriched homogenized model is developed based on a proposed homogenization strategy, to describe the wave propagation behaviour through periodic layered composites. The intrinsic parameters characterising the micro-inertia effect and non-local interactions are defined transparently in terms of the constituent materials’ properties and volume fractions. The framework starts with the introduction of an additional kinematic field to characterise the displacement of the stiff layer, before setting up macro kinematic fields to account for the average deformation of the constituent materials within a segmented unit cell. Relationships between these macro average strain fields are determined based on suitable micro-mechanical arguments. The Hill–Mandel condition is next applied to translate the energy statements from micro to macro. A system of coupled governing equations of motion is finally extracted naturally at the macro level via Hamilton’s Principle. Through a series of benchmark examples, it is shown that the proposed model exhibits excellent predictive capabilities over a broad range of loading frequencies.

Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow

A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance $2L$, has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number $Re=\unicode[STIX]{x1D70C}U_{max}L/\unicode[STIX]{x1D702}$ and the elasticity number $E\equiv W/Re=\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}L^{2})$, where $W=\unicode[STIX]{x1D706}U_{max}/L$ is the Weissenberg number. Here, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D702}$ is the fluid viscosity, $\unicode[STIX]{x1D706}$ is the micro-structural relaxation time and $U_{max}$ is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined ($Re\leqslant 10^{4},E\leqslant 0.01$). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the $Re$–$k$ plane (here $k$ is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite $E$) of the Tollmien–Schlichting (TS) instability already present for Newtonian channel flow. As $E$ is increased beyond $0.0016$, a new elastic mode appears at $Re\sim 10^{4}$, which coalesces with the continuation of the TS mode for a range of $Re$, thereby yielding a single unstable mode in this range. This trend persists until $E\sim 0.0021$, beyond which this neutral curve splits into two separate ones in the $Re$–$k$ plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number $Re_{c}\approx 1210.9$ for $E=0.0066$. The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in $E$, which eventually vanish at sufficiently high $E$. For $E\ll 1$, the critical Reynolds number and wavenumber scale as $Re_{c}\sim E^{-1}$ and $k_{c}\sim E^{-1/2}$ for the first two of the symmetric modal families, and as $Re_{c}\sim E^{-5/4}$ for first two of the antisymmetric modal families; $k_{c}\sim E^{-1/4}$ for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as $c_{r,c}\sim E^{1/2}$ for $E\ll 1$, implying that the modes belong to a class of ‘wall modes’ in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.

Dry friction distributed over a contact patch between a rigid body and a visco-elastic plane

We consider the dynamics of an absolutely rigid body moving along a rough horizontal plane. We assume that the plane deforms during the motion so that the contact patch is non-planar and has non-zero, but comparatively small area. In the contact patch, the vertical reactions are proportional to the vertical deformations and their rates, that is, we consider Kelvin–Voigt model. The tangent forces are assumed to be classic Coulomb dry friction in sliding regime (no stiction in contact patch). Due to the viscous part of Kelvin–Voigt law, the contact patch, the plane’s normal reaction, friction force and torque depend on the position, orientation, velocity of the center of mass and the angular velocity of the body. The model does not involve any additional dynamic parameter and gives the friction force and torque directly for any given state of the body. To show the advances of the model, we recall the analytical solution of Cauchy’s initial problem with general initial conditions of ODE governing the dynamics of the homogeneous sphere on a horizontal plane (results of Zobova and Treschev in Proc. Steklov Inst. Math. 281:91–118, 2013). Here we generalize the model for arbitrary convex bodies and study its main properties. The model is compared to the continuous velocity-base friction model for pure sliding (Brown and McPhee in ASME J. Comput. Nonlinear Dyn. 11(5):054502, 2016) and to the combined dry friction (Awrejcewicz and Kudra in Multibody Syst. Dyn., 2018, https://doi.org/10.1007/s11044-018-9624-9 ) in case of sliding with spinning. We illustrate the model considering the results of numerical integration of the Cauchy problem for a controlled differential-drive vehicle on a horizontal plane.

Dynamic Response of a Cracked Viscoelastic Anisotropic Plane Using Boundary Elements and Fractional Derivatives

Abstract The aim of this study is to develop an efficient numerical technique using the non-hypersingular, traction boundary integral equation method (BIEM) for solving wave propagation problems in an anisotropic, viscoelastic plane with cracks. The methodology can be extended from the macro-scale with certain modifications to the nano-scale. Furthermore, the proposed approach can be applied to any type of anisotropic material insofar as the BIEM formulation is based on the fundamental solution of the governing wave equation derived for the case of general anisotropy. The following examples are solved: (i) a straight crack in a viscoelastic orthotropic plane, and (ii) a blunt nano-crack inside a material of the same type. The mathematical modelling effort starts from linear fracture mechanics, and adds the fractional derivative concept for viscoelastic wave propagation, plus the surface elasticity model of M. E. Gurtin and A. I. Murdoch, which leads to nonclassical boundary conditions at the nano-scale. Conditions of plane strain are assumed to hold. Following verification of the numerical scheme through comparison studies, further numerical simulations serve to investigate the dependence of the stress intensity factor (SIF) and of the stress concentration factor (SCF) that develop in a cracked inhomogeneous plane on (i) the degree of anisotropy, (ii) the presence of viscoelasticity, (iii) the size effect with the associated surface elasticity phenomena, and (iv) finally the type of the dynamic disturbance propagating through the bulk material.

Tippe-top on visco-elastic plane: steady-state motions, generalized smale diagrams and overturns

We consider the dynamics of a tippe-top on a visco-elastic plane with dry friction. Tippetop is a rigid dynamically symmetric sphere with the center of mass that lies on the dynamical axis of symmetry but does not coincide with the center of the sphere. It was shown earlier [1] that the full mechanical energy conserves only on the steady-state motions of the tippe-top while the linear on the pseudovelocities Jellet integral remains constant for a quite general model of friction, for example, viscous friction. It allows studying the stability of the steady-state motions and constructing the generalized Smale diagrams [1]. Here we consider another model of friction—so-called distributed dry friction proposed in [2] that is more suitable for the convex bodies on a rough plane. In this case, both energy and Jellet function change their value on solutions. The earlier investigated case is considered as generating. Steady state motions transform in pseudo-steady motions with slow changing parameters. Calculations illustrate the analytical investigation.

Stability of plane Couette flow of a power-law fluid past a neo-Hookean solid at arbitrary Reynolds number

The linear stability of plane Couette flow of a power-law fluid past a deformable solid is analyzed at arbitrary Reynolds number (Re). For flow of a Newtonian fluid past a deformable solid, at high Re, there are two different modes of instability: (i) “wall modes” (Γ∝Re−1∕3) and (ii) “inviscid modes” (Γ∝Re−1) where Γ=VμfGR is the non-dimensional shear-rate in the fluid (V, μf, G, and R denote the top-plate velocity, fluid viscosity, shear modulus of the solid, and fluid thickness, respectively). In this work, we consider the power-law model for the fluid to elucidate the effect of shear-thickening/shear-thinning behaviour on the modes of instability present in the flow, especially at moderate and high Re. At high Re, our numerical results show that wall modes exhibit different scalings in Γ (VηfGR, where ηf is Newtonian-like constant viscosity) vs Re for different values of the power-law index (n), and the scaling exponents are different from that for a Newtonian fluid. This drastic modification in the scaling of wall modes is not observed in viscoelastic (modelled as upper-convected Maxwell or Oldroyd-B fluids) plane Couette flow past a deformable solid. We show that the difference in scaling exponents can be explained by postulating that the wall modes in a power-law fluid are determined by the actual viscosity corresponding to the shear rate of the laminar flow denoted by ηapp. A non-dimensional shear rate based on this viscosity Γapp=VηappGR can be defined, and we show that the postulate Γapp∼Re−1∕3 (motivated by the wall-mode scaling in a Newtonian fluid) captures all the numerically observed scalings for Γ vs Re for different values of n>0.3, which is found to be Γ∼Re−12n+1. Further, we numerically evaluated the wall layer thickness and this agreed with the theoretical scaling of δ∼Re−n2n+1. Interestingly, the theoretical and numerical prediction of wall modes is found to be valid for power-law index, n≥ 0.3. For n≤ 0.3, there is a marked departure from the wall-mode scalings, and our results show a scaling of Γ ∼ Re−1 corresponding to inviscid modes. The variation of the power-law index (n) can stabilise/destabilise the inviscid mode when compared with Newtonian fluid, and this result is observed only in the power-law model and is not seen in the flow of viscoelastic fluid past deformable surfaces. Thus, the present study shows that the shear-rate dependence of viscosity has a significant impact on both the qualitative and quantitative aspects of stability of non-Newtonian fluid flow past deformable surfaces.

Total Lagrangian FEM formulation for nonlinear dynamics of sliding connections in viscoelastic plane structures and mechanisms

In this work we extend a total Lagrangian formulation developed for dynamical nonlinear analysis of elastic plane frames to include structural energy dissipation and sliding connections (as prismatic and cylindrical joints). The joints move in a path that may present an arbitrary roughness profile along its trajectory. Their exact kinematical restraints are introduced in the total mechanical energy of the system by the method of Lagrange multipliers. The structural energy dissipation is considered by a modified Kelvin-Voigt rheological model, for which a pure numerical procedure for the strain rate approximation (finite difference) is proposed. The dynamical equilibrium equation is obtained by a variational principle and the nonlinear solution procedure is done by the Newton-Raphson method. Several examples are presented to demonstrate the efficiency of the proposed formulations.

Generalized fractional derivatives and their applications to mechanical systems

New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are introduced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer- order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed.

Ball on a viscoelastic plane

We consider dynamical problems arising in connection with the interaction of an absolutely rigid ball and a viscoelastic support plane. The support is a relatively stiff viscoelastic Kelvin-Voigt medium that coincides with the horizontal plane in the undeformed state. We also assume that under the deformation the support induces dry friction forces that are locally governed by the Coulomb law. We study the impact appearing when a ball falls on the plane. Another problem of our interest is the motion of a ball “along the plane.” A detailed analysis of various stages of the motion is presented. We also compare this model with classical models of interaction of solid bodies.

Dispersion curves of viscoelastic plane waves and Rayleigh surface wave in high frequency range with fractional derivatives

The moduli of conventional elastic structural materials are extended to one of the viscoelastic materials through a modification whereby the dynamic moduli converge to the static moduli of elasticity as the fractional order approaches zero. By plotting phase velocity curves and group velocity curves of plane waves and Rayleigh surface wave for a viscoelastic material (polyvinyl chloride foam), the influence of the fractional order of viscoelasticity is examined. The phase and group velocity curves in the high frequency range were derived for longitudinal, transverse, and Rayleigh waves inherent to the viscoelastic material. In addition, the equation for the phase velocity was mathematically derived on the complex plane, too, and graphically illustrated. A phenomenon was found that, at the moment when the fractional order of the time derivative reaches an integer value 1, the curve on the complex plane becomes completely different, exhibiting snap-through behavior. We examined the mechanism of the snap-through mathematically. Numerical calculation examples were solved, and good agreement was confirmed between the numerical calculation and the analytical expression mentioned above. From the results of the numerical example, regularities were derived for the absolute value of the complex phase and group velocities on the complex plane.

Твердый цилиндр на вязкоупругой плоскости

Твердый цилиндр на вязкоупругой плоскости

A fast multipole boundary element method for 2D viscoelastic problems

A fast multipole formulation for 2D linear viscoelastic problems is presented in this paper by incorporating the elastic–viscoelastic correspondence principle. Systems of multipole expansion equations are formed and solved analytically in Laplace transform domain. Three commonly used viscoelastic models are introduced to characterize the time-dependent behavior of the materials. Since the transformed multipole formulations are identical to those for the 2D elastic problems, it is quite easy to implement the 2D viscoelastic fast multipole boundary element method. Besides, all the integrals are evaluated analytically, leading to highly accurate results and fast convergence of the numerical scheme. Several numerical examples, including planar viscoelastic composites with single inclusion or randomly distributed multi-inclusions, as well as the problem of a crack in a pressured viscoelastic plane, are presented. The results are verified by comparison with the developed analytical solutions to illustrate the accuracy and efficiency of the approach.

State of stress in slabs as a function of large‐scale plate kinematics

The state of stress of slabs subducting worldwide, as revealed by seismicity, is extremely variable both with depth and between different subduction zones. Although in principle, slab pull should enhance downdip extension in the slab, the reconstructed stress fields for intermediate depths (between 100 and 300 km) range from downdip compression to downdip extension. Using 2‐D viscoelastic plane strain models we investigate the dependency of the stress field of slabs on geometry (dip of the slab) and kinematics (velocity of convergence between upper and lower plates and their absolute velocity with respect to the underlying mantle) of subduction zones. We conclude that although the state of stress in slabs is also controlled by other processes, downdip compression in the subducting slab is enhanced by mantle flow opposing the direction of the dip of the slab, whereas downdip extension is favored by mantle flow in the same direction of the slab dip (i.e., sustaining it). These predictions are in agreement with available geophysical observations, although exceptions to this simple pattern are observed worldwide. In addition, if the slab is decoupled from the upper plate, convergence between upper and lower plates induces a downdip compressional component of stress within the slab, decreasing the magnitude of extension in models characterized by mantle flow sustaining the slab and increasing compression in models with mantle flow opposing subduction. However, these are second‐order variations when compared to the control exerted by absolute plate kinematics and by the magnitude of slab pull. Sensitivity analysis of rheological parameters allows us to conclude that these results are generally consistent, although low values of viscosity of the lithospheric mantle render this prediction less stable.