Compressible Viscoelastic Fluid Research Articles

The pointwise estimates of solutions for the 3D compressible viscoelastic fluids

In this paper, we investigate the Cauchy problem of the 3D compressible viscoelastic hydrodynamic model. By decomposing the compressible viscoelastic fluid equations, according to Duhamel's principle, we make full use of the decomposition of divergence and curl, and obtain the expression of the corresponding solution. By refined integral calculation and classification discussion, we analyze the interaction of different waves in detail, and finally obtain the pointwise time-space estimates of the solutions.

Linear Rayleigh-Taylor instability for compressible viscoelastic fluids

<abstract><p>In this paper, we consider the linear Rayleigh-Taylor instability of an equilibrium state of 3D gravity-driven compressible viscoelastic fluid with the elasticity coefficient $ \kappa $ is less than a critical number $ \kappa_{c} $ in a moving horizontal periodic domain. We first construct the maximal growing mode solutions to the linearized equations by studying a family of modified variational problems, and then we prove an estimate for arbitrary solutions to the linearized equations.</p></abstract>

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate the regularity of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of the flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids, which is different from the case studied by Mei et al. (2018) for the free boundary compressible Navier-Stokes system.

Unjamming and emergent nonreciprocity in active ploughing through a compressible viscoelastic fluid

A dilute suspension of active Brownian particles in a dense compressible viscoelastic fluid, forms a natural setting to study the emergence of nonreciprocity during a dynamical phase transition. At these densities, the transport of active particles is strongly influenced by the passive medium and shows a dynamical jamming transition as a function of activity and medium density. In the process, the compressible medium is actively churned up – for low activity, the active particle gets self-trapped in a cavity of its own making, while for large activity, the active particle ploughs through the medium, either accompanied by a moving anisotropic wake, or leaving a porous trail. A hydrodynamic approach makes it evident that the active particle generates a long-range density wake which breaks fore-aft symmetry, consistent with the simulations. Accounting for the back-reaction of the compressible medium leads to (i) dynamical jamming of the active particle, and (ii) a dynamical non-reciprocal attraction between two active particles moving along the same direction, with the trailing particle catching up with the leading one in finite time. We emphasize that these nonreciprocal effects appear only when the active particles are moving and so manifest in the vicinity of the jamming-unjamming transition.

On the initial-boundary value problem for the three-dimensional compressible viscoelastic fluids with the electrostatic effect

We investigate the initial-boundary value problem of the three-dimensional compressible viscoelastic fluids with the electrostatic effect, in which the Dirichlet-Neumann mixed boundary condition for the electrostatic potential is imposed. We prove that there exists a unique global-in-time small strong solution in H2 Sobolev space. Moreover, we show that such a solution converges to the constant equilibrium state with an exponential decay rate as time tends to infinity.

Nonlinear dynamics of a solid particle in an acoustically excited viscoelastic fluid. II. Acoustic radiation force.

An analytical formula is derived for acoustic radiation force exerted by an axisymmetric acoustic wave on an isotropic solid spherical particle in a compressible viscoelastic fluid. The particle is assumed to undergo pulsation, translation, and shape deformations of all orders. The fluid motion is described by the compressible Oldroyd-B model. No restrictions are imposed on the particle size with respect to the acoustic wavelength and the viscous penetration depth. The obtained analytical formula is used in numerical simulations. The results show that the acoustic radiation force can be increased by more than a factor of 10 due to the viscoelastic nature of the fluid, depending on the parameters. It is also observed that the sign of the force can change so that the force becomes directed to the velocity node instead of the velocity antinode if the driving frequency, the particle size, and the relaxation time of the polymer solution are relatively small.

Nonlinear dynamics of a solid particle in an acoustically excited viscoelastic fluid. I. Acoustic streaming.

An analytical theory is developed for acoustic streaming induced by an axisymmetric acoustic wave field around an isotropic solid spherical particle in a compressible viscoelastic fluid. The particle is assumed to undergo pulsation, translation, and shape deformations of all orders. The fluid motion is described by the compressible Oldroyd-B model. No restrictions are imposed on the particle size with respect to the acoustic wavelength and the viscous penetration depth. The obtained analytical solutions are used in numerical simulations. It is shown that in the general case, the streaming velocity magnitude decreases with increasing polymer viscosity. Increasing relaxation time (elasticity) of the polymer solution leads to increasing streaming velocity magnitude as long as the relaxation time remains relatively small. It is also observed that the variation of the polymer viscosity and the relaxation time can change the pattern of streamlines.

Diffusion wave phenomena and Lp decay estimates of solutions of compressible viscoelastic system

We consider the system of equations describing motion of compressible viscoelastic fluids in a whole space. We investigate the large time behavior of solutions around a motionless state, and obtain the Lp decay estimates of solutions for 1<p≤∞, provided that the initial data is sufficiently close to the motionless state. In addition, we clarify the diffusion wave phenomena caused by the sound wave and the elastic shear wave.

Mathematical modeling and qualitative analysis of viscoelastic conductive fluids

We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier–Stokes–Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

On the Rayleigh–Taylor instability in compressible viscoelastic fluids under [formula omitted]-norm

In Wang and Zhao (2018), the authors investigated the Rayleigh–Taylor (abbr. RT) instability in the compressible viscoelastic fluid driven by the gravity in a bounded domain Ω based on Oldroyd-B model. In particular, the authors proved that the steady-state is nonlinearly unstable in the sense of H3(Ω)-norm to the viscoelastic RT (abbr. VRT) problem. In this paper, by switching our analysis to Lagrangian coordinates, we can show the nonlinear instability result in the sense of L1(Ω)-norm based on a bootstrap instability method. By an inverse transformation of Lagrangian coordinates, we can further get the nonlinear instability result for the VRT problem in the sense of L1(Ω)-norm in Eulerian coordinates, and thus improve the instability result in Wang and Zhao (2018).

From quasi-incompressible to semi-compressible fluids

&lt;p style='text-indent:20px;'&gt;A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit. After surveying and modifying models for the quasi-incompressible fluids, we eventually devise some fully convective models complying with energy conservation and capturing phenomena as pressure-wave propagation with wave-length (and possibly also pressure) dependent velocity.&lt;/p&gt;

Dispersive effect and global well-posedness of the compressible viscoelastic fluids

This paper is concerned with the global well-posedness issue of the compressible viscoelastic fluids in the whole space RN,N≥2. The proof relies on the dispersive estimates to the linearized hyperbolic system combined with energy estimates to the hyperbolic-parabolic system. By exploiting the intrinsic structure of the system under some physical restrictions, we find that the compressible viscoelastic fluids admits a unique global solution with a class of large initial data in the sense of critical L2 framework.

A note on the spheroidal modes vibration of an elastic sphere in linear viscoelastic fluid

Vibration characteristics of elastic nanostructures embedded in fluid medium have been used for biological and mechanical sensing, and also to investigate materials mechanical properties. An analytical approach based on the exact theory has been developed in this paper, to establish a new accurate and simple generalized frequency equation to predict spheroidal vibration of an elastic nanosphere, in a compressible viscoelastic fluid using linear Maxwell fluid model. To demonstrate the accuracy of the present approach, a comparison is made with the published theoretical results in the literature in some particular cases, which shows a very good agreement. Thus, the obtained frequency equation can be very useful to interpret the experimental measurements of vibrational dynamics of nanospheres and can serve as benchmark solution in design of liquid sensors.

On a class of compressible viscoelastic rate-type fluids with stress-diffusion

We develop a mathematical theory for a class of compressible viscoelastic rate-type fluids with stress diffusion. Our approach is based on the concepts used in the nowadays standard theory of compressible Newtonian fluids as renormalization, effective viscous flux identity, compensated compactness. The presence of the extra stress, however, requires substantial modification of these techniques, in particular, a new version of the effective viscous flux identity is derived. With help of these tools, we show the existence of global-in-time weak solutions for any finite energy initial data.

A constitutive model for bentonite–water mixture and the effect of wall slip boundary conditions on its mechanical response

This paper presents the development of a constitutive model for describing the nonlinear viscoelastic behavior of deep-seabed sediments. To quantify the nonlinear response, shear rheological tests are carried out on deep-sea sediments substitute at different shear rates. These substitute samples are prepared by mixing bentonite with water based on the in-situ vane shear strength of deep-sea sediments. A compressible viscoelastic fluid model is formulated based on the thermodynamic framework developed by Rajagopal and Srinivasa (2000) to describe the experimental response of bentonite–water mixture. Experimental responses indicate that the effects of slip are significant in the shear rheometry of bentonite–water mixture. Hence a slip model is proposed, which relates the shear stress to slip velocity at the wall and the imposed shear rate values. The slip boundary conditions coupled with the viscoelastic model is validated using experimental data and is observed to be in good agreement. Further, the influence of shear rate on interfacial slip has been numerically analyzed and slip effects are found to be significant in defining rheological behavior with increasing shear rates.

On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework

The generalized bracket framework is used to derive a family of compressible viscoelastic models. The framework accounts for both reversible and non-reversible dynamics and ensures that the derived models are consistent with the laws of thermodynamics. The most general compressible forms of the UCM and Oldroyd B models are derived. For these models the elastic strain energy is taken to be that for a Hookean material. Nonlinear elastic strain energy functionals are also considered and used to derive new viscoelastic models. The viscometric behaviour of these models is investigated and model predictions are compared with experimental data for Boger fluids and mLLDPE.

Global Existence of Solutions of the Compressible Viscoelastic Fluid Around a Parallel Flow

We consider a system of equations for a motion of a compressible viscoelastic fluid in an infinite layer. When the external force has a suitable form, the system has a solution of parallel flow type. It is shown that the solution of the system exists globally in time if the initial data is sufficiently close to the one of the parallel flow, provided that the Reynolds number and the Mach number are sufficiently small, and that the speed of the propagation of shear wave is sufficiently large.

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.

The 3D compressible viscoelastic fluid in a bounded domain

The 3D compressible viscoelastic fluid in a bounded domain

Global existence for a semi-linear Volterra parabolic equation and neutral system with infinite delay

This paper studies the global and local existence of classical solutions for a semilinear Volterra integro-differential equation of parabolic type: (u+k*u)′=A(u+k*u)+f(u)+g, where A is a (not necessarily densely defined) sectorial operator with its spectrum contained in the left half plane. We transform the Volterra equation into a neutral system with infinite delay assuming the history φ of the system is known. The inverse function theorem is then employed to prove the global existence of classical solution to the system for appropriate “small” data (g, φ) if 0 belongs to the resolvent set of A. An example of the linear part being non-densely defined elliptic operators is shown to illustrate the existence theorems, and an application of our results to compressible viscoelastic fluids with hereditary viscosity is also addressed.