Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Functional and generalized separable solutions to unsteady Navier–Stokes equations

The paper studies unsteady Navier–Stokes equations with two space variables. It shows that the non-linear fourth-order equation for the stream function with three independent variables admits functional separable solutions described by a system of three partial differential equations with two independent variables. The system is found to have a number of exact solutions, which generate new classes of exact solutions to the Navier–Stokes equations. All these solutions involve two or more arbitrary functions of a single argument as well as a few free parameters. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary; such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for solving certain model problems and testing numerical and approximate analytical hydrodynamic methods. The paper uses the obtained results to describe some model unsteady flows of viscous incompressible fluids, including flows through a strip with permeable walls, flows through a strip with extrusion at the boundaries, flows onto a shrinking plane, and others. Some blow-up modes, which correspond to singular solutions, are discussed.     

An anisotropic elastic–viscoplastic model for soft clays

Experimental evidences have shown deficiencies of the existing overstress and creep models for viscous behaviour of natural soft clay. The purpose of this paper is to develop a modelling method for viscous behaviour of soft clays without these deficiencies. A new anisotropic elastic–viscoplastic model is extended from overstress theory of Perzyna. A scaling function based on the experimental results of constant strain-rate oedometer tests is adopted, which allows viscoplastic strain-rate occurring whether the stress state is inside or outside of the yielding surface. The inherent and induced anisotropy is modelled using the formulations of yield surface with kinematic hardening and rotation (S-CLAY1). The parameter determination is straightforward and no additional experimental test is needed, compared to the Modified Cam Clay model. Parameters determined from two types of tests (i.e., the constant strain-rate oedometer test and the 24 h standard oedometer test) are examined. Experimental verifications are carried out using the constant strain-rate and creep tests on St. Herblain clay. All comparisons between predicted and measured results demonstrate that the proposed model can successfully reproduce the anisotropic and viscous behaviours of natural soft clays under different loading conditions.     

Stokes–Darcy coupling for periodically curved interfaces

We investigate the boundary condition between a free fluid and a porous medium, where the interface between the two is given as a periodically curved structure. Using a coordinate transformation, we can employ methods of periodic homogenisation to derive effective boundary conditions for the transformed system. In the porous medium, the fluid velocity is given by Darcy's law with a non-constant permeability matrix. In tangential direction as well as for the pressure, a jump appears. Its magnitudes can be calculated with the help of a generalised boundary layer function. The results can be interpreted as a generalised law of Beavers and Joseph for curved interfaces.     

Vortices for a Rotating Toroidal Bose–Einstein Condensate

We construct local minimizers of the Gross–Pitaevskii energy, introduced to model Bose–Einstein condensates (BEC) in the Thomas–Fermi regime which are subject to a uniform rotation. Our sample domain is taken to be a solid torus of revolution in with starshaped cross-section. We show that for angular speeds ω_ε = O(|ln ε|) there exist local minimizers of the energy which exhibit vortices, for small enough values of the parameter ε. These vortices concentrate at one or several planar arcs (represented by integer multiplicity rectifiable currents) which minimize a line energy, obtained as a Γ-limit of the Gross–Pitaevskii functional. The location of these limiting vortex lines can be described under certain geometrical hypotheses on the cross-sections of the torus.     

Stokes＇ first problem for a viscoelastic fluid with the generalized Oldroyd-B model

The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions. The project supported by the National Natural Science Foundation of China (10272067), the Doctoral Program Foundation of the Education Ministry of China (20030422046), the Natural Science Foundation of Shandong Province, China (Y2006A14) and the Research Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.     

Two-dimensional anisotropic vortex–bright soliton and its dynamics in dipolar Bose–Einstein condensates in optical lattice

We study construction and dynamics of two-dimensional (2D) anisotropic vortex–bright (VB) soliton in spinor dipolar Bose–Einstein condensates confined in a 2D optical lattice (OL), with two localized components linearly mixed by the spin–orbit coupling and long-range dipole–dipole interaction (DDI). It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of norm N. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked VB soliton is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance and OL, and they can realize the transition from the bright component to the vortex component. Our work may provide a convenient way to prepare and manipulate anisotropic VB soliton in high-dimensional space.

     

A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

Navier–Stokes solutions for steady parallel-sided pendent rivulets

We investigate exact solutions of the Navier–Stokes equations for steady rectilinear pendent rivulets running under inclined surfaces. First we show how to find exact solutions for sessile or hanging rivulets for any profile of the substrate (transversally to the direction of flow) and with no restrictions on the contact angles. The free surface is a cylindrical meniscus whose shape is determined by the static equilibrium between gravity and surface tension, by the shape of the solid surface, and by the contact angles on both contact lines. Given this, the velocity field can be obtained by integrating numerically a Poisson equation. We then perform a systematic study of rivulets hanging below an inclined plane, computing some of their global properties, and discussing their stability.     

Weak–Strong Uniqueness Property for the Full Navier–Stokes–Fourier System

The Navier–Stokes–Fourier system describing the motion of a compressible, viscous and heat conducting fluid is known to possess global-in-time weak solutions for any initial data of finite energy. We show that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists. In particular, strong solutions are unique within the class of weak solutions.     

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.     

High-order numerical methods of fractional-order Stokes’ first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes’ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes.     

On Kato’s Method for Navier–Stokes Equations

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR³{\mathbb{R}}^{3} and irregular domains in \mathbbRⁿ{\mathbb{R}}^{n}.     

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

Under investigation is a completely generalized Hirota–Satsuma–Ito equation in (2 + 1)-dimensional. Multiple lump solutions are obtained based on three test functions, including 1-, 2- and 3-order lump solutions. Subsequently, the interaction between lump wave and solitary waves, and the interaction solution between lump wave and periodic wave are studied by using the bilinear form. Final, the stability and phase velocity are investigated. In order to analyze the dynamic behavior of these solutions, some 3D plots and contour plots are given by Mathematica.

     

Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System

The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.     

Cahn–Hilliard/Navier–Stokes Model for the Simulation of Three-Phase Flows

In this article, we describe some aspects of the diffuse interface modelling of incompressible flows, composed of three immiscible components, without phase change. In the diffuse interface methods, system evolution is driven by the minimisation of a free energy. The originality of our approach, derived from the Cahn–Hilliard model, comes from the particular form of energy we proposed in Boyer and Lapuerta (M2AN Math Model Numer Anal, 40:653–987,2006), which, among other interesting properties, ensures consistency with the two-phase model. The modelling of three-phase flows is further completed by coupling the Cahn–Hilliard system and the Navier–Stokes equations where surface tensions are taken into account through volume capillary forces. These equations are discretized in time and space paying attention to the fact that most of the main properties of the original model (volume conservation and energy estimate) have to be maintained at the discrete level. An adaptive refinement method is finally used to obtain an accurate resolution of very thin moving internal layers, while limiting the total number of cells in the grids all along the simulation. Different numerical results are given, from the validation case of the lens spreading between two phases (contact angles and pressure jumps), to the study of mass transfer through a liquid/liquid interface crossed by a single rising gas bubble. The numerical applications are performed with large ratio between densities and viscosities and three different surface tensions.