Symmetries and global solvability of the isothermal gas dynamics equations

We study the Cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics, in which the pressure and the density are related by the $\gamma$-law equation $p(\rho) \sim \rho^\gamma$ with $\gamma =1$. Our results complete those obtained earlier for $\gamma >1$. We prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method. The proof relies on compensated compactness arguments and symmetry group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density. This property enables us to reduce our problem to that with a small initial density. One symmetry group associated with the linear hyperbolic equations describing all entropies of the Euler equations gives rise to a fundamental solution with initial data imposed to the line $\rho=1$. This is in contrast to the common approach (when $\gamma >1$) which prescribes initial data on the vacuum line $\rho =0$. The entropies we construct here are weak entropies, i.e. they vanish when the density vanishes. Another feature of our proof lies in the reduction theorem which makes use of the family of weak entropies to show that a Young measure must reduce to a Dirac mass. This step is based on new convergence results for regularized products of measures and functions of bounded variation.Comment: 29 page

Flows of Linear Polymer Solutions and Other Suspensions of Rod-like Particles: Anisotropic Micropolar-Fluid Theory Approach

We formulate equations governing flows of suspensions of rod-like particles. Such suspensions include linear polymer solutions, FD-virus, and worm-like micelles. To take into account the particles that form and their rotation, we treat the suspension as a Cosserat continuum and apply the theory of micropolar fluids. Anisotropy of suspensions is determined through the inclusion of the microinertia tensor in the rheological constitutive equations. We check that the model is consistent with the basic principles of thermodynamics. In addition to anisotropy, the theory also captures gradient banding instability, coexistence of isotropic and nematic phases, sustained temporal oscillations of macroscopic viscosity, shear thinning and hysteresis. For the flow between two planes, we also establish that the total flow rate depends not only on the pressure gradient, but on the history of its variation as well

Lateral-Concentration Inhomogeneities in Flows of Suspensions of Rod-like Particles: The Approach of the Theory of Anisotropic Micropolar Fluid

To tackle suspensions of particles of any shape, the thermodynamics of a Cosserat continuum are developed by the method suggested by Landau and Khalatnikov for the mathematical description of the super-fluidity of liquid 2He. Such an approach allows us to take into account the rotation of particles and their form. The flows of suspensions of neutrally buoyant rod-like particles are considered in detail. These suspensions include linear polymer solutions, FD-virus and worm-like micelles. The anisotropy of the suspensions is determined through the inclusion of the micro-inertia tensor in the rheological constitutive equations. The theory predicts gradient banding, temporal volatility of apparent viscosity and hysteresis of the flux-pressure curve. The transition from the isotropic phase to the nematic phase is also captured. Our mathematical model predicts the formation of flock-like inhomogeneities of concentration jointly with the hindrance effect

Micropolar Bingham fluids

Non UBCUnreviewedAuthor affiliation: Lavrentyev Institute of HydrodynamicsFacult

Recursive Settling of Particles in Shear Thinning Polymer Solutions: Two Velocity Mathematical Model

Processing of the available experimental data on particles settling in shear-thinning polymer solutions is performed. Conclusions imply that sedimentation should be recursive, since settling also occurs within the sediment. To capture such an effect, a mathematical model of two continua has been developed, which corresponds to experimental data. The model is consistent with basic thermodynamics laws. The rheological component of this model is a correlation formula for gravitational mobility. This closure is justified by comparison with known experimental data available for particles settling in vertical vessels. In addition, the closure is validated by comparison with analytical solutions to the Kynch one-dimensional equation, which governs dynamics of particle concentration. An explanation is given for the Boycott effect and it is proven that sedimentation is enhanced in a 2D inclined vessel. In tilted vessels, the flow is essentially two-dimensional and the one-dimensional Kynch theory is not applicable; vortices play an important role in sedimentation

Electroosmosis law via homogenization of electrolyte flow equations in porous media

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