The dynamics of flexible fibers and vesicles in unbounded planar Poiseuille flow at low Reynolds number is shown to exhibit similar basic features, when their equilibrium (moderate) aspect ratio is the same and vesicle viscosity contrast is relatively high. Tumbling, lateral migration, accumulation and shape evolution of these two types of flexible objects are analyzed numerically. The linear dependence of the accumulation position on relative bending rigidity, and other universal scalings are derived from the local shear flow approximation.

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domainwhere v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (v ≡ [1, 0, 0],ρ ≡ 1). We also show that this solution is unique in a class of small perturbations of (v,ρ). The term u · ∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (v n , ρ n ) that is bounded in W 2 p (Ω) × W 1 p (Ω) and satisfies the Cauchy condition in a larger space L ∞ (0, L; L 2 (Ω 0 )) what enables us to deduce that the weak limit of a subsequence of (v n , ρ n ) is in fact a strong solution to our problem.

We investigate a system describing the flow of a compressible two-component mixture. The system is composed of the compressible Navier-Stokes equations coupled with non-symmetric reactiondiffusion equations describing the evolution of fractional masses. We show the local existence and, under certain smallness assumptions, also the global existence of unique strong solutions in Lp − Lq framework. Our approach is based on so called entropic variables which enable to rewrite the system in a symmetric form. Then, applying Lagrangian coordinates, we show the local existence of solutions applying the Lp-Lq maximal regularity estimate. Next, applying exponential decay estimate we show that the solution exists globally in time provided the initial data is sufficiently close to some constants. The nonlinear estimates impose restrictions 2 < p < ∞, 3 < q < ∞. However, for the purpose of generality we show the linear estimates for wider range of p and q. MSC Classification: 76N10, 35Q30Under the assumption (1.6), global in time strong (unique) solutions around the constant equilibrium for the Cauchy problem was proven by Giovangigli in [19]. He introduced the entropic and normal variables to symmetrize the system (1.1) and applied the Kawashima and Shizuta theory [24,25] for symmetric hyperbolic-parabolic systems of conservation laws. For the local in time existence result to the species mass balances equations in the isobaric, isothermal case we refer to [3], see also [21]. Later on, Jüngel and Stelzer generalized this result and combined it with the entropy dissipation method to prove the global in time existence of weak solutions [23], still in the case of constant pressure and temperature. The detailed description of the method and its applicability for a range of models we refer to [22]. For the qualitative and quantitative analysis of the ternary gaseous system together with numerical simulations we refer to [7]. One should note that constant pressure assumption in (1.5) not only significantly simplifies the cross-diffusion equations but basically decouples the fluid and the reactiondiffusion parts of the system (1.1). Stationary problems for compressible mixtures were considered in [49] under the assumption of Fick law and later in [20,35,36] with cross diffusion, however for different molar masses. Existence of weak solutions for the mixture of non-newtonian fluids has been shown in [8]. Let us also mention results on multi-phase systems [16,26] and incompressible mixtures [27,9,5]. We would also like to mention the theoretical results for the systems describing the compressible reacting electrolytes [11], where the authors prove the existence of global in time weak solutions to the Nernst-Planck-Poisson model originating from the modelling approach developed by Bothe and Dreyer in the previous paper [4]. The classical mixture models in the sense of [19] were studied in the series of papers [50,51,30,31,32], where the global in time existence of weak solutions was proved without any simplification of (1.7). This was...

We consider a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part. We show the existence of a strong solution provided the data are close to a constant, but nontrivial flow with sufficiently large dissipation in the energy equation.