Cultivating oleaginous microalgae in specific culturing devices such as raceways is seen as a future way to produce biofuel. The complexity of this process coupling non linear biological activity to hydrodynamics makes the optimization problem very delicate. The large amount of parameters to be taken into account paves the way for a useful mathematical modeling. Due to the heterogeneity of raceways along the depth dimension regarding temperature, light intensity or nutrients availability, we adopt...

Let $X$ be a compact subset of a separable Hilbert space $H$ with finite fractal dimension $d_F\left(X\right)$, and $P_0$ an orthogonal projection in $H$ of rank greater than or equal to $2d_F\left(X\right)+1$. For every $\delta >0$, there exists an orthogonal projection $P$ in $H$ of the same rank as $P_0$, which is injective when restricted to $X$ and such that $\parallel P-P_0\parallel <\delta$. This result follows from Mañé’s paper. Thus the inverse ${\left(P{|}_X\right)}^{-1}$ of the restricted mapping ${P|}_{X}X\to PX$ is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s...

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for...

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $limsu{p}_{R\to 0⁺}1/R{\int }_{Q_R(x₀,t₀)}|curlu\times u/|u\left|\right|²dxdt\le {\epsilon }_{*}$ for a sufficiently small ${\epsilon }_{*}>0$.

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.

We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH...

We generalize a classical result of T. Kato on the existence of global solutions to the Navier-Stokes system in C([0,∞);L3(R3)). More precisely, we show that if the initial data are sufficiently oscillating, in a suitable Besov space, then Kato's solution exists globally. As a corollary to this result, we obtain a theory of existence of self-similar solutions for the Navier-Stokes equations.

We deal with a suitable weak solution $(𝐯,p)$ to the Navier-Stokes equations in a domain $\Omega \subset {ℝ}^3$. We refine the criterion for the local regularity of this solution at the point $(𝐟x_0,t_0)$, which uses the $L^3$-norm of $𝐯$ and the $L^{3/2}$-norm of $p$ in a shrinking backward parabolic neighbourhood of $({𝐱}_0,t_0)$. The refinement consists in the fact that only the values of $𝐯$, respectively $p$, in the exterior of a space-time paraboloid with vertex at $({𝐱}_0,t_0)$, respectively in a ”small” subset of this exterior, are considered. The consequence is that...

We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker–Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier–Stokes–Fokker–Planck system for dilute polymeric fluids. In this context the Fokker–Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for...

This paper proves a logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain with the Navier-type boundary condition.

Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking...

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the...