arXiv subjects
Publications and source records attributed to Toan T. Nguyen.
In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.
In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.
This paper is a first step toward understanding the effect of toroidal geometry on the rigorous stability theory of plasmas. We consider a collisionless plasma inside a torus, modeled by the relativistic Vlasov-Maxwell system. The surface of the torus is perfectly conducting and it reflects the particles specularly. We provide sharp criteria for the stability of equilibria under the assumption that the particle distributions and the electromagnetic fields depend only on the cross-sectional variables of the torus.
In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for the Green's function, which allow one to close a nonlinear iteration scheme.
We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and Métivier treating the lake equations with a fixed topography and by Gérard-Varet and Lacave treating the Euler equations in singular domains.
Budding from the plasma membrane of the host cell is an indispensable step in the life cycle of the Human Immunodeficiency Virus (HIV), which belongs to a large family of enveloped RNA viruses, retroviruses. Unlike regular enveloped viruses, retrovirus budding happens {\em concurrently} with the self-assembly of retrovirus protein subunits (Gags) into spherical virus capsids on the cell membrane. Led by this unique budding and assembly mechanism, we study the free energy profile of retrovirus budding, taking into account of the Gag-Gag attraction energy and the membrane elastic energy. We find that if the Gag-Gag attraction is strong, budding always proceeds to completion. During early stage of budding, the zenith angle of partial budded capsids, $α$, increases with time as $α\propto t^{1/3}$. However, when Gag-Gag attraction is weak, a metastable state of partial budding appears. The zenith angle of these partially spherical capsids is given by $α_0\simeq(τ^2/κσ)^{1/4}$ in a linear approximation, where $κ$ and $σ$ are the bending modulus and the surface tension of the membrane, and $τ$ is a line tension of the capsid proportional to the strength of Gag-Gag attraction. Numerically, we find $α_0<0.3π$ without any approximations. Using experimental parameters, we show that HIV budding and assembly always proceed to completion in normal biological conditions. On the other hand, by changing Gag-Gag interaction strength or membrane rigidity, it is relatively easy to tune it back and forth between complete budding and partial budding. Our model agrees reasonably well with experiments observing partial budding of retroviruses including HIV.
The phase diagram of a water solution of DNA and oppositely charged spherical macroions is studied. DNA winds around spheres to form beads-on-a-string complexes resembling the chromatin 10 nm fiber. At small enough concentration of spheres these "artificial chromatin" complexes are negative, while at large enough concentrations of spheres the charge of DNA is inverted by the adsorbed spheres. Charges of complexes stabilize their solutions. In the plane of concentrations of DNA and spheres the phases with positive and negative complexes are separated by another phase, which contains the condensate of neutral DNA-spheres complexes. Thus when the concentration of spheres grows, DNA-spheres complexes experience condensation and resolubilization (or reentrant condensation). Phenomenological theory of the phase diagram of reentrant condensation and charge inversion is suggested. Parameters of this theory are calculated by microscopic theory. It is shown that an important part of the effect of a monovalent salt on the phase diagram can be described by the nontrivial renormalization of the effective linear charge density of DNA wound around a sphere, due to the Onsager-Manning condensation. We argue that our phenomenological phase diagram or reentrant condensation is generic to a large class of strongly asymmetric electrolytes. Possible implication of these results for the natural chromatin are discussed.
Adsorption of multivalent counterions on the charged surface of a macroion is known to lead to inversion of the macroion charge due to the strong lateral correlations of counterions. We consider a nontrivial role of the excluded volume of counterions on this effect. It is shown analytically that when the bare charge of macroion increases, its net charge including the adsorbed counterions oscillates with the number of their layers. Charge inversion vanishes every time the top layer of counterions is completely full and becomes incompressible. These oscillations of charge inversion are confirmed by Monte-Carlo simulations. Another version of this phenomenon is studied for a metallic electrode screened by multivalent counterions when potential of the electrode is controlled instead of its charge. In this case, oscillations of the compressibility and charge inversion lead to oscillations of capacitance of this electrode with the number of adsorbed layers of multivalent counterions.
Complexation of a long flexible polyelectrolyte (PE) molecule with oppositely charged spherical particles such as colloids, micelles, or globular proteins in a salty water solution is studied. PE binds spheres winding around them, while spheres repel each other and form almost periodic necklace. If the total charge of PE is larger than the total charge of spheres, repulsive correlations of PE turns on a sphere lead to inversion of the net charge of each sphere. In the opposite case, we predict another correlation effect: under-screened by PE spheres bind to PE in such a great number that they invert charge of PE. The inverted charge by absolute value can be larger than the bare charge of PE even when screening by monovalent salt is weak. At larger concentrations of monovalent salt, the inverted charge can reach giant proportions. Our theory is in qualitative agreement with recent experiments on micelles-PE systems.