Navier's Condition Research Articles

Existence of infinitely many spike solutions for a critical Hénon type biharmonic equation

Following the idea of Wei and Yan [32], with new ingredients to take care of the critical dimension n=5, we construct infinitely many solutions of the biharmonic equation Δ2v=|x|αvn+4n−4 in the unit ball of Rn (n⩾5, α>0) with the Navier conditions v=Δv=0 on the boundary.

Existence and Multiplicity for Quasi-Critical Fourth Order Quasilinear Problems with Generalized Vanishing Potentials

We study the following fourth order quasilinear elliptic equation under Navier boundary conditions in the whole space ℝN $$\left\{ \begin{array}{l} - \Delta {\left( { - \Delta u} \right)^{\frac{1}{{p - 1}}}} + \rho \left( x \right){u^{\frac{1}{{p - 1}}}} = \sigma \left( x \right)f\left( u \right)in{^N} \\ u,\Delta u \to 0\quad as\left| x \right| \to + \infty \\ \end{array} \right.$$ with weights ρ and σ which are allowed to be vanishing at infinity and in presence of quasi-critical power nonlinearities f}. Existence and multiplicity results are proved by means of Mountain Pass Theorem and its symmetric version thanks to some compact imbeddings in σ-weighted Lebesgue spaces. These results apply to an equivalent nonlinear elliptic system of Lane-Emden type and in particular to a biharmonic equation under Navier conditions in ℝN.

Some results on weighted subquadratic Lane–Emden Elliptic Systems in unbounded domains

We study a nonlinear elliptic system of Lane–Emden type in \mathbb R^N, N ≥ 3 , which is equivalent to a fourth order quasilinear elliptic equation involving a suitable ‘‘sublinear’’ term. Thanks to some compact imbeddings in weighted Sobolev spaces, existence and multiplicity results are proved by means of a generalized Weierstrass Theorem and a variant of the Symmetric Mountain Pass Theorem. These results apply in particular to a biharmonic equation under Navier conditions in \mathbb R^N .

Infinitely Many Spiky Solutions for a Hénon Type Biharmonic Equation with Critical Growth

Abstract Following the constructive method of Wei and Yan [29], with new ingredients to take care of high-space dimensions, we prove the existence of infinitely many solutions of the nonlinear biharmonic equation in the unit ball of ℝn (n ≥ 6, α > 0) with the Navier conditions v = Δv = 0 on the boundary.

Motion of a spherical particle in a viscous fluid along a slip wall

Summary The motion of a solid spherical particle in the flow of a viscous fluid is calculated in the framework of Stokes equations, when the Navier condition of slippage applies on the wall and the no-slip condition applies on the particle. The problem for the flow around a sphere translating and rotating in a pure shear flow is solved, from the linearity of Stokes equations, as the sum of elementary problems: a sphere held fixed in a shear flow close to a wall, a sphere translating and rotating along a wall in a fluid at rest. The creeping flow is calculated analytically using the bispherical coordinates technique that provides the fluid velocity and pressure as series. The unknown coupled infinite series of coefficients are reduced by combinations from seven to four. This simplifies the formulation compared with earlier works on the translation and rotation problems. The infinite linear system for the coefficients is then solved using an extension of Thomas algorithm that allows to calculate O(10 5 ) terms in the series. Such a high number is necessary to obtain a good accuracy at small sphere to wall gaps down to 10 −5 sphere radius, in particular for a large slip on the wall. Accurate results for the force and torque on the particle are then obtained for each of the three elementary problems. On this basis, results are provided for the diffusion coefficient of a dilute suspension of freely rotating spheres parallel to a slipping wall and for the translational and rotational velocities of a freely moving sphere in a pure shear flow along such a wall. Simplified formulae are provided for application to various physical problems, like for instance the interpretation of velocity measurements at micro-scales by tracking of particles close to a hydrophobic wall.

Boundary streaming with Navier boundary condition

In microfluidic applications involving high-frequency acoustic waves over a solid boundary, the Stokes boundary-layer thickness δ is so small that some non-negligible slip may occur at the fluid-solid interface. This paper assesses the impact of this slip by revisiting the classical problem of steady acoustic streaming over a flat boundary, replacing the no-slip boundary condition with the Navier condition u|_{y=0}=L_{s}∂_{y}u|_{y=0}, where u is the velocity tangent to the boundary y=0, and the parameter L_{s} is the slip length. A general expression is obtained for the streaming velocity across the boundary layer as a function of the dimensionless parameter L_{s}/δ. The limit outside the boundary layer provides an effective slip velocity satisfied by the interior mean flow. Particularizing to traveling and standing waves shows that the boundary slip respectively increases and decreases the streaming velocity.

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H1 bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.

Effective boundary condition for a quasi‐Newtonian viscous fluid at a slightly rough boundary starting from a Navier condition

We consider the quasi‐Newtonian flow in a domain with a periodic rough bottom Γϵ of period of order the small parameter ϵ and amplitude δϵ, such that δϵ << ϵ. The flow is described by the 3D incompressible non‐Newtonian Navier‐Stokes system where the viscosity is given by the non linear power law which is widely used for dilatant fluids (shear thickening). Assuming that the fluid satisfies the Navier slip condition on Γϵ and letting ϵ → 0, we obtain three different macroscopic models depending on the magnitude of δϵ with respect to ϵ2p‐1/p, with p > 2. In the case δϵ>> ϵ2p‐1/p the effective boundary condition in the limit ϵ = 0 is the no‐slip condition, while if δϵ<< ϵ2p‐1/p there is no roughness‐induced effect. In the critical case when δϵ ∼ ϵ2p‐1/p we provide a more accurate effective boundary condition of Navier type. Finally, we also obtain corrector result for the pressure and velocity in every cases.

On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).

Liquid flow inside a cylindrical capillary with walls covered with a porous layer (Gel)

Viscous incompressible liquid flow in a long cylindrical capillary, the internal surface of which is covered with a permeable porous layer, is studied within the frameworks of three mathematical models. In the first model, the liquid flow in the porous layer is described by the Brinkman equation; according to the second one, the presence of the porous layer is taken into account using the Navier slip boundary conditions; and, in the third model, the Navier condition is imposed on the porous layer-liquid interface, with the flow inside the porous layer being excluded. The theoretical predictions are compared with the experimental data that one of us has obtained for liquid flow rates in porous capillaries. The validity and appropriateness of the appli� cation of the proposed models are discussed.

Asymptotic Behavior of the Navier--Stokes System in a Thin Domain with Navier Condition on a Slightly Rough Boundary

We study the asymptotic behavior of the solutions of the Navier--Stokes system in a thin domain $\Omega_\varepsilon$ of thickness $\varepsilon$ satisfying the Navier boundary condition on a periodic rough set $\Gamma_\varepsilon\subset \partial\Omega_\varepsilon$ of period $r_\varepsilon$ and amplitude $\delta_\varepsilon$, with $\delta_\varepsilon \ll r_\varepsilon \ll \varepsilon$. We prove that the limit behavior as $\varepsilon$ goes to zero depends on the limit $\lambda$ of $\delta_\varepsilon \varepsilon^{1\over 2}/r_\varepsilon^{3\over 2}$. Namely, if $\lambda=+\infty$, the roughness is so strong that the fluid behaves as if we had imposed the adherence condition on $\Gamma_\varepsilon$. If $\lambda=0$, the roughness is too weak and the fluid behaves as if $\Gamma_\varepsilon$ were a plane. Finally, if $\lambda\in(0,+\infty)$, the roughness is strong enough to make a new friction term appear in the limit.

On the Navier boundary condition for viscous fluids in rough domains

In this paper we review some recent results concerning the study of the asymptotic behavior of viscous fluids in rough domains assuming Navier boundary conditions on the rough boundary. Our main interest is to study the relation between both the adherence and the Navier boundary conditions in the case of a boundary with weak rugosities. We show that the roughness acts on the fluid as a friction term. In particular, if the roughness is sufficiently strong, Navier condition implies adherence condition. This generalizes previous results of other authors.

On Leray’s problem under Navier boundary conditions

We consider the stationary motions of a viscous incompressible fluid, of kinematical viscosity ν, in domains with cylindrical outlets to infinity of section &Sigma, and axis e 1 (distorted cylinders). We require that the fluid satisfies the classical boundary Navier condition u·n=0, [&gamma u+T (u,p)·n] &tau =0 (&gamma&ge 0) and tends at infinity to the laminar flow u N in the cylinder with section &Sigma having an assigned flux through &Sigma (Leray problem). We prove that there is a computable positive constant c 0 depending only on &Sigma, such that if ||u N || L &infin (&Sigma) 0 &nu, then the problem has at least a weak solution. Moreover, for &gamma=0 we can alternatively require that u tends at infinity to an assigned constant vector &mu e 1 , &mu&ne 0. In contrast with the most part of the known existence theorems for the steady--state Navier--Stokes problem we show that a solution (u,p) of such a problem exists for every &mu and &nu, is unique for large &nu and converges to &mu e 1 at infinity according to u(x)=&mu e 1 +O(e -&zeta |x 1 | ) for some positive constant &zeta .

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.

The Navier Wall Law at a Boundary with Random Roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size $${\varepsilon \ll 1}$$ . In the parent paper [8], we derived a homogenized boundary condition of Navier type as $${\varepsilon \rightarrow 0}$$ . We show here that for a large class of boundaries, this Navier condition provides a $${O(\varepsilon^{3/2} |\ln \varepsilon|^{1/2})}$$ approximation in L 2, instead of $${O(\varepsilon^{3/2})}$$ for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

Positivity, change of sign and buckling eigenvalues in a one-dimensional fourth order model problem

We study a one dimensional Dirichlet problem of fourth order and a corresponding "buckling eigenvalue problem" under Dirichlet boundary conditions. These problems may serve as model problems for "Orr-Sommerfeld" like boundary and eigenvalue problems. It turns out that eigenvalue curves in appropriate parameter domains look completely different than for the same equation under so called Navier boundary conditions. Further emphasis is laid on positivity properties, and also here, fundamental differences with Navier conditions arise: It may e.g. happen that one has infinitely many linearly independent positive eigenfunctions. Connections and analogies with the clamped plate boundary value problem on families of deformed domains are discussed.

Stokes flow due to the sliding of a smooth plate over a slotted plate

The Stokes flow due to the sliding of a smooth plate over a slotted plate and a bottom fixed plate is solved by eigenfunction expansions and matching. The streamlines show recirculating bubbles near the slots. The drag depends on the void ratio and the relative positions of the three plates. It is found that the Navier condition is in general not satisfied for a slotted plate.

Numerical analysis of axisymmetric oscillations of an orthotropic cylindrical shell

The frequency spectrum of shells is analyzed by numerical methods using three-dimensional and applied theories. The axisymmetric oscillations of an orthotropic cylindrical shell with Navier conditions on the edges are investigated. The effect of boundary conditions on the frequency spectrum is also studied numerically.