Navier's Condition Research Articles

Perturbed eigenvalues of polyharmonic operators in domains with small holes

We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy–Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary.

Smooth Controllability of the Navier–Stokes Equation with Navier Conditions: Application to Lagrangian Controllability

We deal with the 3D Navier–Stokes equation in a smooth simply connected bounded domain, with controls on a non-empty open part of the boundary and a Navier slip-with-friction boundary condition on the remaining, uncontrolled, part of the boundary. We extend the small-time global exact null controllability result in Coron et al. (J Eur Math Soc 22:1625–1673, 2020) from Leray weak solutions to the case of smooth solutions. Our strategy relies on a refinement of the method of well-prepared dissipation of the viscous boundary layers which appear near the uncontrolled part of the boundary, which allows to handle the multi-scale features in a finer topology. As a byproduct of our analysis we also obtain a small-time global approximate Lagrangian controllability result, extending to the case of the Navier–Stokes equations the recent results (Glass and Horsin in J Math Pures Appl (9) 93:61–90, 2010; Glass and Horsin in SIAM J Control Optim 50: 2726–2742, 2012; Horsin and Kavian in ESAIM Control Optim Calc Var 23:1179–1200, 2017) in the case of the Euler equations and the result (Glass and Horsin in ESAIM Control Optim Calc Var 22:1040–1053, 2016) in the case of the steady Stokes equations.

Thermally assisted mobility of nanodroplets on surfaces with weak defects

HypothesisThermal activation plays an essential role in contact line dynamics on nanorough surfaces. However, the relation between the aforementioned concept and the sliding motion of nanodroplets remains unclear. As a result, thermally assisted motion of nanodroplets on nanorough surfaces is investigated in this work. ExperimentsSteady slide and random motion of nanodroplets on surfaces with weak defects are investigated by Many-body Dissipative Particle Dynamics. The surface roughness is characterized by the slip length acquired from the velocity profile associated with the flowing film. FindingsThe slip length is found to decline with increasing the defect density. The linear relationship between the sliding velocity and driving force gives the mobility and reveals the absence of contact line pinning. On the basis of the Navier condition, a simple relation is derived and states that the mobility is proportional to the slip length and the reciprocal of the product of viscosity and contact area. Our simulation results agree excellently with the theoretical prediction. In the absence of external forces, a two-dimensional Brownian motion of nanodroplets is observed and its mean square displacement decreases with increasing the defect density. The diffusivity is proportional to the mobility, consistent with the Einstein relation. This consequence suggests that thermal fluctuations are able to overcome contact line pinning caused by weak defects.

A consistent computational approach for general fluid-poroelasticity-structure-contact interaction problems

The focus of this contribution is the numerical treatment of interface-coupled problems concerned with the interaction of impermeable or permeable elastic bodies1 and the surrounding fluid including the contact interaction of deformable bodies. The presented formulation is based on the approach for fluid-structural-contact interaction in [1], and consistently takes into account fluid-filled poroelastic bodies including the conditions of solid-poroelastic contact, poroelastic-poroelastic contact, and viscous flow poroelasticity interaction. The interface conditions in normal and tangential orientation to formulate this type of coupled problem, based on the fundamental balances of mass and linear momentum, the no-slip condition, the Beavers Joseph condition, and the conditions for frictionless contact on the respective interfaces, are discussed. A continuous transition of the different types of tangential conditions is enabled by application of the general Navier condition with varying slip length. The fluid stress in the zone of closed contact, which is essential for the lift-off behavior of contacting bodies, is obtained by an extension approach augmented by the porous fluid state to ensure continuity and physical accuracy. To account for topological changes of the fluid domain, the numerical approach utilizes non-interface fitted computational meshes for the fluid domain enabled by the Cut Finite Element Method. All interface conditions are incorporated in a weak sense by Nitsche-based approaches. Different numerical examples analyze the proper contacting and lift-off behavior for contact between the different pairs of impermeable and permeable bodies and demonstrate the robustness for more challenging configurations, which include topological changes of the fluid domain, large contacting areas, and 3D configurations.

Self-propelled motion of a rigid body inside a density dependent incompressible fluid

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole ℝ3. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.

A consistent and versatile computational approach for general fluid‐structure‐contact interaction problems

AbstractWe present a consistent approach that allows to solve challenging general nonlinear fluid‐structure‐contact interaction (FSCI) problems. The underlying formulation includes both “no‐slip” fluid‐structure interaction as well as frictionless contact between multiple elastic bodies. The respective interface conditions in normal and tangential orientation and especially the role of the fluid stress within the region of closed contact are discussed for the general problem of FSCI. A continuous transition of tangential constraints from no‐slip to frictionless contact is enabled by using the general Navier condition with varying slip length. Moreover, the fluid stress in the contact zone is obtained by an extension approach as it plays a crucial role for the lift‐off behavior of contacting bodies. With the given continuity of the formulation, continuity of the discrete system of equations for any variation of the coupled system state (which is essential for the convergence of Newton's method) is reached naturally. As topological changes of the fluid domain are an inherent challenge in FSCI configurations, a noninterface fitted cut finite element method (CutFEM) is applied to discretize the fluid domain. All interface conditions, that is the “no‐slip” FSI, the general Navier condition, and frictionless contact are incorporated using Nitsche based methods, thus retaining the consistency of the model. To account for the strong interaction between the fluid and solid discretization, the overall coupled discrete system is solved monolithically. Numerical examples of varying complexity are presented to corroborate the developments. In a first example, the fundamental properties of the presented formulation such as the contacting and lift‐off behavior, the mass conservation, and the influence of the slip length for the general Navier interface condition are analyzed. Beyond that, two more general examples demonstrate challenging aspects such as topological changes of the fluid domain, large contacting areas, and underline the general applicability of the presented method.

Active Damping of Nonstationary Vibrations of a Three-Layer Electro-Viscoelastic Composite Plate

A method for constructing nonstationary vibration equations for electro-viscoelastic composite plates with their active damping is described. The problem is solved in an integral Fourier transform frequency space. To construct a multilayer plate model, a discrete structural approach was used, i.e., the plate is divided through the thickness into computational layers, within which the displacements and electric potentials are approximated by means of quadratic Lagrange polynomials. At the edges of the plate, the Navier conditions are met. The semi-analytical nonstationary vibration equations are derived by means of the variational convolutional Lagrange equation. To represent them in a frequency space, a direct integral Fourier transform is used, which makes it possible to use experimentally determined frequency-dependent complex characteristics directly in calculations of nonstationary vibrations of plates without additional transformations of integral physical relations. Solutions of the problem of active damping of nonstationary vibrations for the frequency images of displacements and electric potentials have been found, their convergence is analyzed. The procedure has been evaluated by analyzing the active damping of nonstationary vibrations of a three-layer edge-hinged square plate under the action of a blast wave impulse. The middle layer is made of a composite material, and the outer electro-viscoelastic layers act as a sensor and an actuator. The active damping of vibrations is provided by connecting these layers of the plate into a negative feedback circuit with the proportional-differential control law. The viscoelastic and electro-viscoelastic characteristics of the materials are taken into account by means of complex elastic, dielectric and piezoelectric moduli. The results of analyzing the active damping of nonstationary vibrations of a plate are presented. The values of the proportional and differential components of the controller as characteristics of this damping were chosen taking into account the permissible electric field strength of the actuator. The research results for the case of passive damping of vibrations are also given for comparison. The dependences of the decrement of plate vibrations on the differential component of the controller and of the maximum plate deflection on the reinforcement angle of the middle layer material have been established. The optimum reinforcement angle according to the minimum plate deflection criterion has been determined, which provides an increase in the efficiency of active damping of nonstationary vibrations of the plate owing to an increase in the differential component of the controller.

On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy

In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body’s boundary. The whole “viscous incompressible fluid + rigid body” system is assumed to occupy the full plane ℝ 2 . We prove the existence of global-in-time weak solutions with constant non-zero circulation at infinity.

On a Critical Fourth Order PDE with Navier Boundary Condition

We consider a fourth order nonlinear PDE involving the critical Sobolev exponent on a bounded domain of ℝn, n ≥ 5 with Navier condition on the boundary. We study the lack of compactness of the problem and we provide an existence theorem through a new index formula.

On the Kinematics of Contact Line Motion

AbstractWe investigate the moving contact line problem for two‐phase incompressible flows by a kinematic approach. The key idea is to derive an evolution equation for the contact angle assuming the transporting velocity field to be given. It turns out that the resulting equation expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. In this paper we consider the Navier slip boundary condition, which is frequently used for the modeling of moving contact lines. Exploiting furthermore the interfacial transmission condition for the viscous stress, we derive an explicit form of the contact angle evolution for sufficiently regular solutions. In the absence of phase change it only involves the contact line velocity and the slip length from the Navier condition. From this equation we can read off the qualitative behavior of the contact angle evolution for sufficiently regular solutions to these models, which turns out to be unphysical.

Wall slip of complex fluids: Interfacial friction versus slip length

Using a dynamic Surface Force Apparatus, we demonstrate that the notion of slip length used to describe the boundary flow of simple liquids, is not appropriate for viscoelastic liquids. Rather, the appropriate description lies in the original Navier's partial slip boundary condition, formulated in terms of an interfacial friction coefficient. We establish an exact analytical expression to extract the interfacial friction coefficient from oscillatory drainage forces between a sphere and a plane, suitable for dynamic SFA or Atomic Force Microscopy non-contact measurements. We use this model to investigate the boundary friction of viscoelastic polymer solutions over 5 decades of film thicknesses and one decade in frequency. The proper use of the original Navier's condition describes accurately the complex hydrodynamic force up to scales of tens of micrometers, with a simple "Newtonian-like" friction coefficient, not frequency dependent, and reflecting closely the dynamics of an interfacial depletion layer at the solution/solid interface.

Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles

Shear-driven flows over superhydrophobic surfaces formed of closely spaced circular bubbles are characterized by giant longitudinal slip lengths, viz., large compared with the periodicity (Schnitzer, Phys Rev Fluids 1(5):052101, 2016). This hints towards a strong superhydrophobic effect in the concomitant scenario of pressure-driven flow between two such surfaces, particularly for non-wide channels where bubble-to-bubble pitch and bubble radius are commensurate with channel width. We show here that such pressure-driven flows can be analyzed asymptotically and in closed form based on the smallness of the gaps separating the bubbles relative to the channel width (and bubble radius). We find that the flow adopts an unconventional plug profile away from the inter-bubble gaps, with the uniform velocity being asymptotically larger than the corresponding Poiseuille scale. For a given solid fraction and channel width, the net volumetric flux is maximized when the length of each semi-circular bubble-liquid interface is equal to the channel width. The plug flow identified herein cannot be obtained via a naive implementation of a Navier condition, which is indeed inapplicable for non-wide channels.

Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary

This paper concerns the low Mach number limit of weak solutions to the compressible Navier-Stokes equations for isentropic fluids in a bounded domain with a Navier-slip boundary condition. In [ 2 ], it has been proved that if the velocity is imposed the homogeneous Dirichlet boundary condition, as the Mach number goes to 0, the velocity of the compressible flow converges strongly in \begin{document}$ L^2$\end{document} under the geometrical assumption (H) on the domain. We justify the same strong convergence when the slip length in the Navier condition is the reciprocal of the square root of the Mach number.

An Optimal Control Problem for the Steady Nonhomogeneous Asymmetric Fluids

We study an optimal boundary control problem for the two-dimensional stationary micropolar fluids system with variable density. We control the system by considering boundary controls, for the velocity vector and angular velocity of rotation of particles, on parts of the boundary of the flow domain. On the remaining part of the boundary, we consider mixed boundary conditions for the vector velocity (Dirichlet and Navier conditions) and Dirichlet boundary conditions for the angular velocity. We analyze the existence of a weak solution obtaining the fluid density as a scalar function in terms of the stream function. We prove the existence of an optimal solution and, by using the Lagrange multipliers theorem, we state first-order optimality conditions. We also derive, through a penalty method, some optimality conditions satisfied by the optimal controls.

A Nitsche cut finite element method for the Oseen problem with general Navier boundary conditions

In this work a Nitsche-based imposition of generalized Navier conditions on cut meshes for the Oseen problem is presented. Other methods from literature dealing with the generalized Navier condition impose this condition by means of substituting the tangential Robin condition in a classical Galerkin way. These methods work fine for a large slip length coefficient but lead to conditioning and stability issues when it approaches zero. We introduce a novel method for the weak imposition of the generalized Navier condition which remains well-posed and stable for arbitrary choice of slip length, including zero. The method proposed here builds on the formulation done by Juntunen and Stenberg (2009). They impose a Robin condition for the Poisson problem by means of Nitsche’s method for an arbitrary combination of the Dirichlet and Neumann parts of the condition. The analysis conducted for the proposed method is done in a similar fashion as in Massing et al. (2018), but is done here for a more general type of boundary condition. The analysis proves stability for all flow regimes and all choices of slip lengths. Also an L2-optimal estimate for the velocity error is shown, which was not conducted in the previously mentioned work. A numerical example is carried out for varying slip lengths to verify the robustness and stability of the method with respect to the choice of slip length. Even though proofs and formulations are presented for the more general case of an unfitted grid method, they can easily be reduced to the simpler case of a boundary-fitted grid with the removal of the ghost-penalty stabilization terms.

The method of lower and upper solutions for fourth order equations with the Navier condition

The aim of this paper is to explore the method of lower and upper solutions in order to give some existence results for equations of the form \t\t\ty(4)(x)+(k1+k2)y″(x)+k1k2y(x)=f(x,y(x)),x∈(0,1),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$y^{(4)}(x)+(k_{1}+k_{2}) y''(x)+k_{1}k_{2} y(x)=f\\bigl(x,y(x)\\bigr), \\quad x\\in(0,1), $$\\end{document} with the Navier condition \t\t\ty(0)=y(1)=y″(0)=y″(1)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$y(0) = y(1) = y''(0) = y''(1) = 0 $$\\end{document} under the condition k_{1}<0<k_{2}<pi^{2}. The main tool is the Schauder fixed point theorem.

On Critical Spaces for the Navier–Stokes Equations

The abstract theory of critical spaces developed in [22] and [20] is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the $L_p$-$L_q$ setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $\mathcal{H}^\infty$-calculus with $\mathcal{H}^\infty$-angle 0, and the real and complex interpolation spaces of these operators are identified.

Some New Results on Subquadratic Lane–Emden Elliptic Systems

We deal with a nonlinear elliptic weighted system of Lane–Emden type in \(\mathbb R^N\), \(N \ge 3\), by exploiting its equivalence with a fourth-order quasilinear elliptic equation involving a suitable “sublinear” term. By overcoming the loss of compactness of the problem with some compact imbeddings in weighted \(L^p\)-spaces, we establish existence and multiplicity results by means of a generalized Weierstrass Theorem and a variant of the Symmetric Mountain Pass Theorem stated by R. Kajikiya for subquadratic functionals. These results, which generalize previous ones stated by the same authors, apply in particular to a biharmonic equation under Navier conditions in \(\mathbb R^N\).

The motion of the rigid body in the viscous fluid including collisions. Global solvability result

We shall consider the problem of the motion of a rigid body in an incompressible viscous fluid filling a bounded domain. This problem was studied by several authors. They mostly considered classical non-slip boundary conditions, which gave them very paradoxical result of no collisions of the body with the boundary of the domain. Only recently there are results when the Navier type of boundary is considered.In our paper we shall consider the Navier condition on the boundary of the body and the non-slip condition on the boundary of the domain. This case admits collisions of the body with the boundary of the domain. We shall prove the global existence of weak solution of the problem.

Free vibrations of an anisotropic cylindrical fiberglass shell reinforced by annular ribs and containing fluid flow

This paper presents the results of determining the free vibration frequency of a structurally anisotropic, cylindrical fiberglass shell reinforced by annular ribs and containing flowing fluid. Boundary Navier conditions are imposed on the ends of the shell. Natural vibration frequencies are calculated as dependences of the frequency on the fiberglass winding angle and fluid flow velocity for different values of the wave formation parameters and the parameters characterizing the geometric dimensions of the shell.