Navier Boundary Condition Research Articles

On a singular epitaxial thin‐film growth equation involving logarithmic nonlinearity

This paper is concerned with the well‐posedness and asymptotic behavior for a singular epitaxial thin‐film growth equation with logarithmic nonlinearity under the Navier boundary condition. Based on the technique of cut‐off and combining with Hardy–Sobolev inequality, the technique of Faedo–Galerkin, and multiplier, we establish the local solvability. Meantime, by virtue of the family of potential wells, we obtain the threshold between the existence and nonexistence of the global solution (including the critical case) and give the upper bound of lifespan and the estimate of blow‐up rate. Furthermore, the results of blow‐up with arbitrary initial energy and the lifespan are derived.

An improved phase-field algorithm for simulating the impact of a drop on a substrate in the presence of surfactants

This paper is devoted to the numerical study of droplet impact on solid substrates in the presence of surfactants. We formulate the problem in an energetically variational framework and introduce an incompressible Cahn-Hilliard-Navier-Stokes system for the phase-field modeling of two-phase flows. Flory-Huggins potential and generalized Navier boundary condition are used to account for soluble surfactants and moving contact lines. Based on the convex splitting and pressure stabilization technique, we develop unconditionally energy stable schemes for this model. The discrete energy dissipation law for the original energy is rigorously proved for the first-order scheme. The numerical methods are implemented using finite difference method in three-dimensional cylindrical coordinates with axisymmetry. Using the proposed methods for this model, we systematically study the impact dynamics of both clean and contaminated droplets (with surfactants) in a series of numerical experiments. In general, the dissipation in the impact dynamics of a contaminated drop is smaller than that in the clean case, and topological changes are more likely to occur for contaminated drops. Adding surfactants can significantly influence the impact phenomena, leading to the enhancement of adherence effect on hydrophilic surfaces and splashing on hydrophobic surfaces. Some qualitative agreements with experiments are also obtained.

Existence of two non-zero weak solutions for a $p(\cdot)$-biharmonic problem with Navier boundary conditions

In this paper, the existence of non-trivial weak solutions for some problems with Navier boundary conditions driven by the p(\cdot) -biharmonic operator is investigated. The proofs combine variational methods with topological arguments.

Slip boundary condition near the wall-interface contact line in axial stratified two-phase flow

The behavior of the velocity and shear stresses in two-phase laminar stratified flow near a triple point (TP), formed in the flow cross-section by the intersection of the interface with the conduit wall, is reconsidered. Differently from the no-slip boundary condition that showed a possibility of diverging shear stresses upon approaching the TP, we allow a slip of the fluids at the wall that can be prominent in the TP region. The Navier's boundary condition is used, which considers a linear relationship between the slip velocity and the strain rate component perpendicular to the surface. The solution for the velocity field in the TP region is obtained in terms of a power series of the distance from the TP having a discrete set of powers. Similarly to the solution obtained with no-slip B.C., a so-called ‘characteristic equation’ is obtained. Its solution yields a discrete set of non-integer powers that are dependent on the dynamic viscosity ratio and the contact angle. However, contrary to the no-slip solution, with Navier's B.C, each of these powers introduces a tower of terms resulting from the relation between the slip velocity and the velocity derivative at the wall. In addition, the solution also includes a tower of integer powers. The solution yields the velocity and the shear stress fields, as well as the flux of the force per length that acts on an arbitrary surface in the flow field. The latter is obtained by considering an equivalent 2D representation of the stress tensor. These are examined in comparison to the corresponding results obtained when no-slip at the wall is imposed. The relaxation of the wall shear stresses in cases where the no-slip condition predicts diverging wall shear stresses at the TP is demonstrated. It is shown that if the local velocity in the TP region is a limit of a global no-slip solution, the scaled velocity field in the TP region is independent of the details of the global flow field.

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

Identification of a boundary obstacle in a Stokes fluid with Dirichlet–Navier boundary conditions: External measurements

The problem of identifying an obstruction within a fluid duct has several applications, one of which is in medicine, where the presence of stenosis in coronary vessels poses a life-threatening disease. In this paper, we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D duct where a Stokes flow hits the boundary (subject to Dirichlet and Navier-slip boundary conditions), generating an acoustic wave. To be precise, using acoustic wave measurements at certain points on the exterior of the duct, we can identify the location, extent, and height of the obstruction. Thus, our framework offers an external approach to solving this inverse-obstacle problem. Synthetic examples are used to verify the effectiveness of the proposed numerical formulation.

Stream function solutions for some contact line boundary conditions: Navier slip, super slip and the generalized Navier boundary condition

The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter λ , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a C 0 continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a C 1 velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a C 2 velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well.

Energy stable finite element strategy for simulating spreading, sliding and rolling flow dynamics of viscoelastic droplets

Moving mesh finite element method (FEM) for simulating dynamics of viscoelastic droplets on inclined surfaces is proposed. Viscoelasticity is incorporated into the governing system employing the Oldroyd–B constitutive model. The supporting (inclined) surface is allowed to have non–homogeneous properties incorporated into the mathematical model through the generalized Navier boundary conditions (GNBC). The droplet motion (sliding and/or rolling) is handled by employing arbitrary Lagrangian Eulerian (ALE) framework. Energy balance is derived from the governing system and it is a starting point in the derivation of the numerical scheme. The overall numerical strategy is designed in such a way that a counterpart of the (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system which, in turn, guarantees the stability of the scheme in the energy norm. The framework proposed is very general and encapsulates both two and three dimensional scenarios. Numerical studies in this work focus primarily on the three dimensional scenarios since such are significantly more challenging and seem to be much scarcer in the literature. The newly proposed numerical strategy is validated on several examples to confirm the theoretical predictions. The role of viscoelasticity in the overall droplet dynamics is briefly investigated and the behaviors of Newtonian and non–Newtonian droplets are compared.

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

We consider the non-linear eigenvalue equations characterizing Lp into Lq Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)2. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if 1p+1q=1. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

The evolution Navier–Stokes equations in a cube under Navier boundary conditions: rarefaction and uniqueness of global solutions

We study the evolution Navier–Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations.

On buoyancy‐driven viscous incompressible flows with various types of boundary conditions

AbstractIn this paper, we study the existence and uniqueness of solutions to the initial‐boundary‐value problem for time‐dependent flows of heat‐conducting incompressible fluids through the two‐dimensional channel. The boundary conditions are of two types: the so‐called “do nothing” boundary condition on the outflow and the so‐called Navier boundary conditions on the solid walls of the channel. A priori estimates play a crucial role in existential analysis, however, the considered mixed boundary conditions do not enable us to derive an energy‐type estimate of the solution. Our aim is to prove the existence and uniqueness of a solution on a sufficiently short time interval for arbitrarily large data.

Note on a higher order pseudo‐parabolic equation with variable exponents

In this paper, we study a higher order pseudo‐parabolic equation involving ‐Laplacian with the Navier boundary condition. We use the energy method, the Sobolev embedding inequalities and the Galerkin's approximation to show the classification of singular solutions, including the existence and nonexistence of global, blow‐up, and extinction solutions. Extinction rates and time of extinction solutions, blow‐up time of blow‐up solutions, and decay estimates of global solutions are discussed.

Evolution of channel flow and Darcy’s law beyond the critical Reynolds number

For incompressible channel flow, there is a critical state, characterized by a critical Reynolds number Rec and a critical wavevector mc along the channel direction, beyond which the channel flow becomes unstable in the linear regime. In this work, we investigate the channel flow beyond the critical state and find the existence of a new fluctuating, quasi-stationary flow that comprises the laminar Poiseuille flow superposed with a counter-flow component, accompanied by vortices and anti-vortices. The net flow rate is reduced by ~ 15% from the linear, laminar regime. Our study is facilitated by the analytical solution of the linearized, incompressible, three-dimensional (3D) Navier–Stokes (NS) equation in the channel geometry, with the Navier boundary condition, alternatively denoted as the hydrodynamic modes (HMs). By using the HMs as the complete mathematical basis for expanding the velocity in the NS equation, the Rec is evaluated to 5-digit accuracy when compared to the well-known Orszag result, without invoking the standard Orr-Sommerfeld equation. Beyond Rec, the analytical solution is indispensable in offering physical insight to those features of the counter-flow component that differs from any of the pressure-driven channel flows. In particular, the counter flow is found to comprise multiple HMs, some with opposite flow direction, that can lead to a net boundary reaction force along the counter-flow direction. The latter is analyzed to be necessary for satisfying Newton’s law. Experimental verification of the predictions is discussed.Graphical

Construction of Boundary Conditions for Navier–Stokes Equations from the Moment System

This work concerns with boundary conditions (BCs) of the linearized moment system for rarefied gases. Under the acoustic scaling, we analyze the boundary-layer behaviors of the moment system by resorting to a three-scale asymptotic expansion. The asymptotic analysis casts the flows into the outer solution, the viscous layer, and the Knudsen layer. Starting from the BCs of the moment system, we construct BCs for the Navier–Stokes equations by a matching requirement. The obtained BCs contain the effect of second-order terms on the velocity slip and temperature jump. For the illustrative case of the Rayleigh problem, we prove the validity of the constructed BCs through error estimates. Meanwhile, numerical tests are presented to show the performance of the constructed BCs.

Solutions to a (p(x),q(x))-biharmonic elliptic problem on a bounded domain

Using variational methods and critical point results, we prove the existence and multiplicity of weak solutions of a (p(x),q(x))-biharmonic elliptic equation along with a singular term under Navier boundary conditions.

Singular limit solutions for a four‐dimensional semilinear elliptic Kuramoto–Sivashinsky system in some general case

We consider the existence of singular limit solutions for a nonlinear elliptic Kuramoto–Sivashinsky system with Navier boundary conditions. We use the nonlinear domain decomposition method and a Pohozaev type identity.

Characteristic-Based Volume Penalization-Imposed Wall Function Method for Turbulent Boundary Layer Modeling

A method to approximate near-wall boundary conditions for the compressible Reynolds-Averaged Navier–Stokes equations is proposed. The differential formulation to match the external and the wall function solutions is reformulated in a form of the generalized characteristic-based volume penalization method to model the transfer of the shear stress from the outer region of the boundary layer to the wall. The exchange location is specified implicitly in terms of a localized source term in the boundary layer equation written as a function of the distance from the wall normalized by the viscous length scale. The shear stress on the wall is determined by solving an auxiliary equation for the wall-stress imposing the analytical wall function solution through the characteristic-based volume penalization method. The proposed method noticeably reduces the near-wall mesh resolution requirements without a significant modification of the numerical algorithm and completely eliminates the ill-defined explicit solution matching procedure. The developed approach is numerically implemented using the vertex-centered control volume method on structured meshes. Its effectiveness is demonstrated by solving two test problems: the two-dimensional channel flow and turbulent flow over an infinitely thin plate.

Positive solutions for a class of biharmonic equations: Existence and uniqueness

In this paper, we analyze the existence of positive solution to a biharmonic equation with Navier boundary conditions, and formulate simple uniqueness conditions for the positive solution. In addition, a necessary condition for the existence of the positive solution is also derived.

Singular Limit Solutions for a $4$-dimensional Emden–Fowler System of Liouville Type in Some General Case

We prove the existence of singular limit solutions for a nonlinear elliptic Emden–Fowler system with Navier boundary conditions by using the nonlinear domain decomposition method and the Pohozaev identity.

Uniqueness and uniform structural stability of Poiseuille flows in an infinitely long pipe with Navier boundary conditions

In this paper, uniqueness and uniform structural stability of Poiseuille flows in an infinitely long pipe with Navier boundary conditions are established for axisymmetric solutions of steady Navier-Stokes system. This is a key step to study general Leray problem for flows in a general infinitely long nozzle. The crucial point is that the estimate is uniform with respect to both the flux of flows and friction coefficient which appeared in Navier boundary conditions. With the aid of special structure of Navier-Stokes system and the refined estimate for some quantities such as radial velocity, the uniqueness and existence of steady solutions of Navier-Stokes system can be obtained even when the external forces are large as long as the fluxes are large. The delicate partition of the two dimensional plane for parameters, friction coefficient and Fourier variable associated with the axial coordinate, plays a key role to achieve these estimates.