Navier Boundary Research Articles

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.

Prediction of the Absorption Characteristics of Non-Uniform Acoustic Absorbers with Grazing Flow

In this paper, planar and the cylindrical broadband non-uniform acoustic absorbers were constructed, both of which use broadband absorption units (BAUs) as their building blocks. The impedance boundary Navier–Stokes equation (IBNSE) method was developed to predict the absorption characteristics of the lined duct with non-uniform acoustic absorbers, in which each small piece of perforated plate is acoustically equivalent to a semi-empirical impedance model through the boundary condition. A total of four semi-empirical impedance models were compared under different control parameters. The full Navier–Stokes equation (FNSE) method was used to verify the accuracy of these impedance models. It was found that the IBNSE method with the Goodrich model had the highest prediction accuracy. Finally, the planar and the cylindrical non-uniform acoustic absorbers were constructed through spatial extensions of the BAU. The transmission losses and the absorption coefficients of the rectangular duct–planar acoustic absorber (RDPAA) and annular duct–cylindrical acoustic absorber (ADCAA) systems under grazing flow were predicted, respectively. The results demonstrated that the broadband absorption of the designed non-uniform acoustic absorbers was achieved. The developed IBNSE method with Goodrich model was accurate and computationally efficient, and can be used to predict the absorption characteristics of an acoustically treated duct in the presence of grazing flow.

Influences of serrated trailing edge on the aerodynamic and aeroacoustic performance of a flapping wing during hovering flight

The influences of serrated trailing edge on the aerodynamic and aeroacoustic performance of a flapping wing during hovering flight are investigated using a hybrid framework of an immersed boundary Navier–Stokes solver for the flow field and the Ffowcs Williams–Hawkings (FW–H) analogy for the sound field. A rigid rectangular wing with an aspect ratio of 2 undergoes pitching and stroke motions at a Reynolds number (Re) of 310 and a Mach number (M) of 0.012. Simulations are conducted by varying the dimensionless wavenumber k* from 2π to 10π and wave amplitude 2h* from 0.25 to 1.0. We find that at k*=8π and 2h*=1.5 (D4), the average sound power level is reduced by up to 6.8 dB within the Strouhal number (St) between 2.0 and 4.0 compared to that of a plain trailing edge while the lift coefficient is maintained. The directivity at St = 0.2, St = 0.4, St = 2.2, and St = 2.4 is discussed. It is found that the serrations of D4 do not affect the directivity for the first two frequencies and significantly reduce the magnitude of the directivity for the last two frequencies. The serrations of D4 considerably alter the flow field near the wing surface and reduced the surface pressure fluctuations near the wing tip, leading to the noise reduction. The lift coefficient of D4 is not significantly changed, because the reduction in the pressure-contributed lift is compensated by an increase in the shear stress-contributed lift. The serrations with higher 2h* and k* have larger shear stress-contributed lift.

On the compressible viscous barotropic flows subject to large external potential forces in a half space with Navier's boundary conditions

This paper is concerned with an initial and boundary value problem of the Navier‐Stokes equations for compressible viscous barotropic flow subject to large external potential forces in a half space with Navier's boundary conditions. The global well‐posedness of strong solutions with large oscillations and vacuum is established, provided that the initial energy is suitably small and that the unique steady state is strictly away from vacuum. As a by‐product, the stability of stationary solution is obtained.

A Pressure Associated with a Weak Solution to the Navier–Stokes Equations with Navier’s Boundary Conditions

We show that if u is a weak solution to the Navier-Stokes initial-boundary value problem with Navier's slip boundary conditions in $Q_T:=\Omega\times(0,T)$, where $\Omega$ is a domain in $R^3$, then an associated pressure $p$ exists as a distribution with a certain structure. Furthermore, we also show that if $\Omega$ is a "smooth" domain in $R^3$ then the pressure is represented by a function in $Q_T$ with a certain rate of integrability. Finally, we study the regularity of the pressure in sub-domains of $Q_T$, where $u$ satisfies Serrin's integrability conditions.

Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field

This article presented a solution for torsion vibration a nanotube made of pores material. Based on the porosity distribution, the material properties are assumed to vary according to a function along radius of nanotube. Moreover, the cross-section area of nanotube varied in the longitudinal direction by nonlinear function. As for the torque effect of the axial magnetic field, the well-known Maxwell’s relation is used. Couple stress theory is employed to study the influence of small-scale on torsional vibration of nanotube. The Navier equation and boundary conditions of the size-dependent porous nanotube were derived by the Hamilton principle. These equations were solved by employing the generalized differential quadrature method. Comparison between the results of the present work with the other paper reveals the accuracy of this study. To the best of authors’ knowledge, so far all previous torsional vibration of nanotube address the case of ignoring porosity. The novelty of this work is to present a solution by taking into account the existence of porosity. Finally, numerical results are presented to study the small scale effect and porosity on the frequency of the porous nanotube.

Uniform regularity for free-boundary Navier–Stokes equations with surface tension

We study the zero-viscosity limit of free boundary Navier–Stokes equations with surface tension in unbounded domain thus extending the work of Masmoudi and Rousset [Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal. (2016), doi:10.1007/s00205-016-1036-5] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. Our main idea is to use Dirichlet–Neumann operator and time-derivatives.

Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations

We study the inviscid limit of the free boundary Navier–Stokes equations. We prove the existence of solutions on a uniform time interval by using a suitable functional framework based on Sobolev conormal spaces. This allows us to use a strong compactness argument to justify the inviscid limit. Our approach does not rely on the justification of asymptotic expansions. In particular, we get a new existence result for the Euler equations with free surface from the one for Navier–Stokes.

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.

On a generalized Stokes system with slip boundary conditions in the half-space

We are interested in a system of Stokes type, where the divergence-free constraint is modified by adding a term proportional to the pressure. The domain is the half-space with nonhomogeneous Navier's boundary conditions. The weighted Sobolev spaces yield a natural functional framework to envisage a wide class of behavior at infinity for data and solutions. So, we can give a range of solutions from strong to very weak depending on the regularity of the data. All along this study, we take the bridge between this system and the linear elasticity system.

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 .

STRETCHING FLOWS WITH GENERAL SLIP BOUNDARY CONDITION

General slip boundary condition is used to solve the viscous incompressible flows induced by a stretching sheet. These flow problems corresponds to the planar and axisymmetric stretching. A similarity solution is developed by shooting method using Runge–Kutta algorithm. The results are graphically displayed and discussed under the influence of slip parameter and critical shear rate. The comparison of stretching flow problem subject to Navier's boundary condition in the planar case is made with the available numerical results in the literature.

Local in time strong solvability of the non-steady Navier–Stokes equations with Navier's boundary condition and the question of the inviscid limit

In this Note, we prove the existence of strong solutions to the Navier–Stokes equations for incompressible viscous fluids in a general regular bounded domain of R 3 on a “short” time interval ( 0 , T 0 ) , independent of the viscosity and of the friction between the fluid and the boundary. The solutions to the Navier–Stokes problem satisfy the inhomogeneous Navier's boundary condition and they reveal a remarkable structure of approximation of the solution to the Euler problem, which enables us to solve completely the question of the inviscid limit of the family of obtained solutions on the time interval ( 0 , T 0 ) .

On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions

We study the low Mach number limit for the compressible Navier-Stokes system supplemented with Navier's boundary condition on an unbounded domain with compact boundary. Our main result asserts that the velocities converge pointwise to a solenoidal vector field - a weak solution of the incompressible Navier-Stokes system - while the fluid density becomes constant. The proof is based on a variant of local energy decay property for the underlying acoustic equation established by Kato.

The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions

In this Note, we treat the Navier–Stokes equation with slip Navier's boundary condition in a time variable domain around a finite system of compact bodies moving in a container. The motion of the bodies is assumed to be a priori known. The bodies may collide at a finite number of time instants. We present the theorem on the global in time existence of a weak solution. It is remarkable that Navier's boundary condition enables us to consider a larger class of possible collisions of bodies with C 2 front surfaces in comparison with the no-slip Dirichlet condition. To cite this article: J. Neustupa, P. Penel, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Influence of wall roughness on the slip behaviour of viscous fluids

We consider the stationary equations of a general viscous fluid in an infinite (periodic) slab supplemented with Navier's boundary condition with a friction term on the upper part of the boundary. In addition, we assume that the upper part of the boundary is described by a graph of a function φε, where φε oscillates in a specific direction with amplitude proportional to ε. We identify the limit problem when ε → 0, in particular, the effective boundary conditions.

The linearized non-stationary problem for the permeable boundary Navier–Stokes flows

We consider the following linearized model for the permeable boundary Navier–Stokes flows, representing the non-stationary state: ρ ∂ t v (x,t)=μ Δ v (x,t)− ∇p(x,t)+ f (x,t); v ∈H 2(Ω); (1a) x∈Ω⊂R 3; f ∈L 2(Ω), t>0, subject to (1b) ∇· v (x,t)=0; γ 0 v (y,t)=−η v(y,t) n (y); y∈Γ. σ ∂ t γ 0 v (y,t) +γ 0p(y,t)+2μκη v(y,t)=p 0(t). where, μ is the fluid viscosity; assumed constant, ρ, the fluid density; assumed constant, v( x, t), the fluid velocity field, p( x, t), the fluid pressure field, f( x, t), the external force; assumed to be of potential type. We study the existence and uniqueness of the solution; investigate the stability of the null solution using the Ljapunov's stability theory. This problem is analyzed by Ladyzhenskaya [O.L. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1963] in a different setting. In our setting of a permeable boundary, we introduce the second condition in (1b), which is the Sauer–Maritz boundary permeation model. This model was proposed by Sauer; used by Maritz and Sauer [R. Maritz, N. Sauer, in: H. Amman et al. (Eds.), Navier–Stokes Equations and Related Non-linear Problems, VSP/TEV, Utrecht/Vilnius, 1998, p. 153] in the study of incompressible second grade fluids. It was later used by Hlomuka and Sauer [J.V. Hlomuka, N. Sauer, in: R. Salvi (Ed.), Navier–Stokes Equations: Theory and Numerical Methods, Marcel Dekker, New York, 2001, p. 33] to confirm the stability of the permeable boundary Navier–Stokes flows for the non-linear problem. The third boundary condition is the dynamic boundary condition derived in [J.V. Hlomuka, N. Sauer, in: R. Salvi (Ed.), Navier–Stokes Equations: Theory and Numerical Methods, Marcel Dekker, New York, 2001, p. 33] by making use of the conservation laws.

Slender axisymmetric fluid jets

Fluid fibres and jets are described, in general, by free boundary Navier Stokes problems. If the ratio of the cross section of the jet is small compared to the axial length scale of the jet or fibre, then the free boundary problem can be significantly simplified, and a greatly simplified system of (nonlinear) partial differential equations obtained. In this paper, the broad outline of the method will be discussed, with reference to the problem of tapering optical fibres of arbitrary cross sectional shape and with reference to symmetric fluid jets. In the latter case, further analysis is possible by looking for transformations that reduce the partial differential equations to ordinary differential equations.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.