Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ω_ɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_ɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ^3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H¹(Ω_ɛ), respectively, L²(Ω_ɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.     

Melt-particle mixing in gas-stirred ladles with throughflow

An experimental study is performed on a gas-particle stirred ladle system with throughflow, using a simplified water model. Narrow ladles are used to produce 2-D flows. Flow visualization by the direct photographic method is employed to investigate the effects of ladle geometry, throughflow rate, air flow rate and its injection location on the melt-particle mixing performance. Image processing is applied to aid in determining the mixing performance. It is disclosed that an efficient mixing may be achieved if the gas at a higher flow rate is injected with particles through a nozzle near the bottom corner of the ladle wall on the melt inlet side. The mixing performance is better in a rectangular ladle (aspect ratio of 2) than in a square ladle (aspect ratio of unity). The effect of throughflow rate on mixing is minor. The study has an important application in manufacturing processes, such as continuous casting process, and materials processing.List of symbols AR aspect ratio - B width of water vessel, m - Bn Nozzle location on bottom surface of water vessel, m - H height of water vessel or height between bottom surface and free surface of water vessel, m - Hn Nozzle location on vertical (inlet side) surface of water vessel, m - Q _g volumetric rate of gas, m³/s - Q _l volumetric rate of water, m³/s - Q _s volumetric rate of particle, m³/s - x transverse coordinate, m - y longitudinal coordinate, m Visiting scholar on leave from the Mechanical Engineering Department, Kagoshima University, Japan     

Topography Influence on the Lake Equations in Bounded Domains

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.     

Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t) ^{α, y = x/(t* ~ t)β,α,β>} 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H³(Ω,R³ X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a timet = t*, t* < +∞.     

Effect of the melting system sizes on the temperature and concentration combined melting of a vertical ice plate in a rectangular cavity

This paper is concerned with the melting of a vertical ice plate into a calcium chloride aqueous solution inside a rectangular cavity. The initial temperature of the ice plate and the mixture are both −5°C and the initial concentration of the mixture is 20 wt%. The effect of the liquid height H, the width W, the aspect ratio of the liquid region A (=H/W) and the initial ice plate thickness δ _i on the transient melting mass per unit melting front area, M, is mainly considered numerically. M keeps a similar value in spite of H varied for A = 1 at early melting stage, however, becomes considerably influenced by H as melting progresses. The ice plate melts influenced by A for H = 20 mm fixed at early melting stage due to the fast development of the stagnant region and M decreases with increasing A (=1∼ 10). A dimensionless correlation of the transient melting mass, the aspect ratio and the melting time was presented under the restricted condition of H = 20 mm.     

Direct simulation of isolated elliptic vortices and of their radiated noise

The aerodynamic evolution and the acoustic radiation of elliptic vortices with various aspect ratios and moderate Mach numbers are investigated by solving numerically the full compressible Navier–Stokes equations. Three behaviours are observed according to the aspect ratio σ = a/b where a and b are the major and minor semi-axes of the vortices. At the small aspect ratio σ = 1.2, the vortex rotates at a constant angular velocity and radiates like a rotating quadrupole. At the moderate aspect ratio σ = 5, the vortex is initially unstable. However the growth of instability waves is inhibited by the return to axisymmetry which decreases its aspect ratio. The noise level becomes lower with time and the radiation frequency increases. For vortices with larger aspect ratios σ ≥ 6, the return to axisymmetry does not occur quickly enough to stop the growth of instabilities, which splits the vortices. Various mergers are then found to occur. For instance in the case σ = 6, several successive switches between an elliptic state and a configuration of two co-rotating vortices are observed. The present results show that the initial value of the aspect ratio yields the relative weight between the return to axisymmetry which stabilizes the vortex and the growth of instabilities which tends to split it. Moreover the noise generated by the vortices is also calculated using the analytical solution derived by Howe (J. Fluid Mech. 71:625–673, 1975) and is compared with the reference solution provided by the direct computation. This solution is found to be valid for σ = 1.2. An extended solution is proposed for higher aspect ratios. Finally, the pressure field appears weakly affected by the switches between the two unstable configurations in the case σ = 6, which underlines the difficulty to detect the split or the merger of vortices from the radiated pressure. This study also shows that elliptic vortices can be used as a basic configuration of aerodynamic noise generation.      

Using a piecewise linear bottom to fit the bed variation in a laterally averaged,z‐co‐ordinate hydrodynamic model

In developing a 3D or laterally averaged 2D model for free‐surface flows using the finite difference method, the water depth is generally discretized either with the z‐co‐ordinate (z‐levels) or a transformed co‐ordinate (e.g. the so‐called σ‐co‐ordinate or σ‐levels). In a z‐level model, the water depth is discretized without any transformation, while in a σ‐level model, the water depth is discretized after a so‐called σ‐transformation that converts the water column to a unit, so that the free surface will be 0 (or 1) and the bottom will be ‐1 (or 0) in the stretched co‐ordinate system. Both discretization methods have their own advantages and drawbacks. It is generally not conclusive that one discretization method always works better than the other. The biggest problem for the z‐level model normally stems from the fact that it cannot fit the topography properly, while a σ‐level model does not have this kind of a topography‐fitting problem. To solve the topography‐fitting problem in a laterally averaged, 2D model using z‐levels, a piecewise linear bottom is proposed in this paper. Since the resulting computational cells are not necessarily rectangular looking at the x–z plane, flux‐based finite difference equations are used in the model to solve the governing equations. In addition to the piecewise linear bottom, the model can also be run with full cells or partial cells (both full cell and partial cell options yield a staircase bottom that does not fit the real bottom topography). Two frictionless wave cases were chosen to evaluate the responses of the model to different treatments of the topography. One wave case is a boundary value problem, while the other is an initial value problem. To verify that the piecewise linear bottom does not cause increased diffusions for areas with steep bottom slopes, a barotropic case in a symmetric triangular basin was tested. The model was also applied to a real estuary using various topography treatments. The model results demonstrate that fitting the topography is important for the initial value problem. For the boundary value problem, topography‐fitting may not be very critical if the vertical spacing is appropriate. Copyright © 2004 John Wiley & Sons, Ltd.     

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).     

Convergence Rate for Compressible Euler Equations with Damping and Vacuum

 We study the asymptotic behavior of L ^∞ weak-entropy solutions to the compressible Euler equations with damping and vacuum. Previous works on this topic are mainly concerned with the case away from the vacuum and small initial data. In the present paper, we prove that the entropy-weak solution strongly converges to the similarity solution of the porous media equations in L ^p (R) (2≤p<∞) with decay rates. The initial data can contain vacuum and can be arbitrary large. A new approach is introduced to control the singularity near vacuum for the desired estimates. (Accepted August 31, 2002) Published online January 9, 2003 Communicated by C. M. Dafermos     

Unsteady natural convection in an enclosure with vertical wavy walls

In this paper, unsteady heat transfer and fluid flow characteristics in an enclosure are investigated. The enclosure consists of two vertical wavy and two horizontal straight walls. The top and the bottom walls are considered adiabatic. Two wavy walls are kept isothermal and their boundaries are approximated by a cosine function. Governing equations including continuity, momentum and energy were discretized using the finite-volume method and solved by SIMPLE method in curvilinear coordinate. Simulation was carried out for a range of Grashof number Gr = 10³–10⁶, Prandtl number Pr = 0.5–4.0, wave ratio A (defined by amplitude/wavelength) 0.0–0.35 and aspect ratio W (defined by average width/wavelength) 0.5–1.0. Streamlines and isothermal lines are presented to corresponding flow and thermal fields. Local and average Nusselt number distributions are presented. The obtained results are in good agreement with available numerical and experimental data.     

Viscous-inviscid interacting flow theory

In this paper a viscous-inviscid interacting flow theory (IFT) is developed for an incompressible, two—dimensional laminar flow. IFT's main points are as follows. (1) By introducing a concept of interaction layer where the normal momentum exchange is dominating, a new three-layer structure is established. (2) Through the conventional manipulations and by introducing an interaction model, both the streamwise and normal length scales are proved to be functions of a single parameterm, which is related to the streamwise pressure gradient and Reynolds number. (3) The approximate equations governing the flow of each layer as well as the whole interaction flow are derived. The present IFT is applicable to both attached and attached-separation bubble—reattached flows. The classical boundary layer theory^[1] and Triple-deck theory^[2] are shown to be two special cases of the present theory underm=0 and 1/4, respectively. Furthermore IFT provides new distinctions of both the normal and streamwise length scales for flow-field numerical computation and also gives a new approach to developing the simplified Navier-Stokes (SNS) equations. The project is supported by the National Natural Science Foundation of China.     

Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations

We study the initial boundary value problem of two-dimensional viscous Boussinesq equations over a bounded domain with smooth boundary. We show that the equations have a unique classical solution for H ³ initial data and the no-slip boundary condition. In addition, we show that the kinetic energy is uniformly bounded in time.     

Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L ²∩L ^∞, the initial temperature is close to a constant in L ², and the initial velocity is small in H ^s ∩L ⁴, where s=0 when n=2 and when n=3. (The L ^p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of ⁿ . A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature are relatively smooth in positive time, as is the “effective viscous flux”F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density. (Accepted June 13, 1996)     

Global Behavior of Compressible Navier-Stokes Equations with a Degenerate Viscosity Coefficient

In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity. Precisely, the viscosity coefficient μ is proportional to ρ ^θ with , where ρ is the density, and γ > 1 is the physical constant of polytropic gas. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give the uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we estimate the stabilization rate in L ^∞ norm, (weighted) L ² norm and weighted H ¹ norm of the solution as time tends to infinity.     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

Effect of Discrete Heating on Natural Convection in a Rectangular Porous Enclosure

The main objective of this article is to study the effect of discrete heating on free convection heat transfer in a rectangular porous enclosure containing a heat-generating substance. The left wall of the enclosure has two discrete heat sources and the right wall is isothermally cooled at a lower temperature. The top and bottom walls, and the unheated portions of the left wall are adiabatic. The vorticity–stream function formulation of the governing equations is numerically solved using an implicit finite difference method. The effects of aspect ratio, Darcy number, heat source length, and modified Rayleigh number on the flow and heat transfer are analyzed. The numerical results reveal that the rate of heat transfer increases as the modified Rayleigh number and the Darcy number increases, but decreases on increasing the aspect ratio. The average heat transfer rate is found to be higher at the bottom heater than at the top heater in almost all considered parameter cases except for ε = 0.5. Also, the maximum temperature takes place generally at the top heater except for the case ε = 0.5, where the maximum temperature is found at the bottom heater. Further, the numerical results reveal that the maximum temperature decreases with the modified Rayleigh number and increases with the aspect ratio.     

Global Low-Energy Weak Solutions of the Equations of Three-Dimensional Compressible Magnetohydrodynamics

We prove the global-in-time existence of weak solutions of the equations of compressible magnetohydrodynamics in three space dimensions with initial data small in L ² and initial density positive and essentially bounded. A great deal of information concerning partial regularity is obtained: velocity, vorticity, and magnetic field become relatively smooth in positive time (H ¹ but not H ²) and singularities in the pressure cancel those in a certain multiple of the divergence of the velocity, thus giving concrete expression to conclusions obtained formally from the Rankine–Hugoniot conditions.     

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 H ¹. 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.     

Stability of a fluid-saturated porous medium contained in a vertical cylinder heated from below by forced convection

A linear stability analysis determining the critical Rayleigh number R _c for onset of convection in a bounded vertical cylinder containing a fluid-saturated porous medium is performed for insulated sidewalls, isothermal top surface, and bottom surface heated by forced convection. This Newtonian heating of the bottom surface involves a Biot number Bi that allows consideration of the continuum of boundary conditions ranging from constant heat flux, with global minimum R _min=27.096 found as Bi→0, to isothermal, with global minimum R _min=4π² found as Bi→ ∞. In both cases and for most cylinder aspect ratios, incipient convection sets in as an asymmetric mode, though islands of aspect ratio exist where the onset mode is symmetric. Sample three-dimensional renderings of disturbance temperature distributions showing preferred modes at onset of convection for fixed Bi are provided and an analytical fit to R _min as a function of Bi is given.     

Motion of a sphere in an electrically conducting rotating fluid

The combined effect of rotation and magnetic field is investigated for the axisymmetric flow due to the motion of a sphere in an inviscid, incompressible electrically conducting fluid having uniform rotation far upstream. The steady-state linearized equations contain a single parameter α=1/2βR _m, β being the magnetic pressure number and R _m the magnetic Reynolds number. The complete solution for the flow field and magnetic field is obtained and the distribution of vorticity and current density is found. The induced vorticity is O(α⁴) and the current density is O(R _m) on the sphere.