Existence and Uniqueness of the Flow of Second‐Grade Fluids with Slip Boundary Conditions

. This paper treats the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply‐connected domain. The existence of a unique classical solution, global in time, is proved under suitable regularity and growth restrictions on the initial data, the slip law and the body and surface forces. The method is based on a fixed-point formulation of the problem. (Accepted October 5, 1998)     

Pore-Scale Imaging and Modelling of Reactive Flow in Evolving Porous Media: Tracking the Dynamics of the Fluid–Rock Interface

Transport in Porous Media - Fluid–mineral and fluid–rock interfaces are key parameters controlling the reactivity and fate of fluids in reservoir rocks and aquifers. The interface...     

Darcy–Carreau Model and Nonlinear Natural Convection for Pseudoplastic and Dilatant Fluids in Porous Media

Transport in Porous Media - The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Bénard convection for non-Newtonian inelastic fluids. The...     

Application of the γ-Reθ Transition Model to Simulations of the Flow Past a Circular Cylinder

Based on the finite volume method, the flow past a two-dimensional circular cylinder at a critical Reynolds number (Re = 8.5 × 10⁵) was simulated using the Navier-Stokes equations and the γ-Re_θ transition model coupled with the SST k ? ω turbulence model (hereinafter abbreviated as γ-Re_θ model). Considering the effect of free-stream turbulence intensity decay, the SST k ? ω turbulence model was modified according to the ambient source term method proposed by Spalart and Rumsey, and then the modified SST k ? ω turbulence model is coupled with the γ-Re_θ transition model (hereinafter abbreviated as γ-Re_θ-SR model). The flow past a circular cylinder at different inlet turbulence intensities were simulated by the γ-Re_θ-SR model. At last, the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers were each simulated by the γ-Re_θ-SR model, and the three flow states were analyzed. It was found that compared with the SST k ? ω turbulence model, the γ-Re_θ model could simulate the transition of laminar to turbulent, resulting in better consistency with experimental result. Compared with the γ-Re_θ model, for relatively high inlet turbulence intensities, the γ-Re_θ-SR model could better simulate the flow past a circular cylinder; however the improvement almost diminished for relatively low inlet turbulence intensities The γ-Re_θ-SR model could well simulate the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers.     

Pore-Scale Study to Analyze the Impacts of Porous Media Heterogeneity on Mineral Dissolution and Acid Transport Using Darcy–Brinkmann–Stokes Method

Transport in Porous Media - Inherent pore structure of rocks has a significant impact on the acid–rock interaction during the acidizing process. In this study, a new pore-scale reactive...     

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele�CShaw Cell

We study the radial movement of an incompressible fluid located in a Hele–Shaw cell rotating at a constant angular velocity in the horizontal plane. Within an analytic framework, local existence and uniqueness of solutions is proved, and it is shown that the unique rotationally invariant equilibrium of the flow is unstable. There are, however, other time-independent solutions: using surface tension as a bifurcation parameter we establish the existence of global bifurcation branches consisting of stationary fingering patterns. The same results can be obtained by fixing the surface tension while varying the angular velocity. Finally, it is shown that the equilibria on a global bifurcation branch converge to a circle as the surface tension tends to infinity, provided they stay suitably bounded.     

Starting Darcy–Brinkman Flow in a Sector Duct Using the Method of Eigenfuction Superposition

The powerful method of eigenfunction superposition is applied to the starting flow in a sector duct filled with a porous medium. Using analytic eigenfunctions and eigenvalues of the Helmholtz equation, the solution can be expressed in a simple series. The properties of the velocity and the transient flow rate are found to depend on the sector geometry and a porous medium factor. The starting solution is then used to construct the solution to arbitrary unsteady flows.

     

Asymptotic Behavior of Solutions of the Compressible Navier�CStokes Equation Around the Plane Couette Flow

Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L ² Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L ² norm as solutions of an n − 1 dimensional linear heat equation with a convective term.     

Development and Validation of a Numerical Model of Flow Through Embankment Dams – Comparisons with Experimental Data and Analytical Solutions

The development and validation of a numerical simulation model of the flow through embankment dams is described. The paper focuses on basic verification studies, that is, comparisons with analytical solutions and data from laboratory experiments. Two experimental studies, one dealing with the flow in a Hele–Shaw cell and the other with the flow through a bed of packed glass beads, are also described. Comparisons are carried out with respect to the phreatic surfaces, pressure profiles, seepage levels and discharges. It is concluded that the agreement between experimental, analytical and numerical results is generally satisfactory.     

Flow and Mixing Fields for Transported Scalar PDF Simulations of a Piloted Jet Diffusion Flame (‘Delft Flame III’)

Numerical simulation results are presented for ‘Delft Flame III’, a piloted jet diffusion flame with strong turbulence–chemistry interaction. While pilot flames emerge from 12 separate holes in the experiments, the simulations are performed on a rectangular grid, under the assumption of axisymmetry. In the first part of the paper, flow and mixing field results are presented with a non-linear first order k–ε model, with the transport equation for ε based on a modeled enstrophy transport equation, for cold and reactive flows. For the latter, the turbulence model is applied in combination with pre-assumed β-PDF modeling for the turbulence–chemistry interaction. The mixture fraction serves as conserved scalar. Two chemistry models are considered: chemical equilibrium and a steady laminar flamelet model. The importance of the turbulence model is highlighted. The influence of the chemistry model is noticeable too. A procedure is described to construct appropriate inlet boundary conditions. Still, the generation of accurate inlet boundary conditions is shown to be far less important, their effect being local, close to the nozzle exit. In the second part of the paper, results are presented with the transported scalar PDF approach as turbulence–chemistry interaction model. A C₁ skeletal scheme serves as chemistry model, while the EMST method is applied as micro-mixing model. For the transported PDF simulations, the model for the pilot flames, as an energy source term in the mean enthalpy transport equation, is important with respect to the accuracy of the flow field predictions. It is explained that the strong influence on the flow and mixing field is through the turbulent shear stress force in the region, close to the nozzle exit.     

On a Bound for Amplitudes of Navier�CStokes Flow with almost Periodic Initial Data

For any bounded (real) initial data it is known that there is a unique global solution to the two-dimensional Navier–Stokes equations. This paper is concerned with a bound for the sum of the modulus of amplitudes when initial velocity is spatially almost periodic in 2D. In the case of general dimension, it is bounded on local time of existence shown by Giga et al. (Methods Appl Anal 12:381–393,2005). A class of initial data is given such that the sum of the modulus of amplitudes of a solution is bounded on any finite time interval. It is shown by an explicit example that such a bound may diverge to infinity as the time goes to infinity at least for complex initial data.     

Boundary Value Problems of Robin Type for the Brinkman and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains

The purpose of this paper is to study boundary value problems of Robin type for the Brinkman system and a semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system, on Lipschitz domains in Euclidean setting. In the first part of the paper, we exploit a layer potential analysis and a fixed point theorem to show the existence and uniqueness of the solution to the nonlinear Robin problem for the Darcy–Forchheimer–Brinkman system on a bounded Lipschitz domain in \({\mathbb{R}^n}\) \({(n \in \{2,3\})}\) with small data in L ²-based Sobolev spaces. In the second part, we show an existence result for the mixed Dirichlet–Robin problem for the same semilinear Darcy–Forchheimer-Brinkman system on a bounded creased Lipschitz domain in \({\mathbb{R}^3}\) with small L ²-boundary data. We also study mixed Dirichlet–Robin problems and boundary value problems of mixed Dirichlet–Robin and transmission type for Brinkman systems on bounded creased Lipschitz domains in \({\mathbb{R}^n}\) (n ≥ 3). Finally, we show the well-posedness of the Navier problem for the Brinkman system with boundary data in some L ²-based Sobolev spaces on a bounded Lipschitz domain in \({\mathbb{R}^3}\) .     

Development of a Low-Reynolds-number k-ω Model for FENE-P Fluids

A low-Reynolds-number k-ω model for Newtonian fluids has been developed to predict drag reduction of viscoelastic fluids described by the FENE-P model. The model is an extension to viscoelastic fluids of the model for Newtonian fluids developed by Bredberg et al. (Int J Heat Fluid Flow 23:731–743, 2002). The performance of the model was assessed using results from direct numerical simulations for fully developed turbulent channel flow of FENE-P fluids. It should only be used for drag reductions of up to 50 % (low and intermediate drag reductions), because of the limiting assumption of turbulence isotropy leading to an under-prediction of k, but compares favourably with results from k-ε models in the literature based on turbulence isotropy.     

Logarithmically Improved Regularity Criteria for the Navier�CStokes and MHD Equations

In this paper, logarithmically improved regularity criteria for the Navier–Stokes and the MHD equations are established in terms of both the vorticity field and the pressure.     

An Existence Theorem¶for the Navier-Stokes Flow¶in the Exterior of a Rotating Obstacle

We consider the three-dimensional Navier-Stokes initial value problem in the exterior of a rotating obstacle. It is proved that a unique solution exists locally in time if the initial velocity possesses the regularity L^1/2. This regularity assumption is the same as that in the famous paper of Fujita &; Kato. An essential step for the proof is the deduction of a certain smoothing property together with estimates near t˸ of the semigroup, which is not an analytic one, generated by the operator \Cal Lu = -P[\De u+(\om×x)·\na u-\om×u]\Cal Lu= -P\left[\De u+(\om\times x)\cdot\na u-\om\times u\right] in the space L², where y stands for the angular velocity of the rotating obstacle and P denotes the projection associated with the Helmholtz decomposition.