A 1D-Averaged Model for Stable and Unstable Miscible Flows in Porous Media with Varying Péclet Numbers and Aspect Ratios

This paper deals with reducing the number of spatial dimensions of the models used to solve stable and unstable miscible flows in saturated and homogeneous porous media. Unstable miscible displacements occur when a fluid displaces another fluid of higher viscosity, with which it can fully mix. Stable flows occur if the displaced fluid is less viscous than the displacing one. First, a 1D-averaged model is identified, capable of accurately describing unstable flows at high Péclet numbers. Second, another 1D-averaged model is determined, capable of accurately predicting miscible displacements at low Péclet numbers. Third, a new model is proposed, for any Péclet number and for both stable and unstable flows, as a combination of the previous two models. This combination involves three parameters whose values depend on the dimensionless numbers of the problem, namely, the viscosity ratio M, the Péclet number P_e, the aspect ratio A, and the dispersion length ratio ε. These parameters are computed for several values of M, P_e, A with ε=1 by matching results from direct 2D simulations, obtained from a numerical model previously validated against experimental data. It is found that a 1D-averaged model combining an extended version of the Todd–Longstaff model and the diffusive term of the 1D-miscible model forms an accurate general model for miscible displacements in homogeneous porous media. This paper also provides a large set of data computed from high-resolution 2D simulations of unstable miscible displacements.     

Closed streamline flows past small rotating particles: Heat transfer at high péclet numbers

A heat transfer problem is solved, first for an infinitely long heated cylinder and then for a small heated sphere, each freely suspended in a general linear flow at Reynolds numbers $\text{Re ? 1}$. Asymptotic solutions to the convection problem are developed for very large values of the Péclet number $\text{Pe}$, and expressions are obtained for the asymptotic Nusselt number for two-dimensional flows ranging from solid body rotation to hyperbolic flow. Since the objects in these cases are surrounded by a region of effectively isothermal closed streamlines, the asymptotic Nusselt number becomes independent of the Péclet number in the limit $\text{Pe → ∞}$.     

Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

High Péclet number heat exchange between cocurrent streams

Summary  This paper concentrates on the analysis of the heat transfer between two cocurrent laminar flows in parallel channels. For high values of the Péclet number Pe, a boundary layer arises near the wall separating the streams. Matched asymptotic expansions (MAE) are used to obtain approximate solutions. We consider arbitrary inlet temperatures and derive higher-order corrections of the boundary problem. The separating wall is supposed to be sufficiently thin to neglect the heat conduction in it. Analyticity and adiabatic conditions at the outer walls impose restrictions on the inlet temperatures. It turns out, however, that only the inlet temperatures at the wall separating the two fluids enter the leading-order problem. The Nusselt numbers thus calculated are in the leading order proportional to (Pe/x)^1/3, where x is the stream-wise coordinate. An estimate of the thickness of the separating wall to validate the MAE approach is obtained. It is also demonstrated that the MAE analysis is unable to describe the heat exchange of counterflowing fluids. Received 9 June 1999; accepted for publication 17 November 1999     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

The Onset of Prandtl–Darcy–Prats Convection in a Horizontal Porous Layer

We consider the effect of finite Prandtl–Darcy numbers of the onset of convection in a porous layer heated isothermally from below and which is subject to a horizontal pressure gradient. A dispersion relation is found which relates the critical Darcy–Rayleigh number and the induced phase speed of the cells to the wavenumber and the imposed Péclet and Prandtl–Darcy numbers. Exact numerical solutions are given and these are supplemented by asymptotic solutions for both large and small values of the governing nondimensional parameters. The classical value of the critical Darcy–Rayleigh number is $4\pi ^2$ 4 π 2 , and we show that this value increases whenever the Péclet number is nonzero and the Prandtl–Darcy number is finite simultaneously. The corresponding wavenumber is always less than $\pi $ π and the phase speed of the convection cells is always smaller than the background flux velocity.     

Stability of a nonuniformly heated fluid in a porous horizontal layer

We investigate the stability of a nonuniformly heated fluid in the gravitational field in a plane horizontal porous layer through which vertical forced motion is effected. A similar system was studied in [1, 2]. In the present paper, the nonuniformity of the permeability of the porous layer with respect to the depth and the dependence of the viscosity of the saturating fluid on the temperature are taken into account in addition. As a result of the application of the linear stability theory, an eigenvalue problem arises, which is solved numerically. A family of curves representing the dependence of the critical modified Rayleigh number Ra _k ^⋆ on the injection parameter (the Péclet number Pe) for different degrees of inhomogeneity of the permeability and the viscosity is obtained. It is found that although Pe=0 corresponds to Ra _k ^⋆ for uniform permeability and viscosity and the stability increases monotonically as Pe increases, the presence of nonuniformity of the permeability or the viscosity leads to the appearance of a stability minimum in the region Pe≈1, while under the simultaneous influence of these two factors, the minimum is shifted into the region Pe≈2. The results of the paper can be used, for example, in the investigation of heat transfer in the case of forced fluid motion in the fissures of a permeable rock mass, when, in the case of pumping through a horizontal fissure, the fluid penetrates vertically across its permeable walls into the stratum. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–7, November–December, 1986.     

Nonlinear stability theory of channel flow with heat transfer – an asymptotic approach

A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε _T . Both perturbation parameters, ε and ε _T , are linked by the assumption ε _T =O(ε²) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range. Received on 11 August 1997     

Simulation of the thermal method for reducing turbulent friction

The results of calculating a supersonic turbulent boundary layer on a heated surface on the basis of the algebraic two-parameter (k-ε) and four-parameter (k-ε-θ ²-ε ₆) models of turbulence are compared with experimental data. Emphasis is placed on the ability of the models to predict the behavior of the friction and heat-transfer coefficients on a heated surface. The optimal model of turbulence is chosen. The possibility of improving the efficiency of viscous drag reduction by localizing the regions of heat addition to the boundary layer is demonstrated on the basis of numerical calculations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–68, January–February, 1998. This research was carried out with financial support from the International Scientific and Technological Center (project No. 199).     

Deduction of Saint-Venant"s Principle for the Asymptotic Residue of Elastic Rods

Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ω^ε having cross section with diameter of order ε, and let u ⁽⁰⁾ –Bernoulli–Navier displacement – and u ⁽²⁾ be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u ⁽⁰⁾ + ε² u ⁽²⁾) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the rod. This revised version was published online in August 2006 with corrections to the Cover Date.     

A lower bound for a variational model for pattern formation in shape-memory alloys

The Kohn-Müller model for the formation of domain patterns in martensitic shape-memory alloys consists in minimizing the sum of elastic, surface and boundary energy in a simplified scalar setting, with a nonconvex constraint representing the presence of different variants. Precisely, one minimizes

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* < +∞.     

The wedge subjected to general tractions: A paradox resolved

A wedge subjected to tractions in proportion tor ⁿ (n≥0), is considered. The stresses in the solutions of the classical theory of elasticity become infinite when the angle of the wedge is ρ or 2ρ. The paradox has been resolved by Dempsey^[4] and T.C.T. Ting^[5] whenn=0. The purpose of this paper is to resolve the paradox whenn>0.     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Slow flow past periodic arrays of cylinders with application to heat transfer

Solutions for the slow flow past a square and a hexagonal array of cylinders are determined using a somewhat non-conventional numerical method. The calculated values of the drag on a cylinder as a function of $\text{c}$, the volume fraction of the cylinders, are shown to be in excellent agreement with the corresponding asymptotic expressions for $\text{c ? 1}$ and for $\text{c → c}{}_{\mathrm{<mtext>max</mtext>}}$, the maximum volume fraction. These solutions are then used to calculate the average temperature difference between the bulk and the cylinders which are heated uniformly under conditions of small Reynolds and Péclet numbers.     

A numerical method for steady and nonisothermal viscoelastic fluid flow for high Deborah and Péclet numbers

A combined finite element/streamline integration method is presented for nonisothermal flows of viscoelastic fluids. The attention is focused on some characteristic problems that arise for numerical simulation of flows with high Deborah and Péclet numbers. The two most important problems to handle are the choice of an outflow boundary condition for not completely developed flow and the treatment of the dissipative term in the temperature equation. The ability of the numerical method to handle high Deborah and Péclet numbers will be demonstrated on a contraction flow of an LDPE melt with isotropic and anisotropic heat conductivity. The influence of anisotropic heat conduction and the difference between the stress work and mechanical dissipation will be discussed for contraction flows. Received:  4 February 1997 Accepted: 23 October 1997     

Homogenization of Non-linear Scalar Conservation Laws

We study the limit as ε → 0 of the entropy solutions of the equation . We prove that the sequence u ^ε two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in .     

Nonlocal Effects in Two-Dimensional Conductivity

The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a _ε is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L ¹-boundedness of the sequence a _ε , the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness and L ¹-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions.