Long-time Asymptotic Solution Research Articles

The flow analysis of fiuids in fractal reservoir with the fractional derivative

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the flow models of fluids in fractal reservoirs with the fractional derivative. The flow characteristics of fluids through a fractal reservoir with the fractional order derivative are studied by using the finite integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of fluids flow through an infinite fractal reservoir is studied by using the Stehfest's inversion method of the numerical Laplace transform. It shows that the order of the fractional derivative affect the whole pressure behavior, particularly, the effect of pressure behavior of the early-time stage is larger The new type flow model of fluid in fractal reservoir with fractional derivative is provided a new mathematical model for studying the seepage mechanics of fluid in fractal porous media.

Analysis of the flow of non-Newtonian viscoelastic fluids in fractal reservoir with the fractional derivative

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the relaxation models of non-Newtonian viscoelastic fluids with the fractional derivative in fractal reservoirs. A new type integral transform is introduced, and the flow characteristics of non-Newtonian viscoelastic fluids with the fractional order derivative through a fractal reservoir are studied by using the integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of non-Newtonian viscoelastic fluids flow through an infinite fractal reservoir is studied by using the Stehfest’s inversion method of the numerical Laplace transform. It is shown that the clearer the viscoelastic characteristics of the fluid, the more the fluid is sensitive to the order of the fractional derivative. The new type integral transform provides a new analytical tool for studying the seepage mechanics of fluid in fractal porous media.

Thermally driven reaction fronts in porous media

We examine the reaction fronts that develop when an undersaturated ‘injection fluid’ displaces a saturated ‘formation fluid’ in a chemically reactive porous medium. We allow the injection and formation fluids to differ both in temperature and in chemical composition. The undersaturation of the incoming fluid drives a dissolution reaction and leads to the formation of a ‘depletion’ front. Under certain circumstances, which we describe, the temperature difference drives a separate thermal reaction front. We develop long-time asymptotic solutions of the governing equations which illustrate the interaction between thermal reaction fronts and depletion fronts. Two distinct regimes arise. If the compositional difference between the injection and formation fluids exceeds a critical value, the depletion front travels faster than the thermal front, leaving the porous medium depleted of reactant from the source to a point downstream of the thermal front and no thermal reaction front develops. Conversely, if the compositional difference is smaller than the critical value, a thermal reaction front advances ahead of the depletion front and so there is a double reaction front structure. We illustrate the evolution of the thermal and compositional fields towards these asymptotic solutions with numerical simulations. We discuss the implications of this work for secondary mineralization in subsurface reservoirs.

The Modeling of Unusual Rate Sensitivities Inside and Outside the Dynamic Strain Aging Regime

The viscoplasticity theory based on overstress (VBO) is altered by introducing an augmentation function into the dynamic recovery term of the growth law of the equilibrium stress, which is a tensor valued state variable of the model. It is a measure of the defect structure of metals and alloys. The flow law is unaltered but affected by the equilibrium stress whose growth is modified by the augmentation function. The augmentation function does not affect the initial, quasi-elastic region but exhibits a strong influence on the long-time, asymptotic solution that applies approximately when the flow stress is reached in a laboratory experiment. On the basis of the long-time asymptotic solution positive (stress increases with strain rate), zero (no influence of strain rate) and negative rate sensitivity (stress decreases with an increase in strain rate) can be easily modeled. In the simplest case the augmentation function is a constant and the rate sensitivity remains equal throughout the deformation. The modeling of a change from one kind of rate sensitivity, say negative, to another, say positive can be accomplished by making the augmentation function dependent on the accumulated, effective inelastic strain. The theory is applied to model the varying rate sensitivity of a modified 9Cr-1Mo steel, which changes from negative, to zero and to positive with temperature. The unusual rate sensitivities are modeled well together with relaxation and strain rate change tests.

The Algebraic Growth of Wavenumber One Disturbances in Hurricane-like Vortices

Abstract An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscid vortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptotics for this exact solution have been shown to allow an algebraic instability with unbounded growth even in the absence of exponentially growing modes. The necessary requirement for this instability is that there exist a local maximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturally occurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region. In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics of hurricanes. The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitude grows in time as t1/2, an excitation of the neutral pseudomode (vortex displacement) that is constant in time, and resi...

Adsorption kinetics of ionic surfactants with detailed account for the electrostatic interactions: effect of the added electrolyte

We address the problem about the electro-diffusion-controlled adsorption from a submicellar solution of an ionic surfactant in the presence of a non-amphiphilic electrolyte (salt). We consider a disturbance of the surfactant adsorption layer, which is damped by the electro-diffusive transport of surfactant ions, counterions and coions. The evolution of the electric double layer with time is theoretically examined. The counterion adsorption (binding) is taken into account. Our purpose is to find the long-time asymptotic solution of the full electro-diffusion boundary problem. Assuming small perturbations the problem is linearized and a set of three differential operator equations is obtained. In the asymptotic case of long times these differential equations reduce to hypergeometric equations, whose solutions, fortunately, can be expressed in terms of elementary functions. As a result, explicit expressions for the characteristic relaxation times of the adsorptions of surfactant ions and counterions are obtained. Next we determine the characteristic relaxation time of surface tension, which is different from the adsorption relaxation time because of a specific contribution from the dynamic diffuse electric double layer. The thermodynamic parameters, entering the formula for the relaxation time, are expressed for various surfactant adsorption isotherms, those of Henry, Langmuir, Freundlich, Frumkin and van der Waals. The theory gives a quantitative interpretation of the experimental finding that the addition of salt accelerates the relaxation of the surface tension for the higher surfactant concentrations, but decelerates it for the lower surfactant concentrations. The results could be also useful for a theoretical interpretation of the processes in foams and emulsions under dynamic conditions.

Transient thermal contact resistance

The short-time asymptotic solution for the heat flow between two similar bodies placed in contact has recently been given by Barber. The long-time asymptotic solution is given here: it has exactly the same form as the short-time asymptote Q KT 0 = C 0 + C 1 √(παt) but, in general, has different coefficients.

Adsorption from a well-stirred solution of finite volume: I. Linear adsorption kinetics through a stagnant boundary layer

A macroscopic transport model of adsorption kinetics in well-stirred systems has been developed. The resistance to interfacial mass transfer is accounted for by assuming a stagnant boundary layer—in which transport is by diffusion—separates the adsorbing surface from a well-stirred solution of changing concentration. To generalize the analysis, we consider (i) an adsorbing surface of spherical geometry, (ii) batch and constant-flow operating conditions, (iii) uniform and segregated starting conditions, and (iv) nonequilibrium surface mass action. Explicit series solutions as well as short- and long-time asymptotic solutions of the mass balance equations are given for the important case of linear adsorption kinetics. As illustrated here, the series expressions provide physical insight into the coupling which can occur among transport, adsorption, and constant-flow processes; while the asymptotic solutions provide information on the operating conditions which can lead to rate-limiting behavior. In addition, we show that under certain conditions the stagnant boundary layer model presented here reduces to previously reported linear adsorption models.

Nucleation on substrates from the vapour phase

Abstract Recent developments in the theory of nucleation of vapour deposits on crystalline substrates are reviewed. To facilitate comparison, the theories are formulated in a common dimensionless notation, and an examination of the major underlying assumptions reveals the basic similarity of many of them. The capture and dissociation rates are expressed in terms of the cluster geometry and the pertinent energy parameters, and from these the ‘chemical’ rate equations are set up. The following types of approximate solutions are discussed: (a) long-time asymptotic solutions, from which the conditions for saturation in the cluster concentration may be deduced, (b) a generalization of the type of approximate solution used by Logan (1969), and (c) numerical solutions employing a minimum number of simplifying assumptions. Based on a simple model, agreement between the latter two seems reasonably good. For any given set of fixed parameters (energies, geometrical constants, and arrival rate from the vapour) severa...