Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form u _t–u=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form u/n _a=g(u(t), u(t–r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.     

Diffusion of moisture in deformable porous media. Anisotropic fibrous materials

This paper presents a study on the deformation of anisotropic fibrous porous media subjected to moistening by water in the liquid phase. The deformation of the medium is studied by applying the concept of effective stress. Given the structure of the medium, the displacement of the solid matrix is not taken into account with respect to the displacement of the liquid phase. The transport equations are derived from the model proposed by Narasimhan. The transport coefficients and the relation between the variation in apparent density and effective stress are obtained by test measurements. A numerical model has been established and applied for studying drip moistening of mineral wool samples capable or incapable of deformation.Nomenclature D mass diffusion coefficient [L²t^–1] - e void fraction - g gravity acceleration [Lt^–2] - J mass transfer density [ML^–2t^–1] - K hydraulic conductivity [Lt^–1] - K _s hydraulic conductivity of the solid phase [Lt^–1] - K ^* hydraulic conductivity of the deformable porous medium [Lt^–1] - P pressure of moistening liquid [ML^–1 t^–2] - S degree of saturation - t time [t] - V speed [Lt^–1] - X horizontal coordinate [L] - Z vertical coordinate measured from the bottom of porous medium [L] - z z-coordinate [L] Greek Letters porosity - ₁ total hydric potential [L] - _g gas density [ML^–3] - ₁ liquid density [ML^–3] - ₀ apparent density [ML^–3] - _s density of the solid phase [ML^–3] - density of the moist porous medium [ML^–3] - external load [ML^–1t^–2] - effective stress [ML^–1t^–2] - bishop's parameter - matrix potential or capillary suction [L] Indices g gas - 1 moistening liquid - p direction perpendicular to fiber planes - s solid matrix - t direction parallel to fiber planes - v pore Exponent * movement of solid particles taken into account     

Convergence to equilibrium for delay-diffusion equations with small delay

It is shown that for scalar dissipative delay-diffusion equationsu _t–u=f(u(t),u(t–)) with a small delay, all solutions are asymptotic to the set of equilibria ast tends to infinity.     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Shear stress and normal stress measurements of aqueous polymer solutions in a cone-and-plate rheogoniometer

Summary The rheological behaviour of aqueous solutions of Separan AP-30 and Polyox WSR-301 in a concentration range of 10–10000 wppm is investigated by means of a cone-and-plate rheogoniometer. The relation between the shear stress and the shear rate is for lower shear rates characterized by a timet ₀, which is concentration dependent. Both polymers show for 4000 s^–1 < < 10000 s^–1 a behaviour similar to that of a Bingham material, characterized by a dynamic viscosity ₀ and an apparent yield stress ₀, which also depend on the concentration. The inertial forces are measured for water and some other Newtonian liquids. An explanation is given why the theoretical model developed for these forces does not match the experimental values; the shape of the liquid surface is shear rate dependent. To obtain the first normal stress difference, we have to correct for these inertial forces, the surface tension and the buoyancy. The normal forces, measured for Separan AP-30, appear to be a linear function of the shear rate for 350 s^–1 < < 3300 s^–1.

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Zusammenfassung Das rheologische Verhalten wäßriger Polymerlösungen von Separan AP-30 und Polyox WSR-301 wird in einem Konzentrationsgebiet von 10–10000 wppm in einem Kegel-Platte-Rheogoniometer untersucht. Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit wird für niedrige Schergeschwindigkeiten durch eine konzentrationsabhängige Zeitt ₀ gekennzeichnet. Für Schergeschwindigkeiten 4000 s^–1 < < 10000 s^–1 zeigen beide Polymere ein genähert binghamsches Verhalten, gekennzeichnet durch eine dynamische Viskosität ₀ und eine scheinbare Fließgrenze ₀, welche ebenfalls konzentrationsabhängig sind. Die Trägheitskräfte werden für Wasser und einige newtonsche Öle bestimmt. Die Abweichung der experimentellen Ergebnisse vom theoretischen Modell wird durch die Abhängigkeit der Gestalt der Flüssigkeitsoberfläche von der Schergeschwindigkeit erklärt. Um die Werte der ersten Normalspannungsdifferenz zu erhalten, muß man bezüglich der Trägheitskräfte, der Oberflächenspannung und der Auftriebskräfte korrigieren. Die Normalspannungen für Separan AP-30, gemessen für 350 s^–1 < < 3300 s^–1, zeigen eine lineare Abhängigkeit von der Schergeschwindigkeit.

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c concentration (wppm) - g acceleration of gravity (ms^–2) - K force (N) - K _b buoyant force (N) - K _c force, acting on the cone (N) - K ₀ dimensional constant def. by eq. [24] (N) - K _s force, def. by eq. [22] (N) - M dimensional constant def. by eq. [24] (Ns) - P _s pressure def. by eq. [17] (Nm^–2) - P ₀ average pressure in the liquid atr = 0 (Nm^–2) - P _R average pressure in the liquid atr = R (Nm^–2) - r ₁,r ₂ radii of curved liquid surface (m) - R platen radius (m) - R _w radius of wetted platen area (m) - S _x standard deviation ofx - t ₀ characteristic time def. by eq. [1] (s) - T temperature (°C) - V volume of the submerged part of the cone (m³) - v tangential velocity of liquid (ms^–1) - x distance (m) - angle (rad) - ₀ cone angle (rad) - calibration constant (Nm^–3) - shear rate (s^–1) - dynamic viscosity (mPa · s) - ₀ viscosity def. by eq. [1] (mPa · s) - contact angle (rad) - density (kgm^–3) - static surface tension (Nm^–1) - shear stress (Nm^–2) - ₀ yield stress def. by eq. [1] (Nm^–2) - _c, _p angular velocity (c = cone,p = plate) (s^–1) With 8 figures and 3 tables     

About the Regularized Navier–Stokes Equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier–Stokes system. The Marcinkiewicz space L^3, is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical regularized Navier–Stokes systems. The first one was introduced by J. Leray and consists in mollifying the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (–)^{/ 2}, > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t toward solutions of the original Navier–Stokes system.     

Contaminant Transport in Fractured Media with Sources in the Porous Domain

We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu _t = u _yy – u + f(x,y,t),x,y>0, t>0, u _t = –u _x + u _y – u on y = 0; u(0,0,t) =u₀(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Algebraic-Delay Differential Systems, State-Dependent Delay, and Temporal Order of Reactions

Systems of the form

The asymptotic theory of semilinear perturbed telegraph equation and its application

I.IntroductionConsiderthefollowingsemilinearperturbedtelegraphequationuII-u., P'u==sj(t,x,u,ul,u,,e)(-ooo)(l.l)u(o,x)=u,(x,e)(-ooo,u=u(t,x),fuoandulsatisfycertainsmoo…