Radial Steady State Research Articles

Effects of resonant magnetic perturbations on neutral beam heating in a tokamak

The effects of resonant magnetic perturbations (RMPs) on tangential neutral beam heating in the EAST tokamak are studied numerically. RMPs with linear resistive magnetohydrodynamics response are used in the modeling. A variety of representing configurations of RMP coil currents are examined, and their effects on the neutral beam injection (NBI) heating efficiency are compared, in order to find a parameter window where deleterious effects of RMPs on NBI heating efficiency are minimized. It is found that the internal redistribution of fast ions by RMPs induces local accumulation of fast ions, resulting in higher local fast ion pressure than the case without RMPs. It is also found that the toroidal phasing of the RMP with respect to the fast ion source has slight effects on the steady-state radial profile of fast ions. The dependence of fast ion loss fraction on the RMP up-down phase difference shows a similar behavior as the dependence of the radial width of chaotic magnetic field on the phase difference. A statistical method of identifying resonances between RMPs and lost fast ions is proposed, and the results indicate that some resonances between RMPs and lost passing particles may be of non-integer fractional order, rather than the usual integer order.

Newtonian repulsion and radial confinement: Convergence toward steady state

We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions.

Uniqueness of entire ground states for the fractional plasma problem

We establish uniqueness of vanishing radially decreasing entire solutions, which we call ground states, to some semilinear fractional elliptic equations. In particular, we treat the fractional plasma equation and the supercritical power nonlinearity. As an application, we deduce uniqueness of radial steady states for nonlocal aggregation-diffusion equations of Keller-Segel type, even in the regime that is dominated by aggregation.

Steady-State Radial Flow Modeling through the Production Well in the Confined Aquifer of Monzoungoudo, Benin

This study aims to develop a mathematical analysis for one-dimensional modeling of a radial flow through a production well drilled in a confined aquifer, in the case of steady-state flow conditions. An analytical solution has derived from that expression for estimation of drawdowns according to different flowrates. Through that process, the evaluation of static pressure, the calculation of hydraulic charge due to the waterflow through the well is evaluated, the drawdowns curves are drawn and at last, the obtained curves are analyzed. The curves obtained for the different flow rates have an asymptotic direction, the axis of the hydraulic charges. The variation of the hydraulic charge depends on the radial distance for different flow rates. The P point, is a common point of all curves obtained for different production flowrates in the well. This point is where the well production flowrate is optimum for the optimal hydraulic charge.

Steady State Radial Flow in Anisotropic and Homogenous in Confined Aquifers

The purpose of this paper is to introduce the analysis of steady-state radial flow in an anisotropic and homogenous hydraulic property and discuss who we can estimate aquifer parameters from field pumping tests.

Vibration of a prestressed tube in the presence of plastic zone

The paper is devoted to a problem of radial steady state vibrations for a tube taking into account the inhomogeneous residual stress (RS) field. According to the theory of ideal plasticity the RS field is caused by elastoplastic deformation due to the internal pressure and the further unloading process. Various RS distributions inside both elastic and plastic zones are described. A finding of displacement function is reduced to the numerical solving of the first-order differential equations. Within the framework of the acoustical method, the inverse problem for RS field identification is stated. On the basis of the free radial vibration problem for tube, we derived the formula for a dependency of eigenfrequency on RS distribution by using the linearization method. By using this formula for two different values of eigenfrequencies, we state the system of two equations to identify the plasticity zone radius and RS level. Examples of numerical solution of this system for different values of parameters (RS level and tube thickness) are demonstrated. The estimation accuracy of eigenfrequencies measurement for successful implementation of the proposed approach is given.

Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

Abstract. An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model describing steady-state radial and vertical flows in a two-zone aquifer. Hydraulic parameters in these two zones can be different but are assumed homogeneous in each zone. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the aquifer domain in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the constant-flux pumping have good accuracy if satisfying the criterion.

On the stability of the positive radial steady states for a semilinear Cauchy problem

This paper is a contribution to the analysis of the Cauchy problem (0.1){∂u∂t=Δu+∑i=1kci|x|liupiinRn×(0,T),u(x,0)=φ(x)inRn, with initial function φ≢0. The stability and asymptotic stability of the positive radial steady states, which are positive solutions of (0.2)Δu+∑i=1kci|x|liupi=0, have been discussed with φ in weighted Banach spaces.

Influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport: A computational model

We present a theoretical model for steady-state radial and longitudinal oxygen transport in arterioles containing flowing blood (plasma and red blood cells) and surrounded by living tissue. This model combines a detailed description of convective and diffusive oxygen transport inside the arteriole with a novel boundary condition at the arteriolar lumen surface, and the results provide new mass transfer coefficients for computing arteriolar O 2 losses based on far-field tissue O 2 tension and in the presence of spatially distributed capillaries. A numerical procedure is introduced for calculating O 2 diffusion from an arteriole to a continuous capillary-tissue matrix immediately adjacent to the arteriole. The tissue O 2 consumption rate is assumed to be constant and capillaries act as either O 2 sources or sinks depending on the local O 2 environment. Using the model, O 2 saturation (SO 2) and tension (PO 2) are determined for the intraluminal region of the arteriole, as well as for the extraluminal region in the neighbouring tissue. Our model gives results that are consistent with available experimental data and previous intraluminal transport models, including appreciable radial decreases in intraluminal PO 2 for all vessel diameters considered (12–100 μm) and slower longitudinal decreases in PO 2 for larger vessels than for smaller ones, and predicts substantially less diffusion of O 2 from arteriolar blood than do models with PO 2 specified at the edge of the lumen. The dependence of the new mass transfer coefficients on vessel diameter, SO 2 and far-field PO 2 is calculated allowing their application to a wide range of physiological situations. This novel arteriolar O 2 transport model will be a vital component of future integrated models of microvascular regulation of O 2 supply to capillary beds and the tissue regions they support.

Classification of radial solutions arising in the study of thermal structures with thermal equilibrium or no flux at the boundary

The authors provide a complete classification of the radial solutions to a class of reaction diffusion equations arising in the study of thermal structures such as plasmas with thermal equilibrium or no flux at the boundary. In particular, their study includes rapidly growing nonlinearities, that is, those where an exponent exceeds the critical exponent. They describe the corresponding bifurcation diagrams and determine existence and uniqueness of ground states, which play a central role in characterizing those diagrams. They also provide information on the stability-unstability of the radial steady states.

Convergence of Anisotropically Decaying Solutions of a Supercritical Semilinear Heat Equation

We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence of solutions.

On the Stability of the Positive Radial Steady States for a Semilinear Cauchy Problem Involving Critical Exponents

This article is contributed to the Cauchy problem { ∂ u ∂ t = Δ u + K ( | x | ) u p in R n × ( 0 , T ) , u ( x , 0 ) = ϕ ( x ) in R n ; with initial function φ notequiv; 0. The stability of positive radial steady state, which are positive solutions of Δ u + K ( | x | ) u p = 0 , is obtained when p is critical for general K(|x|).

Analytic Solution to Steady-State Radial Diffusion of a Substrate with First-Order Reaction Kinetics in the Tissue of a Krogh's Cylinder

It is often useful to calculate the concentration profile for a substrate undergoing reaction in the tissue surrounding a capillary. In this paper, we consider a model geometry consisting of a long straight cylinder of tissue surrounding a capillary. Substrate diffuses radially out of the capillary through the tissue, with consumption of substrate in the tissue directly proportional to substrate concentration (i.e., first-order reaction kinetics). The model is extended to include the case where a cylinder of necrotic tissue surrounds a metabolically active inner tissue cylinder. A simple analytic solution is derived, and concentration profiles are generated for various combinations of parameters. Compared to the case where substrate consumption is independent of concentration, this model predicts much more rapid depletion of substrate near the capillary interface. This can have significant implications for the calculation of the hypoxic fraction (e.g., tissue with pO(2)<0.5-5 mmHg) when tumor oxygenation is modeled. The model also permits calculation of the limiting substrate concentration for cell viability when the reaction rate constant is known and vice versa.

Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations

We first study the initial value problem for a general semilinear heat equation. We prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low ( n ⩽ 10 ) or if the steady state is flat enough at infinity: the solution of the heat equation either becomes unbounded as t approaches the lifespan, or eventually stays above or below another bounded radial steady state, depending on if the initial value is above or below the first steady state; moreover, the second steady state must be a constant if n ⩽ 10 . Using this instability result, we then prove that every nonconstant radial steady state of the generalized Fisher equation is a hair-trigger for two kinds of dynamical behavior: extinction and spreading. We also prove more criteria on initial values for these types of behavior. Similar results for a reaction–diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained.

On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem

This paper is contributed to the Cauchy problem { ∂ u ∂ t = Δ u + K ( | x | ) u p + μ f ( | x | ) in R n × ( 0 , T ) , u ( x , 0 ) = φ ( x ) in R n , with initial function φ ≢ 0 . The monotonicity and stability of the positive radial steady states, which are positive solutions of Δ u + K ( | x | ) u p + μ f ( | x | ) = 0 , are discussed with K ( | x | ) , f ( | x | ) and φ in weighted Banach spaces.

Stability of steady states of the Cauchy problem for the exponential reaction–diffusion equation

We consider the Cauchy problem { u t = Δ u + e u , x ∈ R N , t ∈ ( 0 , T ) , u ( x , 0 ) = u 0 , x ∈ R N , where u 0 ∈ C ( R N ) and T > 0 . We first study the radial steady states of the equation and the number of intersections distinguishing four different cases: N = 1 , N = 2 , 3 ⩽ N ⩽ 9 and N ⩾ 10 , writing explicitly every steady state for N = 1 and N = 2 . Then we study the large time behavior of solutions of the parabolic problem.

On the stability of the positive radial steady states for a semilinear Cauchy problem

This paper is contributed to the Cauchy problem (0.1) ∂u ∂t = Δu+K(|x|)u p in R n×(0,T), u(x,0)=ϕ(x) in R n with initial function ϕ≢0. The stability and instability of the positive radial steady states, which are positive solutions of (0.2) Δu+K(|x|)u p=0, has been discussed with different assumption on K(| x|) and ϕ under the norm (0.3) ∥ψ∥ λ= sup x∈ R n |(1+|x| λ)ψ(x)|, where ϕ and ψ are some nonnegative continuous functions in R n , and λ is a real number.

One-dimensional Steady-state Radial Swirling Gas Flow

One-dimensional Steady-state Radial Swirling Gas Flow

Cryosurgery of normal and tumor tissue in the dorsal skin flap chamber: Part I--thermal response.

Current research in cryosurgery is concerned with finding a thermal history that will definitively destroy tissue. In this study, we measured and predicted the thermal history obtained during freezing and thawing in a cryosurgical model. This thermal history was then compared to the injury observed in the tissue of the same cryosurgical model (reported in companion paper (Hoffmann and Bischof, 2001)). The dorsal skin flap chamber, implanted in the Copenhagen rat, was chosen as the cryosurgical model. Cryosurgery was performed in the chamber on either normal skin or tumor tissue propagatedfrom an AT-1 Dunning rat prostate tumor. The freezing was performed by placing a approximately 1 mm diameter liquid-nitrogen-cooled cryoprobe in the center of the chamber and activating it for approximately 1 minute, followed by a passive thaw. This created a 4.2 mm radius iceball. Thermocouples were placed in the tissue around the probe at three locations (r = 2, 3, and 3.8 mm from the center of the window) in order to monitor the thermal history produced in the tissue. The conduction error introduced by the presence of the thermocouples was investigated using an in vitro simulation of the in vivo case and found to be <10 degrees C for all cases. The corrected temperature measurements were used to investigate the validity of two models of freezing behavior within the iceball. The first model used to approximate the freezing and thawing behavior within the DSFC was a two-dimensional transient axisymmetric numerical solution using an enthalpy method and incorporating heating due to blood flow. The second model was a one-dimensional radial steady state analytical solution without blood flow. The models used constant thermal properties for the unfrozen region, and temperature-dependent thermal properties for the frozen region. The two-dimensional transient model presented here is one of the first attempts to model both the freezing and thawing of cryosurgery. The ability of the model to calculate freezing appeared to be superior to the ability to calculate thawing. After demonstrating that the two-dimensional model sufficiently captured the freezing and thawing parameters recorded by the thermocouples, it was used to estimate the thermal history throughout the iceball. This model was used as a basis to compare thermal history to injury assessment (reported in companion paper (Hoffmann and Bischof, 2001)).

Time-related capture zones for radial flow in two dimensional randomly heterogeneous media

We consider the effect of randomly heterogeneous hydraulic conductivity on the spatial location of time-related capture zones (isochrones) for a non-reactive tracer in the steady-state radial flow field due to a pumping well in a confined aquifer. A Monte Carlo (MC) procedure is used in conjunction with FFT-based spectral methods. The log hydraulic conductivity field is assumed to be Gaussian and stationary, with isotropic exponential correlation. Various degrees of domain heterogeneity are considered and stability and accuracy of the MC procedure is examined. The location of an isochrone becomes uncertain due to heterogeneity, and it is strongly influenced by hydraulic conductivity variance. The probability that a particle released at a point in the aquifer is pumped by the well within a given time is identified. We propose a new expression for the probabilistic spatial distribution of isochrones, which is formally similar to the analytical solution for a uniform medium and takes into account the effects of heterogeneity.