Annular Domain Research Articles

Analogs of the Globevnik problem on Riemannian two-point homogeneous spaces

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.

Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients

We consider the Navier–Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature. A global-in-time, spherically or cylindrically symmetric, classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1. These results are of Nishida–Smoller type and extend the work (Liu et al. (2014) [16]) restricted to the one-dimensional flows.

Optimal convection cooling flows in general 2D geometries

We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.

On surfaces that are intrinsically surfaces of revolution

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.

Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains

The purpose of this paper is to investigate positive solutions of integral equations involving Bessel potential. Exploiting the moving plane method in integral form, we give the radial symmetry of both the domain and solutions of our integral equations in exterior domains and in annular domains respectively.

On the onset of instabilities in a Bénard-Marangoni problem in an annular domain with temperature gradient

This manuscript addresses the linear stability analysis of a thermoconvective problem in an annular domain. The flow is heated from below, with a linear decreasing horizontal temperature profile from the inner to the outer wall. The top surface of the domain is open to the atmosphere and the two lateral walls are adiabatic. The effects of several parameters in the flow are evaluated. Three different values for the ratio of the momentum diffusivity and thermal diffusivity are considered: relatively low Prandtl number (Pr = 1), intermediate Prandtl number (Pr = 5) and high Prandtl number (ideally Pr ? ? , namely Pr = 50). The thermal boundary condition on the top surface is changed by imposing different values of the Biot number, Bi. The influence of the aspect ratio (?) is assessed for through by studying several aspect ratios, ?. The study has been performed for two values of the Bond number (namely Bo = 5 and 50), estimating the perturbation given by thermocapillarity effects on buoyancy effects. Different kinds of competing solutions appear on localized zones of the ?-Bi plane. The boundaries of these zones are made up of co-dimension two points. Co-dimension two points are found to be function of Bond number, Marangoni number and boundary conditions but to be independent on the Prandtl number.

ON BERNOULLI'S FREE BOUNDARY PROBLEM WITH A RANDOM BOUNDARY

This article is dedicated to the solution of Bernoulli’s exterior free boundary problem in the situation of a random interior boundary. We provide the theoretical background that ensures the well-posedness of the problem under consideration and describe two different frameworks to define the expectation and the deviation of the resulting annular domain. The first approach is based on the Vorob’ev expectation, which can be defined for arbitrary sets. The second approach is based on the particular parametrization. In order to compare these approaches, we present analytical examples for the case of a circular interior boundary. Additionally, numerical experiments are performed for more general geometric configurations. For the numerical approximation of the expectation and the deviation, we propose a sampling method like the Monte Carlo or the quasi-Monte Carlo quadrature. Each particular realization of the free boundary is then computed by the trial method, which is a fixed-point-like iteration for the solution of Bernoulli’s free boundary problem.

Constructing solutions for a kinetic model of angiogenesis in annular domains

We present an iterative technique to construct stable solutions for an angiogenesis model set in an annular region. Branching, anastomosis and extension of blood vessel tips is described by an integrodifferential kinetic equation of Fokker–Planck type supplemented with nonlocal boundary conditions and coupled to a diffusion problem with Neumann boundary conditions through the force field created by the tumor induced angiogenic factor and the flux of vessel tips. Convergence proofs exploit balance equations, estimates of velocity decay and compactness results for kinetic operators, combined with gradient estimates of heat kernels for Neumann problems in non convex domains.

Conformal Mapping for the Efficient Solution of Poisson Problems with the Kansa-RBF Method

We consider the solution of Poisson Dirichlet problems in simply-connected irregular domains. These domains are conformally mapped onto the unit disk and the resulting Poisson Dirichlet problems are solved efficiently using a Kansa-radial basis function (RBF) method with a matrix decomposition algorithm (MDA). In a similar way, we treat Poisson Dirichlet and Poisson Dirichlet---Neumann problems in doubly-connected domains. These domains are mapped onto annular domains by a conformal mapping and the resulting Poisson Dirichlet and Poisson Dirichlet---Neumann problems are solved efficiently using a Kansa-RBF MDA. Several examples demonstrating the applicability of the proposed technique are presented.

Solving Helmholtz equation at high wave numbers in exterior domains

This paper presents effective difference schemes for solving the Helmholtz equation at high wave numbers. Considering the problems in the polar and spherical coordinates, we show that pollution-free difference schemes can be constructed for annulus and hollow sphere domains, but the “pollution effect” cannot be avoided in the neighborhood of the origin of the coordinate. Using the fact that the “pollution effect” appears in a small neighborhood of the origin of the coordinate, we propose a local mesh refinement approach so that a fine grid is applied only for a small region near the origin of the coordinate and a much larger grid size is employed in the remaining domain. The most attractive feature of this approach is that the computational storage can be significantly reduced while keeping almost the same numerical accuracy. Moreover, by applying a two-level local refinement technique, we can further enhance the performance of the proposed scheme. Numerical simulations are reported to verify the effectiveness of the proposed schemes for the Helmholtz problem in exterior domains.

Synthesis of Constant Torque Compliant Mechanisms

Constant torque compliant mechanisms produce an output torque that does not change in a large range of input rotation. They have wide applications in aerospace, automobile, timing, gardening, medical, and healthcare devices. Unlike constant force compliant mechanisms, the synthesis of constant torque compliant mechanisms has not been extensively investigated yet. In this paper, a method is presented for synthesizing constant torque compliant mechanisms that have coaxial input rotation and output torque. The same shaft is employed for both input rotation and output torque. A synthesized constant torque compliant mechanism is modeled as a set of variable width spline curves within an annular design domain formed between a rotation shaft and a fixed ring. Interpolation circles are used to define variable width spline curves. The synthesis of constant torque compliant mechanisms is systematized as optimizing the control parameters of the interpolation circles of the variable width spline curves. The presented method is demonstrated by the synthesis of constant torque compliant mechanisms that have different number of variable width spline curves in this paper.

Quasi-neutral limit for the Euler–Poisson system in the presence of plasma sheaths with spherical symmetry

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions for the Euler–Poisson equations in a three-dimensional annular domain. We first propose a suitable condition on the velocity at the sheath edge, referred as to Bohm criterion for the annulus, and under this condition together with the constant Dirichlet boundary conditions for the potential, we show that there exists a unique stationary spherical symmetric solution. Moreover, we study the quasi-neutral limit behavior by establishing [Formula: see text] estimate of the difference of the solutions to the Euler–Poisson equations and its quasi-neutral limiting equations, incorporated with the correctors for the boundary layers. The quasi-neutral limit analysis employing the correctors and their pointwise estimates enables us to obtain detailed asymptotic behaviors including the convergence rates in [Formula: see text] and [Formula: see text] norms as well as the thickness of the boundary layers as a consequence of the pointwise estimates.

Inverse Cauchy problem of annulus domains in the framework of spectral meshless radial point interpolation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the Cauchy problems of two-dimensional elliptic PDEs in annulus domains. Unknown data on the inner boundary are obtained while overspecified boundary data are imposed on the outer boundary using the SMRPI. The SMRPI employs monomials and radial basis functions (RBFs) through interpolation and applies them locally with the help of spectral collocation ideas. In this way, localization in SMRPI can reduce the ill-conditioning for Cauchy problem. Furthermore, we improve previous results and overcome the ill-conditioning of Cauchy problem. It is revealed the SMRPI is more accurate and stable by adding strong perturbations.

On radial solutions for singular combined superlinear elliptic systems on annular domains

We prove the existence of a large positive solution to the system{−(rN−1ϕ1(u′))′=λrN−1f1(v),a<r<b,−(rN−1ϕ2(v′))′=λrN−1f2(u),a<r<b,u(a)=0=u(b),v(a)=0=v(b), where a>0, λ is a small positive parameter, fi:(0,∞)→R are continuous and limz→∞⁡ϕ1−1(f1(c(ϕ2−1(f2(z)))z=∞ for all c>0.

An analytic solution to the coupled pressure–temperature equations for modeling of photoacoustic trace gas sensors

Trace gas sensors have a wide range of applications including air quality monitoring, industrial process control, and medical diagnosis via breath biomarkers. Quartz-enhanced photoacoustic spectroscopy and resonant optothermoacoustic detection are two techniques with several promising advantages. Both methods use a quartz tuning fork and modulated laser source to detect trace gases. To date, these complementary methods have been modeled independently and have not accounted for the damping of the tuning fork in a principled manner. In this paper, we discuss a coupled system of equations derived by Morse and Ingard for the pressure, temperature, and velocity of a fluid, which accounts for both thermal effects and viscous damping, and which can be used to model both types of trace gas sensors simultaneously. As a first step toward the development of a more realistic model of these trace gas sensors, we derive an analytic solution to a pressure–temperature subsystem of the Morse–Ingard equations in the special case of cylindrical symmetry. We solve for the pressure and temperature in an infinitely long cylindrical fluid domain with a source function given by a constant-width Gaussian beam that is aligned with the axis of the cylinder. In addition, we surround this cylinder with an infinitely long annular solid domain, and we couple the pressure and temperature in the fluid domain to the temperature in the solid. We show that the temperature in the solid near the fluid–solid interface can be at least an order of magnitude larger than that computed using a simpler model in which the temperature in the fluid is governed by the heat equation rather than by the Morse–Ingard equations. In addition, we verify that the temperature solution of the coupled system exhibits a thermal boundary layer. These results strongly suggest that for computational modeling of resonant optothermoacoustic detection sensors, the temperature in the fluid should be computed by solving the Morse–Ingard equations rather than the heat equation.

Existence and non-existence results for minimizers of the Ginzburg–Landau energy with prescribed degrees

Let [Formula: see text] be a smooth annular type domain. We consider the simplified Ginzburg–Landau energy [Formula: see text], where [Formula: see text], and look for minimizers of [Formula: see text] with prescribed degrees [Formula: see text], [Formula: see text] on the boundaries of the domain. For large [Formula: see text] and for balanced degrees (i.e. [Formula: see text]), we obtain existence of minimizers for domains with large capacity (corresponding to thin annulus). We also prove non-existence of minimizers of [Formula: see text], for large [Formula: see text], if [Formula: see text], [Formula: see text] and if [Formula: see text] is a circular annulus with large capacity. Our approach relies on similar results obtained for the Dirichlet energy [Formula: see text], on a previous existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.

Asymptotically radial solutions to an elliptic problem on expanding annular domains in Riemannian manifolds with radial symmetry

We consider the boundary value problem $$\left \{ \textstyle\begin{array}{l} \Delta_{\mathbf{g}} u +u^{p}= 0 \quad\mbox{in } \Omega_{R}, \\ u=0 \quad \mbox{on } \partial\Omega_{R}, \end{array}\displaystyle \right . $$ $\Omega_{R}$ being a smooth bounded domain diffeomorphic to the expanding domain $A_{R}:=\{x \in M, R< r(x)< R+1\}$ in a Riemannian manifold M of dimension $n \geq2$ endowed with the metric ${\mathbf{g}}=dr^{2}+S^{2}(r)g_{{\mathbb{S}}^{n-1}}$ . After recalling a result about existence, uniqueness, and non-degeneracy of the positive radial solution when $\Omega_{R}=A_{R}$ , we prove that there exists a positive non-radial solution to the aforementioned problem on the domain $\Omega_{R}$ . Such a solution is close to the radial solution to the corresponding problem on $A_{R}$ .

The vacant set of two-dimensional critical random interlacement is infinite

For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s.\ infinite, thus solving an open problem from arXiv:1502.03470. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion

The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet problem for Fast Diffusion Equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Lojasiewicz-Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Lojasiewicz-Simon inequality in a different way.

Elongated bubble shape in inclined air-water slug flow

The shape of elongated bubbles in upward inclined air-water slug flow was studied experimentally by quantitative measurements of the cross sectional distribution of the phases within the pipe, using a wire mesh sensor. Ensemble-averaged shapes of elongated bubbles were determined for a wide range of gas and liquid flow rates, as well as for different pipe inclination angles. The elongated bubble nose can be characterized by an annular domain where liquid is present above the gas. The effect of gas and liquid flow rates, as well as of the pipe inclination angle on the bubble shape (front and tail) is studied. A simplified theoretical model is proposed to determine the bubble front shape. The model predictions compare favorably with the experimental results.