Cylindrical Domain Research Articles

Front propagation and blocking of time periodic bistable reaction-diffusion equations in cylindrical domains

Front propagation and blocking of time periodic bistable reaction-diffusion equations in cylindrical domains

Nonlinear three-dimensional stability characteristics of geometrically imperfect nanoshells under axial compression and surface residual stress

Abstract Through reduction of thickness value in nanostructures, the features of surface elasticity become more prominent due to having a high surface-to-volume ratio. The main aim of this research work was to examine the surface residual stress effect on the three-dimensional nonlinear stability characteristics of geometrically perfect and imperfect cylindrical shells at nanoscale under axial compression. To do so, an unconventional three-dimensional shell model was established via combination of the three-dimensional shell formulations and the Gurtin–Murdoch theory of elasticity. The silicon material is selected as a case study, which is the most utilized material in the design of micro-electromechanically systems. Then, the moving Kriging meshfree approach was applied to take numerically into account the surface free energy effects and the initial geometrical imperfection in the three-dimensional nonlinear stability curves. Accordingly, the considered cylindrical shell domain was discretized via a set of nodes together using the quadratic polynomial type of basis shape functions and an appropriate correlation function. It was found that the surface stress effects lead to an increase the critical axial buckling load of a perfect silicon nanoshell about 82.4 % 82.4 \% for the shell thickness of 2 nm 2{\rm{nm}} , about 32.4 % 32.4 \% for the shell thickness of 5 nm 5{\rm{nm}} , about 15.8 % 15.8 \% for the shell thickness of 10 nm 10{\rm{nm}} , and about 7.5 % 7.5 \% for the shell thickness of 20 nm 20{\rm{nm}} . These enhancements in the value of the critical axial buckling load for a geometrically imperfect silicon nanoshell become about 92.9 % 92.9 \% for the shell thickness of 2 nm 2{\rm{nm}} , about 36.5 % 36.5 \% for the shell thickness of 5 nm 5{\rm{nm}} , about 17.7 % 17.7 \% for the shell thickness of 10 nm 10{\rm{nm}} , and about 8.8 % 8.8 \% for the shell thickness of 20 nm 20{\rm{nm}} .

Gyroscopic polynomials

Gyroscopic alignment gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical discretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and generalized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a relatively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differentiation operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment.

On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition

Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.

A direct numerical simulation study for confined non-isothermal jet impingement at moderate nozzle-to-plate distances: Capturing jet-to-ambient density effects

A direct numerical simulation (DNS) campaign is deployed for a series of confined downward oriented, non-isothermal turbulent impinging jet configurations. A baseline Reynolds number of 9960 is obtained through a precursor DNS pipe flow simulation (Reτ=505). Three jet temperature configurations (confinement height to nozzle diameter of three) enter a cylindrical domain that share ambient and impingement plate temperatures (298.15K). The range of jet temperatures are crafted such that the ratio of inlet to ambient density varies from unity to 0.52, showcasing the effect of density disparity on flow characteristics such as core collapse, radial mixing of momentum and energy, near-wall stagnation behavior, wall-jet profiles, and large-scale vortical structures. Surface quantities provided include mean radial heat flux and wall-shear stress profiles, and heat flux histograms at select radial stations. Results showcase increased radial normal stresses for higher temperature jets that support increased mixing, resulting in large-scale recirculation structures that are smaller, while retaining similar normalized radial wall profiles for shear stress, heat flux and pressure. Radial plots for wall shear stress and Nusselt number showcase strong radial decay as compared to previous configurations that share similar jet and ambient temperatures. For the 373.15 K case, a Gaussian-like histogram for heat fluxes at the impingement plate transitions to a log-normal profile as radial distances increase. In contrast, the 573.15 K configuration displays a bi-modal heat flux characteristic at the impingement plate, and in similar manner to the moderate temperature counterpart, transitions to a log-normal profile at larger radial distances.

Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation

Abstract In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named “adaptive Runge–Kutta regularization”, which does not require any penalization term. Finally, we compare them numerically.

Periodic solutions for the Blackstock–Crighton–Westervelt equation

We investigate the Blackstock–Crighton–Westervelt equation which models nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We prove existence and regularity, in a Lp−Lq setting, of time-periodic solutions for a given sufficiently small time-periodic forcing data, and homogeneous Dirichlet boundary conditions over a cylindrical domain. We show maximal Lp-regularity for the abstract linearized model. We use techniques of operator-valued Fourier multiplier theorems combined with a generalized version of the implicit function theorem.

On a Mathematical Model of Melting Metal–Polymer Composites for Injection Molding in a Cylindrical Domain

On a Mathematical Model of Melting Metal–Polymer Composites for Injection Molding in a Cylindrical Domain

Characterizing Swirl Strength and Recirculation Zone Formation in Tangentially Injected Isothermal Flows

This paper presents a computational study characterizing the swirl intensity distribution and Internal Recirculation Zone (IRZ) formed in a cylindrical domain with tangential injections and isothermal flow. The range of inlet boundary conditions investigated is 5o to 25o for the injection angle and 7190 to 100711 for the bulk flow Reynolds number. The evolution of swirl intensity is presented with and without incorporating effects of the accompanying pressure variations. The Shear Stress Transport (SST) k-ω is used to model turbulence. Results show that the Swirl strength created by such tangential injections strongly depends on the injection angle but does not vary with bulk flow Reynolds number (Re), except for low Re values. The swirling flow is shown to result in IRZ formation at injection angles 6o and above or when asymptotic value of the maximum Swirl Number in the domain exceeds approximately 0.6, same as the transition value of inlet Swirl Number in swirling flows with axial injections. The IRZ length increases with injection angle and varies with Re for lower values of Re at a given injection angle but asymptotes for higher values above 40000. The conventional Swirl Number rises rapidly downstream of the injection plane followed by a slow decline. On the other hand, an alternative Swirl Number, which incorporates the gauge pressure variation, shows slow and consistent decay all the way downstream of the injection plane. The Swirl Number incorporated with gauge pressure term subsumes interconversions between the axial momentum and pressure in the regions of vortex breakdown and IRZ formation, thereby presenting an alternative picture of swirl intensity evolution in swirling flows.

Rayleigh–Faber–Krahn, Lyapunov and Hartmann–Wintner Inequalities for Fractional Elliptic Problems

In this paper, in the cylindrical domain, we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh–Faber–Krahn inequality. Also, we give Lyapunov and Hartmann–Wintner inequalities for the fractional elliptic boundary value problem.

Kinetic Pathway of the Order–Order Transition from Hexagonally Packed Cylinder to Hexagonally Perforated Layer in Polystyrene-block-Poly(2-vinylpyridine) Using Time-Resolved 3D Transmission Electron Microtomography

The kinetic pathway of the order–order transition (OOT) from a hexagonally packed cylinder (HEX) to a hexagonally perforated layer (HPL) phase in polystyrene-block-poly(2-vinylpyridine) (PS-b-P2VP) was studied using time-resolved three-dimensional transmission electron microtomography (3D-TEM). Our results revealed that this OOT proceeds via the nucleation and growth process. During the nucleation stage, the cylindrical domains transition from an initial metastable state at random positions owing to thermal fluctuation. Adjacent undulating cylindrical domains merge into wave-like structures to form perforations in hexagonal packing and act as nuclei. During the growth stage, after each nucleus exceeds the critical size, the neighboring intermediates coalesce into the HPL phase with AB-type stacking (P63/mmc space group). The coherency of the HPL orientation in the grain increased during OOT. Understanding the OOT kinetic pathway is useful for the morphological evolution of block copolymers and for designing suitable transition pathways to obtain ordered structures for diverse nanotechnology applications.

Well-Posedness of the Generalized Samarskii–Ionkin Problem for Elliptic Equations in a Cylindrical Domain

Well-Posedness of the Generalized Samarskii–Ionkin Problem for Elliptic Equations in a Cylindrical Domain

The variational approach to s-fractional heat flows and the limit cases s → 0+ and s → 1−

This paper deals with the limit cases for s-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as s→0+ and s→1−.We describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the L2 metric. To this end, we first provide a Γ-convergence analysis for the s-Gagliardo seminorms as s→0+ and s→1−; then, we exploit an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of Γ-converging uniformly λ-convex energy functionals.We prove that s-fractional heat flows (suitably scaled in time) converge to the standard heat flow as s→1−, and to a degenerate ODE type flow as s→0+. Moreover, looking at the next order term in the asymptotic expansion of the s-fractional Gagliardo seminorm, we show that suitably forced s-fractional heat flows converge, as s→0+, to the parabolic flow of an energy functional that can be seen as a sort of renormalized 0-Gagliardo seminorm: the resulting parabolic equation involves the first variation of such an energy, that can be understood as a zero (or logarithmic) Laplacian.

Advance thermal and mass transports for the fractional flows through a cylindrical domain: a Prabhakar-like generalization

In this study, the flow of a viscous fluid through an infinite vertical cylinder is discussed. The respective flow is induced by pressure gradient and with the contribution of temperature and concentration gradients. The flow modeling is done with the generalized heat and molecular fluxes by utilizing the Prabhakar-like fractional derivative. The fractional partial differential equations which govern the flow model are reduced to the dimensionless form to project the deep understanding of the flow model and then solved analytically by integral transforms namely Hankel and Laplace transforms. The physical aspects of flow regarding involved parameters especially fractional parameter are discussed by sketching the graphs of field variables for the concerned parametric variations.

Stochastic primitive equations with horizontal viscosity and diffusivity

We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain M=(−h,0)×G, G⊂R2 bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in L2, more regular initial data are necessary to establish uniqueness in the anisotropic space Hz1Lxy2 so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.

Nonlocal dispersal equations in domains becoming unbounded

<p style='text-indent:20px;'>This paper is concerned with a class of nonlocal dispersal equation in asymptotically cylindrical domain. Our aim is to investigate the asymptotic behavior of positive solutions in domain becoming infinite in some direction. We prove that the limiting behavior of positive solutions is only determined by the bounded part of whole domain.</p>

Carbon-dioxide Emission Analysis in a Convective Cylindrical Pipe

This article analyzed carbon dioxide (CO2) emission from the combustion of reactive materials modeled in a cylindrical domain. Reactive materials in this case involve carbon-containing substances that react spontaneously with the oxygen of the surrounding environment under the influence of an exothermic chemical reaction. In this analysis, the reactant (oxygen) consumption was neglected. The nonlinear differential equation governing the problem was solved numerically using the Finite Difference Method embedded within the Maple software. It was found that there are kinetic parameters that enhance the emission of CO2, like the rate of reaction, and others, like the heat loss parameter, retard the CO2 emission during the exothermic chemical reaction.

Начально-краевая задача о течении в кровеносном сосуде

The solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed. It is assumed that the side surface of the domain is a free (soft, compliant) boundary on which the kinematic condition is set. The dynamic condition at the boundary is modeled by some constitutive relation – the dependence of the pressure P on the cross-sectional area S of the cylindrical domain. To study the flow an asymptotic model based on the shallow water theory is used. When constructing the solution of the initial-boundary (as well as initial and boundary) problem for a system of two quasilinear partial differential equations of the first order, the hodograph method based on the conservation law is used. A variant of the polynomial constitutive relation is chosen for the study: P~S2β (β>0). In the case of the initial data problem which specified at the initial time the Riemann-Green function, allowing to construct an implicit analytical solution, and an algorithm (numerical) for constructing an explicit solution of the original problem are given. The main attention is focused on the special case P~S2, for which all the formulas necessary to construct a solution are written in explicit form. For P~S2, several variants of conditions (initial and boundary value) are considered for the initial boundary value problem, which allow us to trace the evolution of the solution in detail. Numerical calculations demonstrating the motion of shock waves and wave fronts are given. The results obtained, in practice, can be used to describe flows in blood vessels, as well as reliable tests for testing compu tational algorithms intended to solve such problems, in particular, systems of quasi-linear hyperbolic equations.

On fractional Poincaré inequality for unbounded domains with finite ball conditions: Counter example

In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Mancini and Sandeep (2010) showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain satisfies finite ball condition. We prove that such a result cannot be true in the ‘nonlocal/fractional’ setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all eigenvalues of fractional Dirichlet problems on long cylindrical domains is addressed.

Comparison of spherical harmonics method and discrete ordinates method for radiative transfer in a turbulent jet flame

In this study, we systematically compared the accuracy and computational cost of two popular solution methods for the radiative transfer equation (RTE): the spherical harmonics method (PN) and the discrete ordinates method (DOM). We first investigated convergence characteristics of different orders of PN and DOM in a series of 1D homogeneous configurations with varying optical thicknesses. Both solvers perform better for optically thicker cases. The accuracy of PN methods increases with its order, N, but the gain in accuracy reduces with the increase in N, i.e., improvement of P7 over P5 is less than that of P3 over P1. This decreasing trend becomes more prominent as the optical thickness decreases. On the other hand, DOM’s accuracy increases almost linearly with the increase in the number of ordinates (or polar angles in this study) in all cases. While comparing the directional profile of radiative intensity, both solvers perform better when the radiative intensity is more isotropic. These solvers were then connected with a full spectrum k-distribution (FSK) spectral model and used to perform radiation-coupled simulations of a turbulent jet flame in an axi-symmetric cylindrical domain. Results obtained from P1 to P7 approximations for PN, and 2 × 4, 4 × 4, 4 × 8, 8 × 8 finite angles for DOM are compared with that from an optically thin model, and a reference solution from line-by-line (LBL) photon Monte Carlo (PMC) method. The choice of radiation solver shows a noticeable impact on the temperature distribution of the flame. The PN solvers lead to slightly higher radiant fractions and the DOM solvers lead to slightly lower radiant fractions than the PMC benchmark solution. Finally, the computational costs of each of these solvers are also reported and an intermittent evaluation / time blending scheme to improve the computational efficiency of radiation solvers in radiation-coupled simulations are also demonstrated.