Rayleigh Problem Research Articles

Bounds on the eigenvalues for the circular Rayleigh problem of hydrodynamic stability

Bounds on the eigenvalues for the circular Rayleigh problem of hydrodynamic stability

Rayleigh — Benard problem for Polymer Solution

There are three mathematical models describing the motion of aqueous solutions of polymers: the second grade fluid model (Rivlin — Eriksen), the hereditary model (Voitkunsky — Amfilokhiev — Pavlovsky), and its asymptotic simplification (Pavlovsky). This work considers the problem of fluid equilibrium stability in a horizontal fluid layer heated from below or from above. Also, equations of thermal gravitational convection for all three models are derived. Three types of boundary conditions are considered: two solid boundaries; the lower solid boundary and the upper free boundary; two free boundaries (the Rayleigh problem). For the case of heating from below, the principle of perturbation monotonicity is established that ensures the spectral problem eigenvalues to be of real type. This greatly simplifies the determination of the critical Rayleigh numbers. It turned out that these numbers coincide with the critical Rayleigh numbers in the classical Rayleigh — Benard problem. In the case of heating from above at large temperature gradients, the perturbation decrements become complex, but their real parts are negative. The conclusion that the relaxation properties of a second grade fluid and an aqueous solution of polymers do not lead to a change in the critical Rayleigh number may seem strange at first glance. According to our assumption, it is explained by the base state of the liquid being a state of rest.

Unsteady incompressible laminar boundary layer: time and space variable molecular viscosity

The subject of this work is two approaches to describe the laminar unsteady flow of an incompressible fluid in the boundary layer. In the first approach, the viscosity of the fluid and the acceleration with which the plane is set in motion are considered constant. In essence, this is Rayleigh's problem. The solution obtained on the basis of these assumptions asymptotically converges to the well-known self-similar Stokes solution. It is important that the solutions of Stokes and Rayleigh asymptotically at large values of time correspond to the disappearance of shear stresses between the liquid and moving plane after acceleration. A paradox emerges the equations derived by Stokes to describe internal friction indicate the absence of the same friction between a moving body and fluid. Since research using the calculus of variation methods revealed that the molecular viscosity inside the stationary boundary layer should depend on the distance to the moving surface, the corresponding non-steady problem was considered. As a result, as before for the steady case, solutions describing both non-gradient and gradient flows of incompressible fluid in the boundary layer are obtained. The asymptotic analysis of the transition to steady flow testifies the consistency of these solutions. For the case of non-gradient flow, a comparison of the classical solution with the solution corresponding to the extreme fluid flow rate carried by the moving surface is made. It is shown that according to the solution obtained on the basis of the calculus of variation approach, the shear stress on the surface does not disappear anywhere after the motion becomes steady but, as expected, acquires a constant value. The research methods are purely theoretical and the results are analyzed by comparison with available theoretical and experimental data and compliance with the fundamental laws of physics, in particular the law of conservation of energy. These methods are based on the construction of analytical mathematical models, which are differential equations in partial derivatives supplemented with appropriate physical initial and boundary conditions. In addition, Euler's differential equations for the extreme of functional theory are used (in this paper, this is the extreme of fluid flow rate across the cross-section of the boundary layer). When solving these equations, the well-known Fourier method of variable separation is used. Arbitrary functions of time arising during partial integration (by one of the variables – the spatial coordinate) are determined from the conditions of asymptotic convergence of the solutions of non-steady problems to the corresponding solutions of steady problems. Conclusions. The presented results are of fundamental importance for understanding the physics of the flow around aircraft parts, as they indicate the contradiction of the existing idea of the reversibility of direct and inverse problems: the motion of a body in a still fluid and the flow of a fluid around an immobile body.

Construction of Boundary Conditions for Navier–Stokes Equations from the Moment System

This work concerns with boundary conditions (BCs) of the linearized moment system for rarefied gases. Under the acoustic scaling, we analyze the boundary-layer behaviors of the moment system by resorting to a three-scale asymptotic expansion. The asymptotic analysis casts the flows into the outer solution, the viscous layer, and the Knudsen layer. Starting from the BCs of the moment system, we construct BCs for the Navier–Stokes equations by a matching requirement. The obtained BCs contain the effect of second-order terms on the velocity slip and temperature jump. For the illustrative case of the Rayleigh problem, we prove the validity of the constructed BCs through error estimates. Meanwhile, numerical tests are presented to show the performance of the constructed BCs.

Existence and Stability of Solutions for Linear and Nonlinear Damping of q-Fractional Duffing–Rayleigh Problem

In this current paper, using q-fractional calculus, we study the Duffing–Rayleigh type problem with sequential fractional q-derivative of the Caputo type. We investigate the existence and uniqueness of solutions by applying some classical fixed point theorems. Also we define and study the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of solutions for our problem. An example is presented to illustrate the main results.

The Rayleigh Problem (Random Flights) is Everywhere Redux

The phase noise problem (or Rayleigh problem) occurs in all aspects of radar. It is an effect that a radar engineer or physicist always has to take into account as part of a design or in an attempt to characterize the physics of a problem such as reverberation. Normally, the mathematical difficulties of phase noise characterization are avoided by assuming the phase noise probability distribution function (PDF) is uniformly distributed, and the Central Limit Theorem (CLT) is invoked to argue that the superposition of relatively few random components obey the CLT and hence the superposition can be treated as a normal distribution. By formalizing the characterization of phase noise for an individual random variable, the summation of identically distributed random variables is the product of multiple characteristic functions (CF). The product of the CFs for phase noise has a CF that can be analyzed to understand the limitations CLT when applied to phase noise.

Fractal dimensions in fluid dynamics and their effects on the Rayleigh problem, the Burger's Vortex and the Kelvin–Helmholtz instability

Fractal dimensions in fluid dynamics and their effects on the Rayleigh problem, the Burger's Vortex and the Kelvin–Helmholtz instability

On compressible circular Rayleigh problem of hydrodynamic stability

On compressible circular Rayleigh problem of hydrodynamic stability

Some new applications of β-Laplace integral transform

The β-Laplace integral transform(β-LIT) has been introduced recently by Gaur et. al. which is a generalization of Laplace transform. This new β-LIT shows some special properties over Laplace integral transform. In the current paper, we investigate the applicability of the β-LIT technique to solve partial differential equation such as initial value problem of first order and second order PDE, wave equation, Diffusion equation, Potential and current equation, Heat conduction equation, Axisymmetric Heat conduction equation, Inhomogeneous partial differential equation, inhomogeneous wave equation, Stroke and Rayleigh problem in fluid dynamics and illustrate by some examples.

Quantum Rayleigh problem and thermocoherent Onsager relations

The role of quantum coherence and correlations in heat flow and equilibration is investigated by exploring the Rayleigh's dynamical problem to equilibration in the quantum regime and following Onsager's approach to thermoelectricity. Specifically, we consider a qubit bombarded by two-qubit projectiles from a side. For arbitrary collision times and initial states, we develop the master equation for sequential and collective collisions. By deriving the Fokker-Planck equation out of the master equation, we identify the quantum version of the Rayleigh's heat conduction equation. We find that quantum discord and entanglement shared between the projectiles can contribute to genuine heat flow only when they are associated with so-called heat-exchange coherences. Analogous to Onsager's use of Rayleigh's principle of least dissipation of energy, we use the entropy production rate to identify the coherence current. Both coherence and heat flows can be written in the form of quantum Onsager relations, from which we predict coherent Peltier and coherent Seebeck effects. The effects can be optimized by the collision times and collectivity. Finally, we discuss some of the possible experimental realizations and technological applications of the thermocoherent phenomena in different platforms.

Fundamental tones of clamped plates in nonpositively curved spaces

We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n≥2) Cartan-Hadamard manifold (M,g) with sectional curvature K≤−κ2 for some κ≥0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M,g) is universally bounded from below by (n−1)416κ4 whenever the κ-Cartan-Hadamard conjecture holds on (M,g), e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in (M,g) of volume v>0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature −κ2 provided that v≤cn/κn with c2≈21.031 and c3≈1.721, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. K≡κ=0). Sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of low-dimensional hyperbolic spaces are also given. The sharpness of our results requires the validity of the κ-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on (M,g)) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.

Note on Instability Regions in the Circular Rayleigh Problem of Hydrodynamic Stability

Batchelor and Gill’s semicircular bound on the range of the complex wave velocity of an arbitrary unstable mode in the stability problem of inviscid incompressible homogeneous axial flows with respect to axisymmetric disturbances is further reduced. Significance In the circular Rayleigh problem of hydrodynamic stability, we have obtained a parabolic instability region. The significance of this result is that it depends on the discriminant $$\Psi (r)$$ which plays a vital role in deciding whether a basic axial flow is stable or not to axisymmetric disturbances. However, our parabolic instability region is an unbounded one unlike the semicircular instability region of Batchelor and Gill, which has the weakness that it does not depend on $$\Psi (r)$$ . So we have demonstrated that our parabola intersects the semicircle of Batchelor and Gill for two classes of basic flows, and consequently the semicircular instability region of Batchelor and Gill is further reduced. Moreover, we have stated the stability problem consisting of a second-order ODE and associated boundary conditions in three different but related variables. The significance of this is that further analytical results can be obtained for this problem, and some of these are presented in our subsequent paper.

A Chebyshev pseudo-spectral method for the multi-dimensional fractional Rayleigh problem for a generalized Maxwell fluid with Robin boundary conditions

Many pseudospectral schemes have been developed for time fractional partial differential equations, most of which use the spectral tau method. In this study, we develop an accurate numerical scheme using a combination of Lagrange and Chebyshev polynomials of the first kind for the multi-dimensional fractional Rayleigh problem integrated using the Gauss–Lobatto quadrature. The Chebyshev expansion coefficients are evaluated using the orthogonality condition of the polynomials. Arbitrary temporal derivatives are approximated using shifted Chebyshev polynomials, while the spatial derivatives are approximated using Lagrange basis functions. To establish the accuracy of the proposed scheme, we present a convergence analysis of the absolute errors. The convergence analysis shows that the absolute error tends to zero for sufficiently large number of collocation points.

An improved instability region for the extended Rayleigh problem of hydrodynamic stability

For the extended Rayleigh problem of hydrodynamics stability dealing with homogeneous shear flows with variable cross section, we have obtained a parabolic instability region. This improved parabolic instability region intersects with the semicircular instability region under certain condition. The validity of the result is illustrated with an example of basic flows. Furthermore, we have obtained a bound for the complex part of the phase velocity.

The Stability of a Rotating and Heated From Below Horizontal Cylindrical Layer of a Viscous, Incompressible Liquid with Free Boundaries

The stability of a rotating and heated from below horizontal cylindrical layer of a viscous, incompressible liquid with free boundaries was theoretically investigated. Neglecting the centrifugal forces, the equations of motion, thermal conductivity and incompressibility of the liquid were written, from which the well-known dispersion equation was derived in the linear approximation. The stability of a rotating cylindrical volume of a liquid with no heating from below was considered, provided that the temperature difference between the horizontal boundaries of the liquid was fixed and equal to zero. It was demonstrated, that with no heating from below the temperature difference between the horizontal boundaries of the rotating liquid was not fixed and not maintained from the outside, the perturbed liquid temperature would increase, but its final value did not exceed the phase transition temperature. The obtained result was used to explain the heating of water in Ranque – Hilsch vortex tubes. It was concluded that the water heating in Ranque -Hilsch tubes should be considered as the inverse Rayleigh problem, in which the temperature gradient can be determined from the known distribution of velocities inside the volume. The stability of a rotating cylindrical volume of a liquid when heated from below was analyzed. It was demonstrated, that the value of the specified temperature difference at cylinder boundaries, as well as the initial rate of its variation, determine the final heating temperature of the liquid. A comparison of the proposed theory and experimental data for water heating shows their good qualitative and quantitative agreement.

On sufficient conditions for stability in the circular Rayleigh problem of hydrodynamic stability

Two sufficient conditions for stability are presented for the circular Rayleigh problem of hydrodynamic stability.

Generalized Noh self-similar solutions of the compressible Euler equations for hydrocode verification

A family of exact self-similar solutions of the compressible Euler equations developed for hydrocode verification is described. This family generalizes the classic Noh problem, which has served as a standard verification test of numerical methods for modeling inviscid compressible flows for three decades. This generalization allows finite pressure initial conditions, nearly arbitrary equations of state, and describes shocked compression as well as isentropic expansion and compression of the gas. In particular, the solutions describe a) the propagation of a finite-strength spherical isentropic expansion wave into a moving uniform gas, leaving behind either a core of uniform gas at rest or a vacuum/cavitation; b) the convergence of a finite-strength isentropic compression wave into a uniform gas or a collapse of a cavity in a finite-pressure gas (a compressible analog of the Rayleigh problem); and c) the expansion of a finite-strength accretion shock wave into a converging isentropic flow of stagnating gas. Our proposed verification test seeks to numerically reproduce all three of these stages of gas motion in a single simulation run. The successful verification of a high-order Godunov Eulerian hydrodynamics code is presented as an example of the expected use of this family of exact solutions.

Variational Rayleigh Problem of Gas Lubrication Theory. Low Compressibility Numbers

The two-dimensional variational problem for a gas-lubricated slider bearing is considered. In the gas layer the pressure field is described by the linear Reynolds equation which corresponds to low compressibility numbers. The boundary conditions are the conditions of vanishing the excess pressure on the boundaries of the domain. The load capacity acts as the functional of the variational problem. The system of necessary conditions of extremum which underlies the calculation algorithm is analyzed qualitatively. The present study develops radically and supplements the results of author’s studies at the modern level of theoretical and computational possibilities.

The Rayleigh problem for a yield stress fluid with spurt

We consider a generalized Newtonian fluid which exhibits a yield stress and spurt when the yield stress is reached. The constitutive behavior above the yield stress is assumed linear. We solve the associated Rayleigh problem. The solution is not self-similar and exhibits a propagating front separating yielded and unyielded regions. The behavior near t=0 is crucial for starting the numerical simulation. We find that the position of the front behaves like −tlnt.

VISCOUS DISSIPATION AND HEAT TRANSFER EFFECTS IN MHD RAYLEIGH PROBLEM.

24Feb 2017 VISCOUS DISSIPATION AND HEAT TRANSFER EFFECTS IN MHD RAYLEIGH PROBLEM. P. T. Hemamalini and M. Deivanayaki. Professor, Department of Science and Humanities, Faculty of Engineering, Karpagam University, Coimbatore. Assistant Professor, Department of Science and Humanities, Faculty of Engineering, Karpagam University, Coimbatore.