Rayleigh Dissipation Function Research Articles

Nanofluidics of nematic liquid crystals in hollow capillaries.

The aim of this paper is to investigate the response of a homogeneously aligned nematic nanosized hollow cavity (HANNHC) confined between two charged horizontal coaxial cylinders and subjected to both a radially applied electrostatic field E, arising from the surface charge density κ and the temperature gradient ∇T set between these cylinders. This was done within the framework of an extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation. The physical mechanism responsible for the excitation of the hydrodynamic flow in the HANNHC is based on the interaction of the director and temperature gradients and the static electric field. Calculations show that under the influence of both the ∇T and E, a stationary flow u^{st} is excited in the HANNHC in the horizontal direction. It is shown that the electric force enforced by the flexoelectric polarization plays a crucial role in the excitation of u^{st} between these cylinders.

Shear-Driven Mechanism of Temperature Gradient Formation in Microfluidic Nematic Devices: Theory and Numerical Studies

The purpose of this paper is to show some routes in describing the mechanism responsible for the formation of the temperature difference ΔT at the boundaries of the microfluidic hybrid aligned nematic (HAN) channel, initially equal to zero, if one sets up the stationary hydrodynamic flow vst or under the effect of an externally applied shear stress (SS) to the bounding surfaces. Calculations based on the nonlinear extension of the classical Ericksen–Leslie theory, supplemented by thermomechanical correction of the SS σzx and Rayleigh dissipation function while accounting for the entropy balance equation, show that the ΔT is proportional to the heat flux q across the HAN channel and grows by up to several degrees under the influence of the externally applied SS. The role of vst=ust(z)i^ with a sharp triangular profile ust(z) across the HAN in the production of the highest ΔT is also investigated.

Observability, controllability and stability analysis of discrete time engineering dynamic systems by means of Lagrangian, Hamiltonian and dissipative functions in discrete forms

Purpose The purpose of this paper is to analyze the dynamical state of a discrete time engineering/physical dynamic system. The analysis is performed based on observability, controllability and stability first using difference equations of generalized motion obtained through discrete time equations of dissipative generalized motion derived from discrete Lagrange-dissipative model [{L,D}-model] for short of a discrete time observed dynamic system. As a next step, the same system has also been analyzed related to observability, controllability and stability concepts but this time using discrete dissipative canonical equations derived from a discrete Hamiltonian system together with discrete generalized velocity proportional Rayleigh dissipation function. The methods have been applied to a coupled (electromechanical) example in different formulation types. Design/methodology/approach An observability, controllability and stability analysis of a discrete time observed dynamic system using discrete equations of generalized motion obtained through discrete {L,D}-model and discrete dissipative canonical equations obtained through discrete Hamiltonian together with discrete generalized velocity proportional Rayleigh dissipation function. Findings The related analysis can be carried out easily depending on the values of classical elements. Originality/value Discrete equations of generalized motion and discrete dissipative canonical equations obtained by discrete Lagrangian and discrete Hamiltonian, respectively, together with velocity proportional discrete dissipative function are used to analyze a discrete time observed engineering system by means of observability, controllability and stability using state variable theory and in the method proposed, the physical quantities do not need to be converted one to another.

Two shear driven flow regimes in microfluidic nematic devices: Tumbling and laminar

The purpose of this study is to show how the tangential component of the shear stress (SS) σzx applied to the boundary of the microsized nematic channel containing a temperature gradient ∇T effects the character of director field n̂ evolution to its stationary orientation n̂st. Calculations based on extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the stress tensor and Rayleigh dissipation function, with accounting the entropy balance equation, show that due to coupling among the SS σzx,∇T, and ∇n̂ in confined hybrid aligned nematic (HAN) channel the horizontal nematic flow v may be excited. Calculations also show that there is a difference in the evolution of n̂ to its stationary orientation n̂st, in the “laminar” case of confined nematic phase, composed of 4-cyano-4′-pentylbiphenyl(5CB) molecules, when -γ2/γ1>1, and in the “tumbling” case of confined nematic phase, composed of 4-cyano-4′-octylbiphenyl(8CB) molecules, when -γ2/γ1<1, respectively. It has been also shown the possible way how to set up a temperature difference ΔT=Tup-Tlw across the HAN microfluidic channel, initially equal to zero, under the action of σzx applied to the boundary of the HAN channel.

Convective Heat Transfer of Laminar Swirling Pipe Flow With Viscous Dissipation Effects

Abstract The oil-water-gas separation is a critical aspect of the treatment of production flows in the oil industry. The segregation of gas bubbles and/or water droplets dispersed in viscous oil by an in-line swirling flow separator has been considered by the oil industry for topside and subsea applications. For high viscosity oils, heat transfer processes can be affected. Works addressing these applications are rare in the literature. In this way, the article presents a numerical investigation on heat transfer characteristics in a decaying swirling flow, considering the effects of viscosity dissipation due to the high viscosity of the fluid. The flow has both velocity and temperature profiles developing simultaneously in a tube with a constant diameter having a uniform wall heat flux in a laminar flow regime, particularly the behavior of heat transfer characteristics for strongly swirling numbers considering viscous dissipation. Three swirl numbers (S = 0.0, 0.3, and 0.7) and five Brinkman numbers (Br = 0.0, 0.1, 0.5, 1.0, and 10.0) were investigated and the effects of those parameters on the dimensionless temperature profiles, Nusselt number and viscous dissipation function were examined. The heat transfer analysis indicated that the swirling flow affects the fluid's axial and radial temperature distribution. They promoted increased fluid in wall temperature and bulk temperature and affected the local Nusselt number distribution.

Energy dissipation and behavioral regimes in an autonomous double pendulum subjected to viscous and dry friction damping

The nonlinear dynamics of a system of a compound double pendulum are experimentally analyzed and numerically verified. The experiments are carried out for a combination of initial angles in the range 0 ⩽ θ 0, ϕ 0 ⩽ 180°, with zero initial angular velocity and with an increment of 1°. The motion of the pendula is captured in the video and the individual pendula are tracked using image processing. The energy dissipation in the system is evaluated using the Rayleigh dissipation function to include non-conservative forces as well, as a lumped parameter in the Lagrangian equation. Furthermore, the experimental and the numerical results are brought to agreement by tuning the dissipation factors. The linear, nonlinear and chaotic behavior of the system is classified in the coordinate plane. The behavioral regime represents the dependency of the nature of the system on the total energy and the difference in instantaneous amplitudes of the pendula. Uncertainty in the behavior of the system is observed in the transition region from the nonlinear to the chaotic regime due to the sensitivity of the system to the initial conditions. The dependence of the dissipation factors on the angular displacements is represented as a surface plot, which highlights the dominance of viscosity in the higher amplitude oscillations and the dominance of dry friction in the lower amplitude oscillations. This phenomenon is established by observing the nature of decay in the response plots of the pendula.

Gilbert damping in the layered antiferromagnet CrCl3

We theoretically and experimentally studied the Gilbert damping evolution of both acoustic and optical magnetic resonance modes in the layered flake CrCl3 with an external magnetic field H applied in plane. Based on a Lagrangian equation and a Rayleigh dissipation function, we predicted that the resonance linewidth ΔH as a function of microwave frequency ω is nonlinear for both acoustic and optical modes in the CrCl3 flake, which is significantly different from the linear relationship of ΔH ∝ ω in ferromagnets. Measuring the microwave transmission through the CrCl3 flake, we obtained the ω–H dispersion and damping evolution ΔH–ω for both acoustic and optical modes. Combining both our theoretical prediction and experimental observations, we concluded that the nonlinear damping evolution ΔH–ω is a consequence of the interlayer interaction during the antiferromagnetic resonance, and the interlayer Gilbert dissipation plays an important role in the nonlinear damping evolution because of the asymmetry of the non-collinear magnetizaiton between layers.

Microfluidics of liquid crystals induced by laser radiation.

Several scenarios for the formation of hydrodynamic flows in microsized hybrid aligned nematic (HAN) channels, based on the appropriate nonlinear extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation, are analyzed. Detailed numerical simulations were performed to elucidate the role of the heat flux q caused by laser radiation focused on the lower boundary of the equally warmed up the HAN channel containing a monolayer of azobenzene with the possibility of a trans-cis and cis-trans conformational changes in formation of the vortex flow v. It is shown that a thermally excited vortex flow is maintained with motion in a positive sense (clockwise) in the vicinity of the orientation defect at the lower boundary of the HAN channel caused by the trans-cis and cis-trans conformational changes. In the case of the same HAN channel, but without the azobenzene monolayer at the lower boundary, the heat flux q can also produce the vortical flow in the vicinity of the laser spot at the lower boundary, directed in a negative sense (counterclockwise). At that, the second vortex is characterized by a much slower speed than the vortical flow in the first case.

Heat Driven Flows in Microsized Nematic Volumes: Computational Studies and Analysis

The nematic fluid pumping mechanism responsible for the heat driven flow in microfluidic nematic channels and capillaries is described in a number of applications. This heat driven flow can be generated either by a laser beam focused inside the nematic microvolume and at the nematic channel boundary, or by inhomogeneous heating of the nematic channel or capillary boundaries. As an example, the scenario of the vortex flow excitation in microsized nematic volume, under the influence of a temperature gradient caused by the heat flux through the bounding surface of the channel, is described. In order to clarify the role of heat flux in the formation of the vortex flow in microsized nematic volume, a number of hydrodynamic regimes based on a nonlinear extension of the Ericksen–Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation, are analyzed. It is shown that the features of the vortex flow are affected not only by the power of the laser radiation, but also by the duration of the energy injection into the microsized nematic channel.

Analytical Solution of the MHD Forced Flow and Heat Transfer of a non-Newtonian Visco-Inelastic Fluid between Two Infinite Rotating Disks

In the present article, the model of forced flow and heat transfer in a non-Newtonian Reiner-Rivlin fluid between two infinite rotating disks in the presence of a magnetic field has been studied analytically. We converted the higher-order, partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs) of flow and heat transfer by using the Von Karman similarity transformation. Homotopy Perturbation Method (HPM) is adopted to solve the system of nonlinear ODEs associated with boundary conditions. The viscous dissipation function is included in the energy equation. Therefore, the differential equation of the energy equation involves the Eckert number. The influence of physical parameters like Reynolds number, visco-inelastic parameter, forced parameter, Prandtl number, magnetic field on radial, transverse, axial velocity components, and temperature profile are discussed in detail. A comparison of the results obtained by HPM is made with numerical results, which proves the accuracy of HPM. The Nusselt number on the lower and upper disks are computed and presented graphically. The shear and couple stress on the lower disk is also discussed. The Maple software package is used for numeric and symbolic calculation of velocity components and temperature profiles.

Nonlinear vibrations of a polar-orthotropic thin circular plate subjected to circularly moving point load

Here, we formulate and study nonlinear vibrations of a thin, polar-orthotropic circular plate subjected to a circularly moving point load at a fixed radius and constant angular velocity. The governing equations and boundary conditions are obtained following Kirchhoff’s plate theory, incorporating von Kármán nonlinearity, and employing extended Hamilton’s principle. The external damping is introduced via Rayleigh dissipation function. The governing equations are solved using mode summation procedure. The mode shapes and the natural frequencies of the polar-orthotropic circular plate are found using Frobenius series method. Mode summation procedure results in coupled nonlinear ordinary differential equations. These equations are solved using the Runge–Kutta method for time response and method of harmonic balance with the arc continuation method for frequency response. The spectrum of the undamped linear vibration response of isotropic and polar-orthotropic plates exhibits natural frequencies of plates and angular velocity of the rotating load. The damped response contains the frequency of the angular velocity of the rotating load only. The nonlinear transverse vibrations of the undamped and damped plate due to rotating point load reveal frequency rich spectrums. The frequency response function shows strong modal interactions.

Energy loss and developing length during reciprocating flow in a pipe with a free-end

Direct numerical simulations of reciprocating pipe flow in a straight pipe with a free-end are presented. The range of amplitudes and frequencies studied span the laminar regime and the beginning of transition toward a conditionally turbulent flow. Two primary results are reported: the measurement of the flow development length and the loss of energy, both due to the presence of the free-end. Two regimes of flow are identified with distinct length scales. For low frequencies, the development length scales with the pipe diameter D. However, for higher frequencies, the development length scales with the Stokes layer thickness δ=2ν/ω. The energy loss is studied by calculating the viscous dissipation function, indicating where energy is lost, and allowing the energy lost due to the presence of the free-end to be isolated. While strong vortices are formed and convected from the exit, most of the energy they dissipate is lost within a few pipe diameters of the exit. It is shown that these trends continue even as the amplitude and frequency are increased so that the flow begins to transition from a laminar toward a turbulent state. Two modes of instability are observed in the Stokes layers near the free-end, one short wavelength mode with a wavelength set by the Stokes layer thickness and the other long wavelength mode with a wavelength set by the amplitude of the oscillatory flow. These modes are related to those observed in the fully developed oscillatory flow.

Least action principles for irreversible transport processes

Least action principles (i.e., variational principles) were firstly applied to describe reversible processes, such as geometrical optics and mechanics. Then, they were extended to handle irreversible processes on the basis of entropy production rate, and the principle of least dissipation of energy by Onsager is one of the representative theorems. Nevertheless, it should be noted that the least action principles based on entropy production rate cannot always derive the actual constitutive relation of an irreversible transport process. As an example, the least action principles based on entropy production rate, including the principle of least dissipation of energy, will not derive the Fourier’s heat conduction law that is the fundamental constitutive relation of thermal transport process. In order to resolve this problem, Guo et al. proposed a new quantity, entransy dissipation rate, which is defined as the production of heat flux and negative temperature gradient and proportional to the dissipation rate of the potential energy of thermomass. It has been proven that the action of heat conduction should be the entransy dissipation rate; moreover, a theoretical framework for heat transfer optimization has been built based on it. In the present work, we extend the entransy theory to deal with various irreversible transport processes, including mass and momentum transport, and analyze their least action principles and Lyapunov functions. It is found that the action for a specific irreversible transport process can be constructed using the Rayleigh dissipation function and the product of actual driving force and flux, which can be regarded as the dissipation of some type of energy. For instance, the action for the momentum transport process governed by Newton’s viscosity law can be constructed using the dissipation of mechanic energy, while the action for mass transport process described by Fick’s law is related to the dissipation of chemical potential that is the energy absorbed or released due to a change of the particle number of the given species. Lyapunov function reflects the irreversibility of a non-equilibrium transport process; we find that entropy production rate can serve as a Lyapunov function merely for an irreversible transport process governed by a constitutive relation in which the production of force and flux is equal to the entropy production rate, that is, the constitutive relation of entropy representation. It is noted that the constitutive relation of entropy representation cannot always coincide with the actual constitutive relation in practice. For example, heat conduction process is generally regarded as being governed by Fourier’s law; however, its constitutive relation of entropy representation can agree with the Fourier’s law only when the thermal conductivity is inversely proportional to the square of temperature. In fact, for an actual transport process, its Lyapunov function should be identical to its action that can derive the actual constitutive relation. It is the entransy dissipation rate that serves as both the action and Lyapunov function of thermal transport process. Lastly, we note that the combination of action and energy conservation equation cannot give the governing equation for a transient irreversible transport process; the quasi-least action principle can be obtained by introducing convolution integral.

Cauchy Formalism in the Theory of Acoustic Surface Waves

To describe the propagation of acoustic surface waves in an anisotropic layer, a six-dimensional complex formalism is introduced, a Hamiltonian is constructed, an analogue of the Rayleigh dissipative function, and an exponential fundamental matrix. Dispersion equations are obtained for a multilayer plate with different conditions on the boundary surfaces. Examples of the application of the Cauchy formalism to the analysis of the dispersion of Lamb waves are given.

Chaotic Motion of a Parametrically Excited Dielectric Elastomer

In this paper, an effort is made to study the chaotic motions of a dielectric elastomer (DE). The DE is activated by a time-dependent voltage (AC voltage), which is superimposed on a DC voltage. The Gent strain energy function is employed to model the nonlinear behavior of the elastomeric matter. The nonlinear ordinary differential equation (ODE) in terms of the stretch of the elastomer governing the motion of the system is deduced using the Euler–Lagrange method and the Rayleigh dissipation function. This ODE is solved via the use of a time integration-based solver. The bifurcation diagrams of Poincaré sections are generated to identify the chaotic domains. The largest Lyapunov exponents (LLEs) are illustrated for validation of the results obtained by the bifurcation diagrams. Various types of motion for the system are precisely discussed through the depiction of stretch-time responses, phase-plane diagrams, Poincaré sections and power spectral density (PSD) diagrams. The results reveal that the damping coefficient plays an influential role in suppressing the chaos phenomenon. Besides, the initial stretch of the elastomer could affect the chaotic interval of system parameters.

Phase Mixing of Kink MHD Waves in the Solar Corona: Viscous Dissipation and Heating

Abstract Magnetohydrodynamic (MHD) kink waves have been observed frequently in solar coronal flux tubes, which makes them a great tool for seismology of the solar corona. Here, the effect of viscosity is studied on the evolution of kink waves. To this aim, we solve the initial value problem for the incompressible linearized viscous MHD equations in a radially inhomogeneous flux tube in the limit of long wavelengths. Using a modal expansion technique the spatio-temporal behavior of the perturbations is obtained. We confirm that for large Reynolds numbers representative of the coronal plasma the decrement in the amplitude of the kink oscillations is due to the resonant absorption mechanism that converts the global transverse oscillation to rotational motions in the inhomogeneous layer of the flux tube. We show that viscosity suppresses the rate of phase mixing of the perturbations in the inhomogeneous region of the flux tube and prevents the continuous building up of small scales in the system once a sufficiently small scale is reached. The viscous dissipation function is calculated to investigate plasma heating by viscosity in the inhomogeneous layer of the flux tube. For Reynolds numbers of the order of 106–108, the energy of the kink wave is transformed into heat in two to eight periods of the kink oscillation. For larger and more realistic Reynolds numbers, heating happens, predominantly, after the global kink oscillation is damped, and no significant heating occurs during the observable transverse motion of the flux tube.

Analysis of a coupled physical discrete time system by means of extended Euler-Lagrange difference equation and discrete dissipative canonical equation

PurposeThe purpose of this study is the application of the following concepts to the time discrete form. Variational Calculus, potential and kinetic energies, velocity proportional Rayleigh dissipation function, the Lagrange and Hamilton formalisms, extended Hamiltonians and Poisson brackets are all defined and applied for time-continuous physical processes. Such processes are not always time-continuously observable; they are also sometimes time-discrete.Design/methodology/approachThe classical approach is developed with the benefit of giving only a short table on charge and flux formulation, as they are similar to the classical case just like all other formulation types. Moreover, an electromechanical example is represented as well.FindingsLagrange and Hamilton formalisms together with the velocity proportional (Rayleigh) dissipation function can also be used in the discrete time case, and as a result, dissipative equations of generalized motion and dissipative canonical equations in the discrete time case are obtained. The discrete formalisms are optimal approaches especially to analyze a coupled physical system which cannot be observed continuously. In addition, the method makes it unnecessary to convert the quantities to the other. The numerical solutions of equations of dissipative generalized motion of an electromechanical (coupled) system in continuous and discrete time cases are presented.Originality/valueThe formalisms and the velocity proportional (Rayleigh) dissipation function aforementioned are used and applied to a coupled physical system in time-discrete case for the first time to the best of the author’s knowledge, and systems of difference equations are obtained depending on formulation type.

Влияние вязкой диссипации на естественную конвекцию в квадратной области

This article studies the natural convection of a viscous, incompressible fluid in a square cavity in a gravitational field. The temperature of vertical walls is constant. The temperature of the left wall is higher than temperature of the right wall; the horizontal walls are considered thermally insulated. The initial condition for the temperature of a fluid in a square caviry is the constant and equals the temperature of the right wall. The initial condition for the velocity is zero. We consider only those cases where the obtained flow in the cavity is laminar. All thermophysical characteristics are assumed constant, except for one when the motion equation accounts for the gravity. Mathematical model is the Boussinesq approximation but the equation of conservation of energy contains Rayleigh dissipation function.&lt;br&gt; In this article, the authors have researched the effect of viscous dissipation on natural convection heat transfer in square field. The results show that viscous dissipation significantly affects the heat transfer through the cavity. This problem was solved with the finite volume method by algorithm SIMPLER for Pr=1, Gr=10&lt;sup&gt;4&lt;/sup&gt;, and 10&lt;sup&gt;−5&lt;/sup&gt;≤Ec≤10&lt;sup&gt;−3&lt;/sup&gt;.

Film rupture and partial wetting over flat surfaces with variable distributor width

The effect of film stability and wetting behavior under an imposed fluid distribution width on falling film transfer devices for thermal engineering systems are investigated with reference to a variational thermodynamic formulation. The principle of minimizing the energy of a given streamwise section of the film is applied to identify the stable condition as the most likely among the set of possible flow configurations. The evolution toward this stable condition is estimated while adopting a Lagrangian viewpoint. The Lagrange equation is written with reference to a single generalized wetting coordinate and its time derivative, under the effect of Rayleigh's dissipation function and a generalized force associated with a scalar potential defined as the energy excess with respect to the local energy minimum of the stable rivulet configuration. This methodology is extended to include the hysteresis behavior of the contact angle and wettability hysteresis when increasing or decreasing the mass flow rates. Finally, a qualitative and quantitative validation of the results is presented with reference to the visual data captured on a dedicated experimental test section.

Variational approach to study soliton dynamics in a passive fiber loop resonator with coherently driven phase-modulated external field.

We report a detailed semianalytical treatment to investigate the dynamics of a single cavity soliton (CS) and two copropagating CSs separately in a Kerr mediated passive optical fiber resonator which is driven by a phase-modulated pump. The perturbation is dealt with by introducing Rayleigh's dissipation function in the framework of a variational principle that results in a set of coupled ordinary differential equations describing the evolution of individual soliton parameters. We further derive closed-form expressions for quick estimation of the temporal trajectory, drift velocity, and the phase shift accumulated by the CS due to the externally modulated pump. We also extend the variational approach to solve a two-soliton interaction problem in the absence as well as in the presence of the externally modulated field. In the absence of a phase-modulated field, the two copropagating solitons can attract, repulse, or propagate independently depending on their initial delay. The final state of interaction can be predicted through a second-order differential equation which is derived by the variational method. While in the presence of the phase-modulated field, the two-soliton interaction can result in annihilation, merging, breathing, or a two-soliton state depending on the detuning frequency and the pump power. Variational treatment analytically predicts these states and portrays the related dynamics that agrees with the full numerical simulation carried out by solving the normalized Lugiato-Lefever equation. The results obtained through this variational approach will enrich the understanding of complex pulse dynamics under a phase-modulated driving field in passive dissipative systems.