Nonlocal Velocity Research Articles

Nonlocalized thermal behavior of rotating micromachined beams under dynamic and thermodynamic loads

AbstractRotating micromachined beams are one of the most practical devices with several applications from power generation to aerospace industries. Moreover, recent advances in micromachining technology have led to huge interests in fabricating miniature turbines, gyroscopes and microsensors thanks to their high quality/reliability performances. To this end, this article is organized to examine the axial dynamic reaction of a rotating thermoelastic nanobeam under a constant‐velocity moving load. Using Eringen's nonlocal elasticity in conjunction with Euler–Bernoulli theory and Hamilton's principle, the governing equations are derived. It is assumed that the nanobeam is affected by thermal load and the boundary condition is simply supported. The Laplace transform approach is employed to solve the partial differential equations. A numerical example is presented to analyze the effects of the nonlocal parameter, rotation speed and velocity of the static moving load on the dynamic behavior of the system. The numerical results are graphically illustrated and analyzed to recognize the variations of field variables. Finally, in some special cases, our results are compared to those reported in the literature to demonstrate the reliability of the current model.

Large friction limit of pressureless Euler equations with nonlocal forces

We rigorously show a large friction limit of hydrodynamic models with alignment, attractive, and repulsive effects. More precisely, we consider pressureless Euler equations with nonlocal forces and provide a quantitative estimate of large friction limit to a continuity equation with nonlocal velocity fields, which is often called an aggregation equation. Our main strategy relies on the relative entropy argument combined with the estimate of p -Wasserstein distance between densities.

Wave propagation analysis of the higher-order nanopanel with laminated composite core and magneto-electro-elastic face sheets via nonlocal strain gradient theory

In the current report, wave propagation analysis of a sandwich doubly curved nanopanel considering a laminated composite core and a couple of face sheets of magneto-electro-elastic (MEE) in the framework of nonlocal strain gradient theory (NSGT) is investigated. The higher-order-shear-deformable (HSD) model is employed to obtain the formulation of strain–stress equations. Maxwell equations are employed for modeling the behaviors of MEE materials. By considering Hamiltonian, the current model’s governing equations have been extracted. Afterward, in order to analyze the influence of wavenumber, inserted ampere, inserted voltage, the laminated structure’s geometry, nano-sized mechanics impact on the smart sandwich nano-scaled panel’s phase velocity, a parametric study has been considered. It is also observed that by raising the electric potential, the number of critical waves could be likely to be increased. A useful suggestion of this study is that the influence of the magnetic field on the phase velocity is more than the influence of the electric field of the MEE panel on the wave propagation of the smart panel. Besides, the nonlocal parameter and phase velocity have an exponential negative relation, while the length scale parameter and phase velocity have a linear direct relation.

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

We rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.

Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity

In this paper, we study input-to-state stability (ISS) of an equilibrium for a scalar conservation law with nonlocal velocity and measurement error arising in a highly re-entrant manufacturing system. By using a suitable Lyapunov function, we prove sufficient and necessary conditions on ISS. We propose a numerical discretization of the scalar conservation law with nonlocal velocity and measurement error. A suitable discrete Lyapunov function is analyzed to provide ISS of a discrete equilibrium for the proposed numerical approximation. Finally, we show computational results to validate the theoretical findings.

Numerical analysis of flow and forced convection heat transfer of non-Newtonian fluid in a pipe based on fractional constitutive model

PurposeThis paper aims to use a fractional constitutive model with a nonlocal velocity gradient for replacing the nonlinear constitutive model to characterize its complex rheological behavior, where non-linear characteristics exist, for example, the inherent viscous behavior of the crude oil. The feasibility and flexibility of the fractional model are tested via a case study of non-Newtonian fluid. The finite element method is non-Newtonian used to numerically solve both momentum equation and energy equation to describe the fluid flow and convection heat transfer process.Design/methodology/approachThis paper provides a comprehensive theoretical and numerical study of flow and heat transfer of non-Newtonian fluids in a pipe based on the fractional constitutive model. Contrary to fractional order a, the rheological property of non-Newtonian fluid changes from shear-thinning to shear-thickening with the increase of power-law index n, therefore the flow and heat transfer are hindered to some extent.FindingsThis paper discusses two dimensionless parameters on flow regime and thermal patterns, including Reynolds number (Re) and Nusselt number (Nu) in evaluating the flow rate and heat transfer rate. Analysis results show that the viscosity of the non-Newtonian fluid decreases with the rheological index (order α) increasing. While large fractional (order α) corresponds to the enhancement of heat transfer capacity.Research limitations/implicationsFirst, it is observed that the increase of the Re results in an increase of the local Nusselt number (Nul). It means the heat transfer enhancement ratio increases with Re. Meanwhile, the increasement of the Nul indicating the enhancement in the heat transfer coefficient, produces a higher speed flow of crude oil.Originality/valueThis study presents a new numerical investigation on characteristics of steady-state pipe flow and forced convection heat transfer by using a fractional constitutive model. The influences of various non-dimensional characteristic parameters of fluid on the velocity and temperature fields are analyzed in detail.

Well-posedness and critical thresholds in a nonlocal Euler system with relaxation

<p style='text-indent:20px;'>We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.</p>

Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev space

Logarithmically regularized 2D Euler equations are active scalar equations with the non-local velocity u=∇⊥Δ−1Tγω for the scalar ω. Two types of the regularizing operator Tγ with a parameter γ>0 are considered: Tγ=ln−γ⁡(e+|∇|) and Tγ=ln−γ⁡(e−Δ). These models regularize the 2D Euler equation for the vorticity (conventionally corresponding to the γ=0 case), which results in their local well-posedness in the borderline Sobolev space H1(R2)∩H˙−1(R2) when γ>12. In this paper, we examine the regularized models in the remaining regime γ≤12 and establish the strong ill-posedness in the borderline space. This completely solves the well-posedness problem of the regularized models in the borderline space by closing the gap between the local well-posedness result for γ>12 and the strong ill-posedness for γ=0.

Response of a micro-capillary system exposed to a moving mass in magnetic field using nonlocal strain gradient theory

In this study, the dynamic behaviour of the micro dimensional Timoshenko beam (micro capillary system) exposed to a moving mass in a magnetic field is modelled using the non-local strain gradient theory. The motion equations of the Timoshenko micro-beam, which interacts with the moving mass, are obtained by considering the effects of strain gradient elasticity that provides stiffening and non-local elasticity that provide softening. The transverse Lorentz force acting on the beam in the magnetic field is obtained by Maxwell's equations and combined with the motion equations of the beam. These equations are converted into a weak form finite element equation by applying the exact shape functions of the two-node Timoshenko beam element in the Galerkin's method. Along with the effects of the magnetic field, the effects of the amount of different non-local parameters and the mass ratio and speed of the moving mass on the dynamics of the micro beam are presented. Experimental and theoretical studies have shown that the dynamic behaviour of the capillary structural systems is negatively affected by the effect of moving mass except for the non-local properties. Depending on the velocity and quantity of the mass and the nonlocal elasticity, the natural frequencies of the system drop excessively. When the mass ratio (moving mass/mass of the beam) is around 0.5, the frequencies have been observed to drop by as much as 60%, and the non-local elasticity and velocity of the mass are also effective in this. It has been shown that these disadvantages can be eliminated by the effect of the force obtained using the directed magnetic field. In fact, a desired system stiffness can be obtained by adjusting the magnitude of the magnetic field at the level required by the application.

Continuum simulation for regularized non-local μ(I) model of dense granular flows

In this work, we implement the local μ(I) rheology law and its non-local modification in a PISO-VOF numerical scheme with a sharp-interface-capturing THINC method for two-dimensional dense granular flows in both steady and transient states. The non-local effect on the constitutive relation is addressed by the gradient expansion model to account for additional momentum transport in view of the spatial variation of flow inertial number and the stress model singularity near the static state, I=0, is handled by the Bercovier-Engelman regularization scheme. Our simulations provide the first numerical evidence for the crucial role of non-local momentum transport on degrading the velocity profiles predicted with a local rheology model as those reported from two-dimensional discrete element simulations. For example, we successfully reproduce the sub-Bagnold velocity profile for steady inclined flows of height comparable to the Hstop and capture how the linear velocity profile in a simple shear creeping flow is degraded to a S-shape. More importantly, we exploit the current solver to study the layer-accumulated deviation, ε, between the local and the non-local velocity profile (Bagnold to its degradation) for inclined flows. For steady flows, ε decays monotonically with flow Froude number Fr and a critical Frc≈0.25 is detected below which ε rises rapidly necessitating a non-local constitutive relation. For a layer developing from rest to its steady state, the time evolution of ε(t) shows further dependence on flow height and can be collapsed into a monotonically growing trend with local instantaneous inertial number I(t). A critical Icr is also determined and the local model prediction can be erroneous whenever I(t)<Icr. Interestingly, the result for simple shear flows without gravity also confirms that non-local effect becomes non-negligible when the macroscopic inertial number drops to the order of Icr. Finally, we simulate the two-dimensional column collapse flows with both local and non-local rheology and find a shorter run-out distance with the presence of non-local momentum transport.

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order 1/2 in Wasserstein distance. This result is illustrated by numerical simulations, which show the optimality of the order of convergence.

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.

Global smoothness for a 1D supercritical transport model with nonlocal velocity

We are concerned with a nonlocal transport 1D-model with supercritical dissipation γ ∈ ( 0 , 1 ) \gamma \in (0,1) in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the well-known 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when γ ∈ ( 0 , 1 / 2 ) \gamma \in (0,1/2) . On the other hand, as stated by Kiselev (2010), in the supercritical subrange γ ∈ [ 1 / 2 , 1 ) \gamma \in \lbrack 1/2,1) it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative H 3 / 2 H^{3/2} -strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that γ ∈ [ γ 1 , 1 ) \gamma \in \lbrack \gamma _{1},1) where γ 1 \gamma _{1} depends on the H 3 / 2 H^{3/2} -initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016).

On the global well-posedness of one-dimensional fluid models with nonlocal velocity

On the global well-posedness of one-dimensional fluid models with nonlocal velocity

Mean-field Analysis of Multipopulation Dynamics with Label Switching

The mean-field analysis of a multipopulation agent-based model is performed. The model couples a particle dynamics driven by a nonlocal velocity with a Markov-type jump process on the probability t...

The periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system

In this paper, we study the periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system which includes the Fokas-Olver-Rosenau-Qiao (FORQ) or modified Camassa-Holm (MCH) equation and the two-component MCH system as two special cases. The system is integrable in the sense of possessing a non-isospectral Lax pair with the spectrum depending on time t, and admits multi-peakon solutions in an explicit form. Furthermore, we establish the local well-posedness for the system in the Besov space B2,rs(T) with s>3/2, 1≤r≤∞, where the key ingredients include the Friedrichs regularization method, the Littlewood-Paley decomposition theory, and the transport theory in Besov spaces. Then we derive a precise blow-up criteria, which is dependent of the parameters α(t) and γ(t). Moreover, by the intrinsic structure of the system, we obtain a new blow-up result for strong solutions with sufficient conditions on the initial data and parameters. The entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves.

Controllability and stabilization of a conservation law modeling a highly re-entrant manufacturing system

In this paper, we study the controllability and stabilization for a scalar conservation law modeling a highly re-entrant manufacturing system with local and nonlocal velocity. We prove a local state controllability result, i.e., there exists a control that drives the solution from any given initial condition to any desired final condition in a certain time period, provided that the initial and final data are both close to the origin. A local result on nodal profile controllability is also given, i.e., for any initial data and any given out-flux in a neighborhood of the origin, there exists a control under which the solution starts from any initial data reaches exactly any desired out-flux over a fixed time period. Besides, using a Lyapunov function approach, we can stabilize the system to the origin exponentially by output feedback control.

A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems

In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics is described by a transport equation with non-local velocities which are affine in the control, and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle formulated in the so-called Gamkrelidze form.

On the local and global existence of solutions to 1d transport equations with nonlocal velocity

We consider the 1D transport equation with nonlocal velocity field: \begin{document}$ \begin{equation*} \label{intro eq} \begin{split} &\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta = 0, & u = \mathcal{N}(\theta), \end{split} \end{equation*} $\end{document} where \begin{document}$ \mathcal{N} $\end{document} is a nonlocal operator and \begin{document}$ \Lambda^{\gamma} $\end{document} is a Fourier multiplier defined by \begin{document}$ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $\end{document} . In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.

Memory effects in measure transport equations

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.