scholarly journals Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

2016 ◽  
Vol 7 (2) ◽  
pp. 130-149 ◽  
Author(s):  
Lidia Saluto ◽  
Maria Stella Mongioví
Keyword(s):  
Evolution Equation ◽  
Dimensional Analysis ◽  
Unit Volume ◽  
Vortex Formation ◽  
Second Order ◽  
Line Density ◽  
Superfluid Turbulence ◽  
Turbulence Flow ◽  
Vortex Diffusion

Abstract We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.

Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2

2018 ◽  
Vol 3 (1) ◽  
pp. 14-18 ◽  
Author(s):  
Chun-Chi Lin ◽  
Yang-Kai Lue
Keyword(s):  
Evolution Equation ◽  
Convergent Subsequence ◽  
Second Order ◽  
Plane Curves ◽  
Smooth Solutions ◽  
Fixed Position ◽  
Elastic Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

Abstract For any given C2-smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.

Numerical study of rotational diffusion in sheared semidilute fibre suspension

2011 ◽  
Vol 692 ◽  
pp. 153-182 ◽  
Author(s):  
Asif Salahuddin ◽  
Jingshu Wu ◽  
C. K. Aidun
Keyword(s):  
Evolution Equation ◽  
Chemical Engineering ◽  
Numerical Study ◽  
Rotational Diffusion ◽  
Hybrid Approach ◽  
Structural Evolution ◽  
Second Order ◽  
External Boundary ◽  
Aspect Ratios ◽  
Rigid Rod

AbstractFibre-level computer simulation is carried out to study the rotational diffusion and structural evolution of semidilute suspensions of non-Brownian, rigid-rod-like fibres under shear flow in a Newtonian fluid. The analyses use a hybrid approach where the lattice-Boltzmann method is coupled with the external boundary force method. The probability distribution of the orbit constant, $p({C}_{b} )$, in the semidilute regime is predicted with this method. The paper emphasizes assessment of the characteristics of a rotary diffusion model – anisotropic in nature (Koch, Phys. Fluids, vol. 7, 1995, pp. 2086–2088) – when used in suspensions with fibres of different aspect ratios (ranging from ${r}_{p} = 16$ to $72$) and with different volume concentrations (ranging from ${c}_{v} = 7. 58\ensuremath{\times} 1{0}^{\ensuremath{-} 3} $ to $6. 14\ensuremath{\times} 1{0}^{\ensuremath{-} 2} $). A measure of the scalar Folgar–Tucker constant, ${C}_{I} $, is extracted from the anisotropic diffusivity tensor, $ \mathbisf{C} $. The scalar ${C}_{I} $ is mostly $O(1{0}^{\ensuremath{-} 4} )$ in the semidilute regime and compares very well with the experimental observations of Stover (PhD thesis, School of Chemical Engineering, Cornell University, 1991) and Stover, Koch & Cohen (J. Fluid Mech., vol. 238, 1992, pp. 277–296). The ${C}_{I} $ values provide substantial numerical evidence that the range of ${C}_{I} $ (0.0038–0.0165) obtained by Folgar & Tucker (J. Rein. Plast. Compos., vol. 3, 1984, pp. 98–119) in the semidilute regime is actually overly diffusive. The paper also branches out to incorporate anisotropic diffusion (through the use of the Koch model) in the second-order evolution equation for $ \mathbisf{A} $ (a second-order orientation tensor). The solution of the evolution equation with the Koch model demonstrates unphysical behaviour at low concentrations. The most plausible explanation for this behaviour is error in the closure approximation; and the use of the Koch model in a spherical harmonics-based method (Montgomery-Smith, Jack & Smith, Compos. A: Appl. Sci. Manuf., vol. 41, 2010, pp. 827–835) to solve for the orientation moments corroborates this claim.

scholarly journals Optimal energy decay rates for abstract second order evolution equations with nonautonomous damping

2021 ◽  
Author(s):  
J. R. Luo ◽  
T. J. Xiao
Keyword(s):  
Evolution Equation ◽  
Polynomial Growth ◽  
Evolution Equations ◽  
Energy Decay ◽  
Second Order ◽  
Decay Rates ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Energy Decay Rates ◽  
Autonomous Evolution

We consider an abstract second order non-autonomous evolution equation in a Hilbert space $H:$ $u''+Au+\gamma(t) u'+f(u)=0,$ where $A$ is a self-adjoint and nonnegative operator on $H$, $f$ is a conservative $H$-valued function with polynomial growth (not necessarily to be monotone), and $\gamma(t)u'$ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient $\gamma(t)$ and the exponent associated with the nonlinear term $f$? There seems to be little development on the study of such problems, with regard to {\it non-autonomous} equations, even for strongly positive operator $A$. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of $\gamma(t)$ and $f$.  As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when $f$ is a monotone operator.

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