Viscous Fluids and Quantum Mechanics
Joao Belther Junior
从一个简单的分析粒子轨道和存在或不存在耗散的流体流动，显然不相关的研究领域之间的一些连接。主要结果指出了经典保守系统的量子化和耗散形式之间的深刻联系。
From a simple analysis of particle orbits and fluid flows in presence or not of dissipation, some connections between apparently uncorrelated research areas are made. The main results point out for a deep relation between quantization of classical conservative systems and the dissipative version of these systems.
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Comment on "Flow of temperature-dependent viscous fluid between parallel heated walls: Exact analytical solutions in the presence of viscous dissipation", by K.S. Adegbie and F.I. Alao
在上述论文中，已经进行了分析，以获得具有粘性耗散和温度依赖性粘度的牛顿流体的库埃特流的结果。流体粘度是温度的指数函数。获得了速度和温度计算的精确解析解。然而，有一些基本的错误，在本文中提出的。
Fluid Mixing from Viscous Fingering
粘性指进是一种众所周知的流体动力学不稳定性，当粘性较小的流体取代粘性较大的流体时发生。当这两种流体是可混溶的，粘性指进引入速度场的无序，并对流体混合的速率施加基本的控制。我们展示了粘性不稳定流动中混合过程的流体动力学视频，该视频是使用对任意粘度比稳定的计算策略通过高分辨率数值模拟生成的。我们开发了一个两方程的浓度方差和平均耗散率的动态模型来量化在这样的位移过程中的混合程度。该模型准确地再现了这两个量的演变，在高分辨率的数值模拟中观察到的，并捕捉到的非平凡的沟道和创建的界面区域之间的相互作用，作为粘性指进的结果。
Influence of gravitational forces and fluid flows on a shape of surfaces of a viscous fluid of capillary size
毛细管尺寸的粘性流体的纳维-斯托克斯方程和边界条件在与流体-气体界面的几何形状相关联的曲线坐标中被制定。结果，可以考虑重力和流动对表面形状的影响来描述液滴和液滴的流体动力学。这为相应的数值研究提供了方便的基础。估计的影响的情况下，一个蒸发的座滴。
The process of one fluid pushing another is universally common while involving complex interfacial instabilities. Particularly, occurring in a myriad of natural and industrial processes, wavy fingering patterns frequently emerge when a less viscous fluid pushes another more viscous one, such as water invading oil, in a porous medium. Such finger-shaped interfaces producing partial displacement significantly affect the efficiency of numerous applications, for example, chromatography, printing devices, coating flows, oil-well cementing, as well as large-scale technologies of groundwater and enha…
Falling Jets of Particles in Viscous Fluids
这个流体动力学视频演示了低雷诺数下粘性流体中重力驱动颗粒射流的实验和模拟。一个最初的直射流显示开发静脉曲张调制其直径，因为它的作用下的重力沉积。虽然这种不稳定性是定性的不混溶流体的经典瑞利-高原不稳定性的回忆，其机制尚未被理解为既不惯性也不表面张力发挥作用的情况下，在稀悬浮液在Re=0。
In this work, we provide two novel approaches to show that incompressible fluid flow in a finite domain contains at most a finite number vortices. We use a recently developed geometric theory of incompressible viscous flows along with an existing mathematical analysis concept to establish the finiteness. We also offer a second proof of finiteness by roping in the Kolmogorov's length scale criterion in conjunction with the notion of diametric disks.
Surface waves at the interface between two viscous fluids
利用表面格林函数匹配分析（SGFM）方法，研究了考虑两种流体密度和粘度差异的非混合流体界面振动的简正模。粘性-无粘性系统的极限情况用于比较。分析了粘性和密度比对动量交换和表面模极化的影响。
Viscous spreading of an inertial wave beam in a rotating fluid
本文报道了旋转流体中振荡圆柱产生的惯性波的实验测量结果。二维波发生在静止的十字形波包中。用共转粒子图像测速系统测量了垂直于造波机的垂直平面内的速度场和涡量场。的粘性传播的光束和相关的衰减的速度和涡度包络的特点。发现它们与线性粘性理论的相似解很好地一致，线性粘性理论是在类似于Thomas和Stevenson [J. Fluid Mech.54（3），495-506（1972）]对内波的经典分析的准平行假设下导出的。
Vortex shedding by a swimming sphere in a viscous incompressible fluid is studied for surface modulation characterized by a superposition of dipolar and quadrupolar, as well as for quadrupolar and octupolar displacements, varying harmonically in time. The time-dependent swimming velocity and the flow velocity are calculated to second order in the amplitude of surface modulation for both models. The models are also useful for the discussion of bird flight.
In this paper the small-amplitude motion of multiple superposed viscous fluids is studied as a linearized initial-value problem. The analysis results in a closed set of equations for the Laplace transformed amplitudes of the interfaces that can be inverted numerically. The derived equations also contain the general normal mode equations, which can be used to determine the asymptotic growth-rates of the systems directly. After derivation, the equations are used to study two different problems involving three fluid layer. The first problem is the effect of initial phase difference on the develop…
Fluid-structure model for disks vibrating at ultra-high frequency in a compressible viscous fluid
针对圆盘谐振器在可压缩粘性液体中的超高频率径向振动，本文建立解析流固耦合模型，给出品质因数与频移的闭式表达式，并揭示1 GHz以上可压缩性效应的重要性。
Multiple Coexisting States of a Falling Viscous Thread
本文介绍了一个流体动力学视频提交给2009年画廊的流体运动，流体动力学，美国物理学会。
A rest fluid displaced by a less viscous fluid in a porous medium triggers the so-called Saffman-Taylor instability at their contact front and hence forms complicated finger-like patterns. When the two fluids are miscible, the surface tension at their contact front vanishes, leaving the variation in viscosity dominant the contribution to the instability. The phenomena, named viscous fingering, can be studied by the analogy of a single-phase flow in the Hele-Shaw cell, a quasi-two-dimensional rectilinear plane. Viscous fingering produces complex geometrical patterns. They are important not only…
Symmetrization of laminar viscous fluid flow in a flat diffuser by periodic impaction on the inlet flow velocity
本文通过数值求解Navier-Stokes方程，研究了扁平扩张器中粘性不可压缩流体非对称层流的对称化方法，证明即使振幅小于入口速度1%的弱周期性振动也能实现流动对称化，并展示了Richardson环状效应。
Parametrically excited standing waves are observed on a cylindrical fluid filament. These are the cylindrical analog of Faraday instability in a flat surface or spherical droplet. Using the Floquet technique, linear stability analysis has been investigated on a viscous cylindrical fluid surface, which is subjected to a time-periodic radial acceleration. Viscosity has a significant impact on the critical forcing amplitude as well as the dispersion relation of the non-axis symmetric patterns. The effect of viscosity on onset parameters of the pattern with azimuthal wavenumber, $m=1$, has shown d…
We consider small oscillations of micro-particles and gaseous bubbles in viscous fluid around equilibrium states under the action of a sinusoidal external force. Exact solutions to the governing integro-differential equations containing both Stokes and memory-integral drag forces are obtained. The main aim of this study is to clarify the influence of the memory-integral drag force on the resonance characteristics of oscillating particles or gaseous bubbles in a viscous fluid at small Reynolds numbers. The resonant curves (amplitude versus frequency of external force), as well as phase-frequenc…
The equilibrium state of a flexible fiber settling in a viscous fluid is examined using a combination of macroscopic experiments, numerical simulations and scaling arguments. We identify three regimes having different signatures on this equilibrium configuration of the elastic filament: weak and large deformation regimes wherein the drag is proportional to the settling velocity as expected in Stokes flow and an intermediate elastic reconfiguration regime where the filament deforms to adopt a shape with a smaller drag which is no longer linearly proportional to the velocity.
From the principle of least action the equation of motion for viscous compressible and charged fluid is derived. The viscosity effect is described by the 4-potential of the energy dissipation field, dissipation tensor and dissipation stress-energy tensor. In the weak field limit it is shown that the obtained equation is equivalent to the Navier-Stokes equation. The equation for the power of the kinetic energy loss is provided, the equation of motion is integrated, and the dependence of the velocity magnitude is determined. A complete set of equations is presented, which suffices to solve the p…
Vibrational methods for the symmetrization of asymmetric laminar viscous fluid flow in a plane diffuser
针对平面扩压器内非对称粘性层流问题，通过数值模拟比较入口与壁面周期振动两种对称化方法，证明速度低于主流0.01%的弱谐波振动即可使流动对称化。
A new prediction of wavelength selection in radial viscous fingering involving normal and tangential stresses
我们重新考虑径向Saffman-Taylor不稳定性，当从点源注入的流体在Hele-Shaw单元中取代另一种具有更高粘度的流体时，其中流体被限制在两个相邻的平板之间。前进的流体前缘是不稳定的，并沿圆周形成指状。使用所谓的Brinkman方程来描述流场，该方程还考虑了平面中的粘性应力，而不仅仅是像Darcy方程那样由于限制板而产生的粘性应力。色散关系同意更好地与实验结果比经典的线性稳定性分析的径向指在Hele-Shaw细胞，使用达西定律作为流体运动的模型。
The case of a rotating object traveling through viscous fluid appears in many phenomena like the banana ball and missile movement. In this work, we build a model to predict the trajectory of such rotating objects with near-cylinder geometry. The analytical expression of Magnus force is given and a wind tunnel experiment is carried out, which shows the Magnus force is well proportional to the product of angular velocity and centroid velocity. The trajectory prediction is consistent with the trajectory record experiment of Magnus glider, which implies the validity and robustness of this model.
In the theory of the Navier-Stokes equations, the viscous fluid in incompressible flow is modelled as a homogeneous and dense assemblage of constituent "fluid particles" with viscous stress proportional to rate of strain. The crucial concept of fluid flow is the velocity of the particle that is accelerated by the pressure and viscous interaction around it. In this paper, by virtue of the alternative constituent "micro-finite element", we introduce a set of new intrinsic quantities, called the vortex fields, to characterise the relative orientation between elements and the feature of micro-eddi…
Self-propulsion of a spherical electric or magnetic microbot in a polar viscous fluid
基于铁流体动力学理论，研究了极性液体或磁流体中球体的自推进问题。在电的情况下，位于球体内部的振荡电荷密度产生使流体极化的电场。由于弛豫引起的相对于电场的极化滞后产生作用于流体的与时间无关的电扭矩密度，从而使其移动。由此产生的球的推进速度计算在微扰理论的电荷密度的幂的二阶。
The swimming of a deformable planar slab in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations. A continuum of plane wave displacements, symmetric on both sides of the slab and characterized by a polarization angle, allows optimization of the swimming efficiency with respect to polarization. The mean swimming velocity and mean rate of dissipation are calculated to second order in the amplitude of the stroke. The optimum efficiency depends on the ratio of viscosity and mass density of the fluid. For high viscosity a stroke is found with significantly higher ef…
Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function…
In a recent paper by Casado-Pascual [Phys. Rev. E \textbf{97}, 032219 (2018)], directed motion of a sphere immersed in a viscous fluid and subjected to zero-average biharmonic forces is studied. The author explains the dependence on the relative amplitude of the two harmonic components of the average terminal velocity from the perspective of a general formalism. In this Comment, this explanation is shown to be in general incorrect, while the theory of ratchet universality together with the vibrational mechanics approach provide a satisfactory explanation of major aspects of the observed phenom…
We model the flow of a bi-viscous non-Newtonian fluid in a porous medium by a square lattice where the links obey a piece-wise linear constitutive equation. We find numerically that the flow regime where the network transitions from all links behaving according to the first linear part of the constitutive equation to all links behaving according to the second linear part of the constitutive equation, is characterized by a critical point. We measure two critical exponents associated with this critical point, one of the being the correlation length exponent. We find that both critical exponents …
Collinear velocity relaxation of two spheres immersed in a viscous incompressible fluid is studied on the basis of an approximate expression for the retarded hydrodynamic interaction. After a sudden impulse applied to one sphere, the other one instantaneously starts to move as well, with amplitude determined by the added mass effect. The velocities of both spheres eventually decay with a t^{-3/2} long-time tail, but the relative velocity decays with a t^{-5/2} long-time tail. The three relaxation functions are approximated by simple expressions involving only a small number of poles in the com…
Linear stability analysis on the onset of viscous fingering due to a non monotonic viscosity profile for immiscible fluids
针对不可混溶流体在Hele-Shaw单元中非单调黏度剖面引发的黏性指进，理论分析表明界面黏度梯度使界面失稳并促进扰动扩散，并研究了其对色散关系的影响。
Hadamard-Papapetrou method of field discontinuities is here employed in order to determine the effective metric that describes the propagation of acoustic perturbations in isentropic fluids. It is shown that, when dissipative effects are present, small perturbations in fastly moving fluids have a natural description in terms of an effective acoustic geometry. As an application of our results, a model for an acoustic black hole in viscous fluid is investigated.
Self-organization of ultrasound in viscous fluids
我们报告了声学中斑图形成的理论和实验证明。该系统是包含粘性流体的声学谐振器。当系统受到周期性外力驱动时，腔内的超声场经历不同的模式形成不稳定性，导致周期性结构的出现。该系统也被证明具有超声波领域的发展，其中本地化的状态的制度。粘性流体中的热非线性以及远离平衡的条件是所观察到的效应的原因。
Exactly Integrable Dynamics of Interface between Ideal Fluid and Light Viscous Fluid
结果表明，对于二维流动，在小表面斜率近似下，理想流体和轻粘性流体之间的界面动力学是完全可积的。粘性流体的斯托克斯流动提供了界面处的法向速度和压力之间的关系。由两个复Burgers方程分别确定了理想流体的表面高度和速度势，这两个方程对应于界面处速度势到上、下复半平面的解析延拓。如果复奇点（极点）到达界面，界面将失去光滑性。
Free Surface Deformation and Cusp Formation During the Drainage of a Very Viscous Fluid
我们报告了一个非常粘滞的流体的自由表面的变形的实验研究时排出的容器通过一个圆形孔。在流体的某个临界高度处，自由表面变形并形成凹坑。后来，这个酒窝变成了一个尖点。我们发现，从槽底测量的凹坑高度以及在尖点形成之前凹坑的曲率遵循幂律行为。我们发现，这种行为是由于粘性压力和表面张力之间的相互作用
Scaling and self-similarity in an unforced flow of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell
本文研究了Hele-Shaw盒中粘性流体中无粘流体长直条受表面张力作用的准二维弛豫。结合分析和数值解，我们描述了一个自相似的哑铃形状的出现，并找到非平凡的动态指数，表征哑铃尺寸的标度行为。
Viscous behavior of fluids in the eyes of adults: A global survey
本研究通过匿名在线问卷调查全球406名成人对流体黏性行为的朴素概念，并报告主要发现。
Viscous fingering (VF) is an interfacial instability that occurs in a narrow confinement or porous medium when a less-viscous fluid pushes a more viscous one, producing finger-like patterns. Controlling the VF instability is essential to enhance the efficiency of various technological applications. However, the control of VF instability has been challenging and so far focused on simple Newtonian fluids of constant viscosity. Here, we extend to complex yield-stress fluids and examine the controlling feasibility by carrying out a linear stability analysis using a radial cell with a converging ga…
Being a major limiting factor for the efficiency of various technologies, such as Enhanced Oil Recovery, the viscous fingering (or Saffman--Taylor) instability has been extensively studied, especially for simple Newtonian fluids. Here, we experimentally and theoretically demonstrate a vital control of inhibiting the viscous fingering instability for complex (yield-stress) fluids to generate a complete sweep with a flat interface. Using a rectangular tapered cell, we first experimentally show the feasibility of controlling the primary fingering instability of a complex yield-stress fluid when p…
Dispersive Hydrodynamics in Viscous Fluid Conduits
在小雷诺数下，分离轻的粘性流体通过重的、更粘性的外部流体而粘性地上升的管道的界面的演化由非线性和色散之间的相互作用决定。以往的研究工作者提出了一个基于物理参数的近似模型方程，但对这一具有自由边界的双流体系统缺乏精确的理论处理。在这里，推导出的界面方程通过多尺度，扰动技术。在具有适当边界条件的纳维-斯托克斯方程的范围内，考虑了关于垂直均匀层流管道流状态的扰动。内部与外部粘度的比率是渐近分析中使用的小参数，其系统地导致浮力驱动、非线性自陡和粘性、界面应力诱导、非线性色散之间的最大平衡。这导致标量，非线性偏微分方程描述的侵入流体导管的横截面积的大幅度动态，与以前的推导一致。双流体系统的主导阶行为完全由界面动力学表征。该政权的模型有效性的特点，并与以前的实验研究结果一致。粘性流体管道为非线性、色散波现象的研究提供了一个强大的环境。
On a Camassa-Holm type equation describing the dynamics of viscous fluid conduits
本文推导出描述小Reynolds数下粘性流体界面动力学的新非局部非线性色散方程（$g-$模型），研究了其适定性与周期行波存在性，并通过数值模拟揭示有限时间奇异性形成。
We derive and study a new hyperbolic two-phase model of a porous deformable medium saturated by a viscous fluid. The governing equations of the model are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems and by generalizing the unified hyperbolic model of continuum fluid and solid mechanics. Similarly to the unified model, the presented model takes into account the viscosity of the saturating fluid through a hyperbolic reformulation. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturated flu…
By taking advantage of laser Doppler velocimetry (LDV), we explore the existence of discrete wave cascades on fluid interfaces excited upon monochromatic excitation. We study viscous liquids of variable capillarity spanning a broad range of frictional stresses. The existence of an inertial regime of weak surface wave turbulence (weak-SWT) has been pointed out as a cascade of discrete modes with amplitudes compatible with universal Kolmogorov`s scaling. The transit from weak to strong surface turbulence has been explored with this tool revealing the importance of inertia over surface forces to …
Although the Navier-Stokes equation (NSE) is derived under angular-momentum conservation (AMC), numerical simulation methods often lack it. Here, we reveal that AMC violations result from implementation of the degenerated viscous terms of NSE. To maintain AMC, these degenerated terms must be separately integrated in accordance with their stress origins. As observed in particle-based hydrodynamics methods, the violation causes artificial rotations in multi-component fluids with different viscosities. At the interface between two fluids or with a mobile solid object, AMC must be satisfied, where…
Relativistic bulk viscous fluids of Burgers type and their presence in neutron stars
本研究证明三种独立化学成分经两种反应的混合物是符合Burgers模型的体粘滞流体，且考虑μ子的中子星物质属于此类，突破了传统Maxwell-Cattaneo弛豫的认知。
The flow and heat transfer in a viscous fluid over an unsteady stretching surface
本文研究了粘性流体在水平薄片上的流动与传热。片材的拉伸速率和温度随时间变化。通过相似变换，将动量和热能的控制方程化为常微分方程。这些方程近似求解的最优同伦渐近方法（OHAM），这为我们提供了一个方便的方法来控制逼近解的收敛性，并调整收敛严格时，必要的。算例结果表明，该方法简单有效。
Due to the reversibility of viscous flow it is not expected to obtain a fluidic rectifier simply from geometrical asymmetry without any moving mechanical parts. Here, we found a counter example by using spatial asymmetry combined with electric field to inject memory effects that render the flow irreversible. This stems from the strong dependency of the electrorheological (ER) fluid particle chaining on the flow direction. A funnel-shaped microfluidic rectifier with ER fluid has been shown to be easily and rapidly tuned via the applied electric field to achieve more than an order of magnitude r…
It is shown that the helicity of three dimensional viscous incompressible flow can be identified with the overall linking of the fluid's initial vorticity to the expectation of a stochastic mean field limit. The relevant mean field limit is obtained by following the Lagrangian paths in the stochastic Hamiltonian interacting particle system of [S. Hochgerner, Proc. R. Soc. A 474:20180178].
We consider a model of a third grade viscous Korteweg--type fluid in three space dimensions, and apply the extended Liu procedure in order to explicitly solve the constraints imposed by the entropy principle on the non--local constitutive relations. We detail the algorithm we use, and are able to characterize the material functions involved in the constitutive equations. In a natural way, the application of the extended Liu procedure allows us to recover an extra term in the entropy flux, preserving all the features of third grade viscous Korteweg--type fluids. Moreover, a further constraint, …
We present a novel computational modeling framework to numerically investigate fluid-structure interaction in viscous fluids using the phase field embedding method. Each rigid body or elastic structure immersed in the incompressible viscous fluid matrix, grossly referred to as the particle in this paper, is identified by a volume preserving phase field. The motion of the particle is driven by the fluid velocity in the matrix for passive particles or combined with its self-propelling velocity for active particles. The excluded volume effect between a pair of particles or between a particle and …
Dissipation can be represented in Hamiltonian mechanics in an extended phase space as a symplectic process. The method uses an auxiliary variable which represents the excitation of unresolved dynamics and a Hamiltonian for the interaction between the resolved dynamics and the auxiliary variable. This method is applied to viscous dissipation (including hyper-viscosity) in a two-dimensional fluid, for which the dynamics is non-canonical. We derive a metriplectic representation and suggest a measure for the entropy of the system.
Incompressible viscous fluid flows in a thin spherical shell
本文利用球上二维Navier-Stokes方程斯托克斯方程研究了不可压缩粘性流体在薄球壳中流动的线性化稳定性。球上的定常流在球的北极和南极有两个奇点（汇和源）。本文对线性化的Navier-斯托克斯方程解析地证明了定常流是渐近稳定的。当球形层被截断在两个对称环之间时，我们用幂级数解数值研究了线性化方程的特征值，并表明定常流动对所有雷诺数都保持渐近稳定。
Frequency downshift in a viscous fluid
在本文中，我们从Dias，Dyachenko和Zakharov（2008）引入的欧拉方程的弱粘性推广中推导出Dysthe（1979）系统的粘性推广。这种“粘性Dysthe”系统模拟了深水中弱粘性、近单色波列的演化。它包含一个术语，提供了一种在没有风和波浪破碎的情况下频率下移的机制。该方程不保留谱均值。数值模拟表明，谱平均值通常减少，谱峰减少某些初始条件。粘性Dysthe系统的平面波解的线性稳定性分析表明，波数接近于零的波比波数远离于零的波衰减得慢。NLS，耗散NLS，Dysthe，粘性Dysthe系统的实验数据和数值模拟之间的比较建立，粘性Dysthe系统准确地模拟数据从实验中观察到的频率下移和实验中没有观察到的频率下移。
Viscous fingering patterns in ferrofluids
粘性指进发生在两种不混溶的粘性流体在Hele-Shaw池的板之间的流动中。由于压力梯度或重力，分离两种流体的初始平面界面经历Saffman-Taylor不稳定性并形成指状结构。当其中一种流体是铁磁流体并施加垂直磁场时，该不稳定性补充了通常的粘性指进不稳定性，导致视觉上引人注目的复杂图案。我们认为这个问题在一个矩形流动的几何形状，使用微扰模式耦合分析。我们推导出两个一般结果：流体之间的粘度差驱动界面不对称，而没有来自磁力的贡献;铁磁流体内的磁排斥产生指尖分裂，这在普通流体的矩形几何形状中是不存在的。
Ultra-pure conductors may exhibit hydrodynamic transport where the collective motion of charge carriers resembles the flow of a viscous fluid. In a confined geometry (e.g., in ultra-high quality nanostructures) the electronic fluid assumes a Poiseuille-like flow. Applying an external magnetic field tends to diminish viscous effects leading to large negative magnetoresistance. In two-component systems near charge neutrality the hydrodynamic flow of charge carriers is strongly affected by the mutual friction between the two constituents. At low fields, the magnetoresistance is negative, however …
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