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The settling dynamics of spherical and elliptical particles in a viscous Newtonian fluid are investigated numerically using a finite difference technique. The terminal velocity for spherical particles is calculated for different system sizes and the extrapolated value for an infinite system size is compared to the Oseen approximation. Special attention is given to the settling and tumbling motion of elliptical particles where their terminal velocity is compared with the one of the surface equivalent spherical particle.

An approach to lattice Boltzmann simulation is described, which makes a direct connection between classical kinetic theory and contemporary lattice Boltzmann modeling methods. This approach can lead to greater accuracy, improved numerical stability and significant reductions in computational needs, while giving a new philosophical point of view to lattice Boltzmann calculations for a large range of applications.

Analytical solutions of the Navier–Stokes equation based on a locally fully-developed flow assumption with various gas slip models are presented and comparisons for velocity profile, flow rate, friction factor, and pressure distribution are performed. The effect of the second-order coefficient in the slip boundary condition becomes significant as the Knudsen number increases. Most slip models are limited to slip regime or marginally transition regime and break down around Kn = 0.1 while Sreekanth's model, followed by Mitsuya's model, gives a good agreement with the linearized Boltzmann solutions from slip regime up to Kn = 2 for flow rate and friction factor predictions. These two models should be of great use for slip flow analysis in micro-electro-mechanical systems (MEMS) and, in particular, in situations where the flow rate and flow resistance are of interest.

The motivation for this work was a simple experiment [P. M. C. de Oliveira, S. Moss de Oliveira, F. A. Pereira and J. C. Sartorelli, preprint (2010), arXiv:1005.4086], where a little polystyrene ball is released falling in air. The interesting observation is a speed breaking. After an initial nearly linear time-dependence, the ball speed reaches a maximum value. After this, the speed finally decreases until its final, limit value. The provided explanation is related to the so-called von Kármán street of vortices successively formed behind the falling ball. After completely formed, the whole street extends for some hundred diameters. However, before a certain transient time needed to reach this steady-state, the street is shorter and the drag force is relatively reduced. Thus, at the beginning of the fall, a small and light ball may reach a speed superior to the sustainable steady-state value.

Besides the real experiment, the numerical simulation of a related theoretical problem is also performed. A cylinder (instead of a 3D ball, thus reducing the effective dimension to 2) is positioned at rest inside a wind tunnel initially switched off. Suddenly, at t = 0 it is switched on with a constant and uniform wind velocity far from the cylinder and perpendicular to it. This is the first boundary condition. The second is the cylinder surface, where the wind velocity is null. In between these two boundaries, the velocity field is determined by solving the Navier–Stokes equation, as a function of time. For that, the initial condition is taken as the known Stokes laminar limit V → 0, since initially the tunnel is switched off. The numerical method adopted in this task is the object of the current text.

We present a mapping between a Schrödinger equation with a shifted nonlinear potential and the Navier–Stokes equation. Following a generalization of the Madelung transformations, we show that the inclusion of the Bohm quantum potential plus the laplacian of the phase field in the nonlinear term leads to continuity and momentum equations for a dissipative incompressible Navier–Stokes fluid. An alternative solution, built using a complex quantum diffusion, is also discussed. The present models may capture dissipative effects in quantum fluids, such as Bose–Einstein condensates, as well as facilitate the formulation of quantum algorithms for classical dissipative fluids.

We present a numerical method to deal efficiently with large numbers of particles in incompressible fluids. The interactions between particles and fluid are taken into account by a physically motivated ansatz based on locally defined drag forces. We demonstrate the validity of our approach by performing numerical simulations of sedimenting non-Brownian spheres in two spatial dimensions and compare our results with experiments. Our method reproduces qualitatively important aspects of the experimental findings, in particular the strong anisotropy of the hydrodynamic bulk self-diffusivities.

We show that the renormalization group flows of the massless superstring modes in the presence of fluctuating D-branes satisfy the equations of fluid dynamics. In particular, we show that the D-brane's U(1) field is related to the velocity function in the Navier–Stokes equation while the dilaton plays the role of the passive scalar advected by the turbulent liquid. This leads us to suggest a possible isomorphism between the off-shell superstring theory in the presence of fluctuating D-branes and the fluid mechanical degrees of freedom.

By applying the Hamilton’s quaternion algebra, we propose the generalized electromagnetic-fluid dynamics of dyons governed by the combination of the Dirac–Maxwell, Bernoulli and Navier–Stokes equations. The generalized quaternionic hydro-electromagnetic field of dyonic cold plasma consists of electrons and magnetic monopoles in which there exist dual-mass and dual-charge species in the presence of dyons. We construct the conservation of energy and conservation of momentum equations by equating the quaternionic scalar and vector parts for generalized hydro-electromagnetic field of dyonic cold plasma. We propose the quaternionic form of conservation of energy is related to the Bernoulli-like equation while the conservation of momentum is related to Navier–Stokes-like equation for dynamics of dyonic plasma fluid. Further, the continuity equation, i.e. the conservation of electric and magnetic charges with the dynamics of hydro-electric and hydro-magnetic flow of conducting cold plasma fluid is also analyzed. The quaternionic formalism for dyonic plasma wave emphasizes that there are two types of waves propagation, namely the Langmuir-like wave propagation due to electrons, and the ’t Hooft–Polyakov-like wave propagation due to magnetic monopoles.

Fractal fluid is considered in the framework of continuous models with noninteger dimensional spaces (NIDS). A recently proposed vector calculus in NIDS is used to get a description of fractal fluid flow in pipes with circular cross-sections. The Navier–Stokes equations of fractal incompressible viscous fluids are used to derive a generalization of the Poiseuille equation of steady flow of fractal media in pipe.

Aerodynamic characteristics of an Ejection Seat System at different angles of attack are studied by the numerical method and the flow mechanisms for such flows are carefully analyzed. The governing equations are Reynolds-averaged Navier-Stokes equations which are solved by the unstructured finite volume method. Upwind Osher scheme is used for spatial discretization and five-stage Runge-Kutta scheme is applied for temporal discretization. The DES model based on S-A one equation turbulence model is adopted. Parallel computation is based on the domain decomposition method and multi-block is achieved by using METIS system. The experimental data is used to validate this method. This research is helpful to understand the aerodynamic characteristics and flow mechanisms of Ejection Seat System at different angles of attack and Mach numbers, and can provide reasonable reference for Ejection Seat System design.

We study the large-time behavior of a spherically symmetric motion of isentropic and compressible viscous gas in a field of potential force over an unbounded exterior domain in ℝⁿ(n≥2). First, we show the unique existence of a stationary solution satisfying an adhesion boundary condition and a positive spatial asymptotic condition. Then, it is shown that the stationary solution becomes a time asymptotic state to the initial boundary value problem with the same boundary and spatial asymptotic conditions. Here, the initial data can be chosen arbitrarily large if it belongs to the suitable Sobolev space. Moreover, if the external force is attractive to the center of a sphere, it can also be taken arbitrarily large. The proof of the stability theorem is based on computations, executed by using the Lagrangian coordinate. In the proof, it is the essential step to obtain the pointwise estimate for the density. It is derived through employing a representation formula of the density with the aid of the standard energy method. The Hölder regularity of the initial data is also required for translating the results in the Lagrangian coordinate to those in the Eulerian coordinate.

This paper is devoted to studying the well-posedness and optimal solution techniques for a full discretization scheme, proposed by Lee and Xu (2006), for a large class of viscoelastic flow models. By using some special properties of the scheme such that it is stable in the energy norm and it preserves the positivity of the conformation tensor, the global existence and uniqueness of the solution of the discrete scheme is established. Furthermore, it is shown that the solution of the discrete scheme at each time step can be obtained by an iterative procedure that only requires operations (with N being the number of nodes of the underlying finite element grid).

This paper has two objectives. On one side, we develop and test numerically divergence-free Virtual Elements in three dimensions, for variable “polynomial” order. These are the natural extension of the two-dimensional divergence-free VEM elements, with some modification that allows for a better computational efficiency. We test the element’s performance both for the Stokes and (diffusion dominated) Navier–Stokes equation. The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. We build a pair of virtual discrete spaces based on general polytopal partitions, the first one being scalar and the second one being vector valued, such that when coupled with our velocity and pressure spaces, yield a discrete Stokes complex.

This work is devoted to the study of the hydrodynamic limit for the fluid-particle flows governed by the Vlasov–Fokker–Planck (VFP) equation coupled with the incompressible Navier–Stokes (INS) equation as the Deborah number approaches to zero. The limit is valid globally in time provided that the initial perturbation is small in a neighborhood of a steady state. The proof is based on a formal derivation of the limiting system via the Hilbert approach, followed by a rigorous justification via introducing a novel decomposition involving some macroscopic quantities and a refined energy estimate motivated by macro–micro decomposition. In contrast to the existing results for the same scaled model, the present work provides the first one on the hydrodynamic limits in a strong sense with an explicit convergence rate.

In the paper, we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss–Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of the Hausdorff vector calculus. The formula is proposed as a mathematical tool to describe the real-world problems for the fractal power-law flow equations with the anomalous diffusion equation. A conjecture for the fractal power-law flow equations analogous to the Smale’s 15th problem and one of Millennium Prize Problems for the Navier–Stokes equations is also addressed.

In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier–Stokes equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier–Stokes equation, then this solution satisfies our tamed equation. Moreover, using this tamed equation we can give a new construction for a suitable weak solution of the classical 3D Navier–Stokes equation introduced in Refs. 16 and 2.

A Cartesian grid method for computing flows with complex immersed, stationary and moving boundaries is presented in this paper. We introduce an augmented projection method for the numerical solution of the incompressible Navier-Stokes equations in arbitrary domains. In a projection method an intermediate velocity field is calculated from the momentum equations, which is then projected onto the space of divergence-free vector fields. In the proposed augmented projection method, we add one more step, which effectively eliminates spurious velocity field caused by complex immersed moving boundaries. The methodology is validated by comparing it with analytic, previous numerical and experimental results.

This report deals with the slipping friction in a progressive wave of real fluids propagating over a permeable bottom. Because the "no-slip" condition is usually considered in the wave motion, the horizontal velocity at the seabed is assumed zero. However, based on the numerical simulations and the laboratorial experiment, the slip velocity occurs at the interface as waves pass a permeable bed. It is found that the adherence condition is not suitable for permeable bed. The "overshooting" which is the local maximum horizontal velocity in the wave boundary layer is related to the wave period and the fluid viscosity.

Both the linearized Navier-Stokes equation and the slip boundary condition are applied to solve the problem of progressive waves over a permeable bed of finite thickness. In the interface of soil and fluid, the slip velocity is simply assumed to be proportional to the characteristic velocity. Thus, the slipping effect and the permeability of the bottom on the velocity near the seabed can be considered. The linear solutions indicated that the joint effect of slipping friction, permeability and the effect of bottom thickness are crucial for overshooting. Comparing with the experimental results shows that the overshooting phenomena can be explained by this joint effect.

We show that a certain stochastic perturbation of the flow of perfect incompressible fluid under some special external force on the flat n-dimensional torus yields a solution of Navier–Stokes equation without external force in the tangent space at unit of volume preserving diffeomorphism group. If that external force is absent, the equation turns into the one of Reynolds type. For the flow of diffuse matter this construction yields the Burgers equation.

The Navier–Stokes equation is a key governing equation for the motion of viscous fluid flow. The main target of our work is to obtain the solution to multi-dimensional Navier–Stokes equations in the Yang–Abdel–Cattani fractional sense. The results of the model are in terms of the three-parameter Prabhakar function. Three examples are discussed that depend on the recommended method for a novel Yang–Abdel–Cattani arbitrary order operator. The obtained results are plotted with the help of MATLAB R2016a mathematical software.