Singular Forces Research Articles

A quick-responding technique for parameters of turning maneuver

A novel and fast technique is proposed to predict the parameters of turning maneuvering performance of a surface combatant at mid and high Froude numbers. An empirical model with two parameters (ship advance speed and rudder angle) and three unknown coefficients is introduced. First, resistance of rudder appended hull has been computed by the time-dependent straight-ahead towing simulations. The code has been verified by implementing three different solution verification methods; Grid Convergence Index (GCI), Correction Factor (CF) and Factors of Safety (FS). After this, Direct Maneuvering Simulations (DMS) have been performed by implementing overset grid technique. Singular body forces located at the same position and direction with the propellers have been applied to the vessel for propulsion during DMS. The method has been validated by comparing the results of bare hull straight-ahead towing case and δ = 35° turning circle simulation ones with the available experimental data. The numerical results have then been used to calculate the unknown coefficients by linear regression analysis. It has been found that the proposed CFD-based empirical model is reliable and robust for both interpolated and extrapolated cases as well as the test matrix with reasonable accuracy and also applicable to similar surface ships.

Convergence of a linearly transformed particle method for aggregation equations

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in L^1 and L^infty norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in L^1 cap L^p norm.

Global Regularity for 1D Eulerian Dynamics with Singular Interaction Forces

The Euler--Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in [Do et al., Arch. Ration. Mech. Anal., 228 (2018), pp. 1--37]. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.

Traces of a Revolution

Traces of a Revolution

Nonstationary Navier–Stokes equations with singular time-dependent external forces

We establish a sufficient condition for the existence of solutions to the incompressible Navier–Stokes equations, with singular time-dependent external forces defined in terms of capacity CapH1,2(E).

Thermal fluctuations of an interface near a contact line.

The effect of thermal fluctuations near a contact line of a liquid interface partially wetting an impenetrable substrate is studied analytically and numerically. Promoting both the interface profile and the contact line position to random variables, we explore the equilibrium properties of the corresponding fluctuating contact line problem based on an interfacial Hamiltonian involving a "contact" binding potential. To facilitate an analytical treatment, we consider the case of a one-dimensional interface. The effective boundary condition at the contact line is determined by a dimensionless parameter that encodes the relative importance of thermal energy and substrate energy at the microscopic scale. We find that this parameter controls the transition from a partial wetting to a pseudopartial wetting state, the latter being characterized by a thin prewetting film of fixed thickness. In the partial wetting regime, instead, the profile typically approaches the substrate via an exponentially thinning prewetting film. We show that, independently of the physics at the microscopic scale, Young's angle is recovered sufficiently far from the substrate. The fluctuations of the interface and of the contact line give rise to an effective disjoining pressure, exponentially decreasing with height. Fluctuations therefore provide a regularization of the singular contact forces occurring in the corresponding deterministic problem.

L2-asymptotic stability of singular solutions to the Navier–Stokes system of equations in [formula omitted

We consider global-in-time small solutions of the initial value problem to the incompressible Navier–Stokes equations in R3. Usually, such solutions do not belong to L2(R3) and may be singular if they correspond to singular external forces. However, we prove that we can still establish their asymptotic stability under arbitrarily large initial L2-perturbations.

Accelerator-feasibleN-body nonlinear integrable system

Nonlinear N-body integrable Hamiltonian systems, where N is an arbitrary number, attract the attention of mathematical physicists for the last several decades, following the discovery of some number of these systems. This paper presents a new integrable system, which can be realized in facilities such as particle accelerators. This feature makes it more attractive than many of the previous such systems with singular or unphysical forces.

Numerical study of flapping filaments in a uniform fluid flow

The coupled dynamics of multiple flexible filaments (also called monodimensional flags) flapping in a uniform fluid flow is studied numerically for the cases of a side-by-side arrangement, and an in-line configuration. The modal behaviour and hydrodynamical properties of the sets of filaments are studied using a Lattice Boltzmann–Immersed Boundary method. The fluid momentum equations are solved on a Cartesian uniform lattice while the beating filaments are tracked through a series of markers, whose dynamics are functions of the forces exerted by the fluid, the filaments flexural rigidity and the tension. The instantaneous wall conditions on the filaments are imposed via a system of singular body forces, consistently discretised on the lattice of the Boltzmann equation. The results exhibit several flapping modes for two and three filaments placed side-by-side and are compared with experimental and theoretical studies. The hydrodynamical drafting, observed so far only experimentally on configurations of in-line flexible bodies, is also revisited numerically in this work, and the associated physical mechanism is identified. In certain geometrical and structural configuration, it is found that the upstream body experiences a reduced drag compared to the downstream body, which is the contrary of what is encountered on rigid bodies (cars, bicycles).

A second order virtual node algorithm for Navier–Stokes flow problems with interfacial forces and discontinuous material properties

We present a numerical method for the solution of the Navier–Stokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and velocity boundary conditions on irregular domains. The method improves on results in [1] (which solved the Stokes equations in two dimensions) by providing treatment of fluid inertia as well as a new discretization of jump and boundary conditions that accurately resolves null modes in both two and three dimensions. We discretize the equations using an embedded approach on a uniform MAC grid to yield discretely divergence-free velocities that are second order accurate. We maintain our interface using the level set method or, when more appropriate, the particle level set method. We show how to implement Dirichlet (known velocity), Neumann (known normal stress), and slip velocity boundary conditions as special cases of our interface representation. The method leads to a discrete, symmetric KKT system for velocities, pressures, and Lagrange multipliers. We also present a novel simplification to the standard combination of the second order semi-Lagrangian and BDF schemes for discretizing the inertial terms. Numerical results indicate second order spatial accuracy for the velocities (L∞ and L2) and first order for the pressure (in L∞, second order in L2). Our temporal discretization is also second order accurate.

Particle approximation of Vlasov equations with singular forces: Propagation of chaos

We obtain the mean field limit and the propagation of chaos for a system of particles interacting with a singular interaction force of the type $1/|x|^\alpha$, with $\alpha <1$ in dimension $d \geq 3$. We also provide results for forces with singularity up to $\alpha < d-1$ but with large enough cut-off. This last result thus almost includes the most interesting case of Coulombian or gravitational interaction, but it is also interesting when the strength of the singularity $\alpha$ is larger but close to one, in which case it allows for very small cut-off.

Stationary Navier–Stokes equations with critically singular external forces: Existence and stability results

We show the unique existence of solutions to stationary Navier–Stokes equations with small singular external forces belonging to a critical space. To the best of our knowledge, this is the largest critical space that is available up until now for this kind of existence. This result can be viewed as the stationary counterpart of the existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier–Stokes equations with initial data in BMO−1. The stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup.

A Multigrid Method for a Model of the Implicit Immersed Boundary Equations

AbstractExplicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain stability. Solving the equations that arise from an implicit discretization is difficult. Recently, several different approaches have been proposed, but a complete understanding of this problem is still emerging. A multigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forces on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but the efficiency gain is limited. The multigrid method alone may not be an effective solver, but when used as a preconditioner for Krylov methods, the speed-up over the explicit-time method is substantial. For example, depending on the constitutive law for the boundary force, with a time step 100 times larger than the explicit method, the implicit method is about 15-100 times more efficient than the explicit method. A very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary points.

Modeling mechanical contact and lubrication in Direct Numerical Simulations of colliding particles

We developed a model for inexpensive Direct Numerical Simulations of particle-laden flow by fully resolving the hydrodynamics at all times except when the gap between colliding particles becomes comparable to the grid step. The resolved hydrodynamics were obtained with a previously developed pressure boundary integral method for direct fluid–particle simulations on Cartesian grids. The unresolved part of the lubrication pressure/shear force in the subgrid gap was predicted using theoretical Stokes flow models. Singular lubrication forces were avoided by postulating that contact begins when the gap distance between the particles becomes equal to the size of micro-asperities. The mechanical contact between particles is dynamically resolved and particle interactions are both inelastic and frictional. The proposed numerical model was validated through resolution tests and comparison with experimental data for immersed binary collisions.

Time Evolution of Two Dimensional Systems with Infinitely Many Particles Mutually Interacting via Very Singular Forces

We study the existence and uniqueness of the time evolution via the Newton law of a two dimensional system of infinitely many particles with very singular mutual interactions. It is an improvement of the result by Fritz and Dobrushin given in (Comm. Math. Phys. 57:67–81, 1977) for inverse power-like singular interactions.

A new enrichment space for the treatment of discontinuous pressures in multi‐fluid flows

SUMMARYIn this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared with the classical P1 space and to the space proposed by Ausas et al (Comp. Meth. Appl. Mech. Eng., Vol. 199, 1019–1031, 2010) in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with another enrichment that accommodates discontinuities in the pressure gradient has also been explored, exhibiting excellent results in problems involving jumps in the density or the volume forces. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Malcolm X: a life of reinvention

Selected by The New York Times Book Review as a Notable Book of the Year Years in the making-the definitive biography of the legendary black activist. Of the great figure in twentieth-century American history perhaps none is more complex and controversial than X. Constantly rewriting his own story, he became a criminal, a minister, a leader, and an icon, all before being felled by assassins' bullets at age thirty-nine. Through his tireless work and countless speeches he empowered hundreds of thousands of black Americans to create better lives and stronger communities while establishing the template for the self-actualized, independent African American man. In death he became a broad symbol of both resistance and reconciliation for millions around the world. Manning Marable's new biography of is a stunning achievement. Filled with new information and shocking revelations that go beyond the Autobiography, X unfolds a sweeping story of race and class in America, from the rise of Marcus Garvey and the Ku Klux Klan to the struggles of the civil rights movement in the fifties and sixties. Reaching into Malcolm's troubled youth, it traces a path from his parents' activism through his own engagement with the Nation of Islam, charting his astronomical rise in the world of Black Nationalism and culminating in the never-before-told true story of his assassination. Malcolm X will stand as the definitive work on one of the most singular forces for social change, capturing with revelatory clarity a man who constantly strove, in the great American tradition, to remake himself anew.

A numerical method for two phase flows with insoluble surfactants

In many practical multiphase flow problems, i.e. treatment of gas emboli and various microfluidic applications, the effect of interfacial surfactants, or surface reacting agents, on the surface tension between the fluids is important. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. The equations for the location of the interface and the surfactant concentration on the interface are coupled with the Navier–Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration. A new accurate and inexpensive numerical method for simulating the evolution of insoluble surfactants is presented in this paper. It is based on an explicit yet Eulerian discretization of the interface, which for two dimensional flows allows for the use of uniform one dimensional grids to discretize the equation for the interfacial surfactant concentration. A finite difference method is used to solve the Navier–Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. The timestepping is based on a Strang splitting approach. Drop deformation in shear flows in two dimensions is considered. Specifically, the effect of surfactant concentration on the deformation of the drops is studied for different sets of flow parameters.

Generalised Hele-Shaw flow: A Schwarz function approach

An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz functiong(z,t) of the free boundary. This generalises the well-known equation ∂g/∂t= 2∂w/∂z, wherewis the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

A general solution of equations of equilibrium in linear elasticity

A general solution of equations of equilibrium in linear elasticity is presented in cylindrical coordinates in terms of three harmonic functions describing an arbitrary displacement field. The structure of this solution is similar to the general solution given by Love (Kelvin’s solution) in spherical coordinates. Galerkin vector representation of our solution leads to an integral connecting the harmonic functions. The connections to Papkovich–Neuber and Muki’s general representations are also provided. Suitable choices of the harmonic functions in our new representation yield general solutions for axisymmetric deformations due to Love, Boussinesq and Michell. Some unbounded deformations induced by singular forces are tabulated in terms of the scalar harmonic functions to justify the simple nature of our representation. Exact solution of the half-space boundary value problem is also provided to demonstrate the power of our approach. The stress components computed via our solution are also listed (see the Appendix).