196 research outputs found
Starting solutions for some simple oscillating motions of second-grade fluids
The exact starting solutions corresponding to the motions of a second-grade fluid, due to the cosine and sine oscillations of an infinite edge and of an infinite duct of rectangular cross-section as well as those induced by an oscillating pressure gradient in such a duct, are determined by means of the double Fourier sine transforms. These solutions, presented as sum of the steady-state and transient solutions, satisfy both the governing equations and all associate initial and boundary conditions. In the special case when Ī±1ā0, they reduce to those for a Navier-Stokes fluid
A modified particle method for semilinear hyperbolic systems with oscillatory solutions
We introduce a modified particle method for semi-linear hyperbolic systems with highly oscillatory solutions. The main feature of this modified particle method is that we do not require different families of characteristics to meet at one point. In the modified particle method, we update the ith component of the solution along its own characteristics, and interpolate the other components of the solution from their own characteristic points to the ith characteristic point. We prove the convergence of the modified particle method essentially independent of the small scale for the variable coefficient Carleman model. The same result also applies to the non-resonant Broadwell model. Numerical evidence suggests that the modified particle method also converges essentially independent of the small scale for the original Broadwell model if a cubic spline interpolation is used
First-order aggregation models with alignment
We include alignment interactions in a well-studied first-order
attractive-repulsive macroscopic model for aggregation. The distinctive feature
of the extended model is that the equation that specifies the velocity in terms
of the population density, becomes {\em implicit}, and can have non-unique
solutions. We investigate the well-posedness of the model and show rigorously
how it can be obtained as a macroscopic limit of a second-order kinetic
equation. We work within the space of probability measures with compact support
and use mass transportation ideas and the characteristic method as essential
tools in the analysis. A discretization procedure that parallels the analysis
is formulated and implemented numerically in one and two dimensions
Variational Multisymplectic Formulations of Nonsmooth Continuum Mechanics
This paper develops the foundations of the multisymplectic
formulation of nonsmooth continuum mechanics. It may be regarded as a PDE generalization of previous techniques that developed a variational approach to collision problems. These methods have already proved of value in
computational mechanics, particularly in the development of asynchronous integrators and efficient collision methods. The present formulation also includes solid-fluid interactions and material interfaces and, in addition, lays
the groundwork for a treatment of shocks
Nonsmooth Lagrangian mechanics and variational collision integrators
Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.
Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated
Comments on: "Starting solutions for some unsteady unidirectional flows of a second grade fluid," [Int. J. Eng. Sci. 43 (2005) 781]
A significant mathematical error is identified and corrected in a recent
highly-cited paper on oscillatory flows of second-grade fluids [Fetecau &
Fetecau (2005). Int. J. Eng. Sci., 43, 781--789]. The corrected solutions are
shown to agree identically with numerical ones generated by a finite-difference
scheme, while the original ones of Fetecau & Fetecau do not. A list of other
recent papers in the literature that commit the error corrected in this Comment
is compiled. Finally, a summary of related erroneous papers in this journal is
presented as an Appendix.Comment: 8 pages, 2 figures (4 images), elsarticle class; accepted for
publication in International Journal of Engineering Scienc
Aggregation-diffusion energies on Cartan-Hadamard manifolds of unbounded curvature
We consider an aggregation-diffusion energy on Cartan-Hadamard manifolds with
sectional curvatures that can grow unbounded at infinity. The energy
corresponds to a macroscopic aggregation model that involves nonlocal
interactions and linear diffusion. We establish necessary and sufficient
conditions on the growth at infinity of the attractive interaction potential
for ground states to exist. Specifically, we derive explicit conditions on the
attractive potential in terms of the bounds on the sectional curvatures at
infinity. To prove our results we establish a new comparison theorem in
Riemannian geometry and a logarithmic Hardy-Littlewood inequality on
Cartan-Hadamard manifolds.Comment: arXiv admin note: text overlap with arXiv:2306.0485
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