4,239 research outputs found
Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity
No full textUsing the method of infinitesimal transformations, a 6-parameter family of exact solutions describing nonlinear sheared flows with a free surface are found. These solutions are a hybrid between the earlier self-propagating simple wave solutions of Freeman, and decaying solutions of Sachdev. Simple wave solutions are also derived via the method of infinitesimal transformations. Incomplete beta functions seem to characterize these (nonlinear) sheared flows in the absence of critical levels
Generalized Burgers equations and Euler-Painleve transcendents. II
No full textIt was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler Painlevé equation yy[script `]+ay[script ']2+ f(x)yy[script ']+g(x) y2+by[script ']+c=0 represents the generalized Burgers equations (GBE's) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE ut+uaux+Ju/2t =(gd/2)uxx (the nonplanar Burgers equation) is considered. It is found that its self-similar form is again governed by the Euler Painlevé equation. The ranges of the parameter alpha for which solutions of the connection problem to the self-similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE's. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well-known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler Painlevé equation with respect to GBE's. Journal of Mathematical Physics is copyrighted by The American Institute of Physics
Initial boundary value problems for scalar and vector burgers equations
No full textIn this article we study Burgers equation and vector Burgers equation with initial and boundry conditions. First we consider the Burgers equation in the quarter plane x > 0, t > 0 with Riemann type of initial and boundary conditions and use the HOPf-cole transformation to linearize the problems and explicitily solve them. We study two limits, the small viscosity limit and the large time behaviour of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gredient of a scalar function. We investigate the asymptonic behaviour of this solution as time tends to infinity and generalize a rsult of HOPf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an intitial value problem etending the previous work of Sachdev et al. (1994). We also study the limits as viscosity parameter goes to 0. Finally, we get an explicit solution for boundry value problem in a cylinder
Analytic and Numerical Study of N-Waves Governed by the Nonplanar Burgers Equation
No full textIn this article, evolution of N-waves under the nonplanar Burgers equation, which takes into account geometrical expansion or contraction, is treated analytically. An exact asymptotic solution, generalizing that for the planar Burgers equation, is given for the case of expansion. An approximate treatement, using a balancing argument, gives asymptotic analytic results for both expansion and contraction. The analysis is fortified by an accurate numerical solution of the problem. This study is brought in close conjunction with the earlier work of Crighton and Scott [13] and Sachdev, Joseph and Nair [3]
Generalized Burgers equations and Euler–Painlevé transcendents. III
No full textIt was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986); P. L. Sachdev and K. R. C. Nair, ibid. 28, 977 (1987)] that the Euler–Painlevé equations y(d2y/dη2)+a(dy/dη)2 +f(η)y(dy/dη)+g(η)y2+b(dy/dη) +c=0 represent generalized Burgers equations (GBE’s) in the same way as Painlevé equations represent the Korteweg–de Vries type of equations. The earlier studies were carried out in the context of GBE’s with damping and those with spherical and cylindrical symmetry. In the present paper, GBE’s with variable coefficients of viscosity and those with inhomogeneous terms are considered for their possible connection to Euler–Painlevé equations. It is found that the Euler–Painlevé equation, which represents the GBE ut+uβux=(δ/2)g(t)uxx, g(t)=(1+t)n, β>0, has solutions, which either decay or oscillate at η=±∞, only when −1<n<1. The solutions are shocklike when n=1. On the other hand, they oscillate over the whole real line when n=−1. Furthermore, the solutions monotonically decay both at η=+∞ and η=−∞, that is, they have a single hump form if β≥βn=(1−n)/(1+n). For β<βn, the solutions have an oscillatory behavior either at η=+∞ or at η=−∞, or at η=+∞ and η=−∞. For β=βn, there exists a single parameter family of exact single hump solutions, similar to those found for the nonplanar Burgers equations in Paper II. Thus the parametric value β=βn seems to bifurcate the families of solutions, which remain bounded at η=±∞. Other GBE’s considered here are also found to be reducible to Euler–Painlevé equations
Instabilities induced by variation of Brunt-Vaisala frequency in compressible stratified shear flows
No full textThe stability characteristics of a Helmholtz velocity profile in a stably stratified, compressible fluid in the presence of a lower rigid boundary are studied. A jump in the Brunt-Vaisala frequency at a level different from the shear zone is introduced and the variation of the Brunt-Vaisala frequency with respect to the vertical coordinate in the middle layer of the three-layered model is considered. An analytic solution in each of the layers is obtained, and the dispersion relation is solved numerically for parameters relevant to the model. The effect of shear in the lowermost layer of the three-layered model for a Boussinesq fluid is discussed. The results are compared with the earlier studies of Lindzen and Rosenthal, and Sachdev and Satya Narayanan. In the present model, new unstable modes with larger growth rates are obtained and the most unstable gravity wave modes are found to agree closely with the observed ones at various heights. Physics of Fluids is copyrighted by The American Institute of Physics
Shock waves & explosions
No full textUnderstanding the causes and effects of explosions is important to experts in a broad range of disciplines, including the military, industrial and environmental research, aeronautic engineering, and applied mathematics. Offering an introductory review of historic research, Shock Waves and Explosions brings analytic and computational methods to a wide audience in a clear and thorough way. Beginning with an overview of the research on combustion and gas dynamics in the 1970s and 1980s, the author brings you up to date by covering modeling techniques and asymptotic and perturbative methods and ending with a chapter on computational methods.Most of the book deals with the mathematical analysis of explosions, but computational results are also included wherever they are available. Historical perspectives are provided on the advent of nonlinear science, as well as on the mathematical study of the blast wave phenomenon, both when visualized as a point explosion and when simulated as the expansion of a high-pressure gas.This volume clearly reveals the ingenuity of the human mind to conceptualize, model, and mathematically analyze highly complicated nonlinear phenomena such as nuclear explosions. It presents a solid foundation of knowledge that encourages further research and original ideas
Evolution and decay of spherical and cylindrical N waves
No full textThe Burgers equation, in spherical and cylindrical symmetries, is studied numerically using pseudospectral and implicit finite difference methods, starting from discontinuous initial (N wave) conditions. The study spans long and varied regimes–embryonic shock, Taylor shock, thick evolutionary shock, and (linear) old age. The initial steep-shock regime is covered by the more accurate pseudospectral approach, while the later smooth regime is conveniently handled by the (relatively inexpensive) implicit scheme. We also give some analytic results for both spherically and cylindrically symmetric cases. The analytic forms of the Reynolds number are found. These give results in close agreement with those found from the numerical solutions. The terminal (old age) solutions are also completely determined. Our analysis supplements that of Crighton & Scott (1979) who used a matched asymptotic approach. They found analytic solutions in the embryonic-shock and the Taylor-shock regions for all geometries, and in the evolutionary-shock region, leading to old age, for the spherically symmetric case. The numerical solution of Sachdev & Seebass (1973) is updated in a comprehensive manner; in particular, the embryonic-shock regime and the old-age solution missed by their study are given in detail. We also study numerically the non-planar equation in the form for which the viscous term has a variable coefficient. It is shown that the numerical methods used in the present study are sufficiently versatile to tackle initial-value problems for generalized Burgers equations
The FM and PL Libraries Documentation
No full textBuilding complex SPMD code in an ecient and portable way is nowadays a challenge, especially when there is no uniformity of tools and libraries across platforms. The Fast Messages (FM) and the Portability Library (PL) where both designed to provide the basis of an abstract enough framework for C, so that problems can be coded and ported to any supported platform with no more than a few changes in the makeles and a recompilation. The FM library provides a message passing communications library built around the Berkeley Active Messages library. The PL library provides the primitives for host to node communication for problem initialization and results collection, as well as other miscellaneous and potentially non-portable primitives. This technical report contains the documentation for both libraries.Technical report LCSR-TR-25
Increasing science, technology, engineering, and mathematics skills using Project Lead the Way
No full textIncludes bibliographical references
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