P. L. Sachdev

Uniformly valid analytical solution to the problem of a decaying shock wave

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

Evolution and decay of spherical and cylindrical N waves

The 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.

Exact, self-similar, time-dependent free surface flows under gravity

A class of exact, self-similar, time-dependent solutions describing free surface flows under gravity is found which extends the self-propagating class of solutions discovered earlier by Freeman (1972) to those which decay with time.

Initial boundary value problems for scalar and vector burgers equations

In 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.

Exact N-wave solutions for the non-planar Burgers equation

An exact representation of N-wave solutions for the non-planar Burgers equation ut + uux + ½ju/t = ½δuxx, j = m/n, m < 2n, where m and n are positive integers with no common factors, is given. This solution is asymptotic to the inviscid solution for |x| < √(2Q0t), where Q0 is a function of the initial lobe area, as lobe Reynolds number tends to infinity, and is also asymptotic to the old age linear solution, as t tends to infinity; the formulae for the lobe Reynolds numbers are shown to have the correct behaviour in these limits. The general results apply to all j = m/n, m < 2n, and are rather involved; explicit results are written out for j = 0, 1, ½, 1/3 and 1/4. The case of spherical symmetry j = 2 is found to be ‘singular’ and the general approach set forth here does not work; an alternative approach for this case gives the large time behaviour in two different time regimes. The results of this study are com pared with those of Crighton & Scott (1979).

N-wave solution of modified Burgers equation

Abstract The modified Burgers equation u t + u n u x = δu xx 2 , where n ≥ 2 is even, is treated analytically for N-wave initial conditions. An exact asymptotic solution is presented, extending the validity of the linear solution far back in time.

Integrability and singularity structure of predator‐prey system

The ‘‘extended’’ ARS (Ablowitz, Ramani, and Segur) algorithm is introduced to characterize a dynamical system as Painleve or otherwise; to that end, it is required that the formal series—the Laurent series, logarithmic, algebraic psi series about a movable singularity—are shown to converge in the deleted neighborhood of the singularity. The determinations thus obtained are compared with those following from the α method of Painleve. An attempt is made to relate the structure of solutions about a movable singularity with that of first integrals (when they exist). All these ideas are illustrated by a comprehensive analysis of the general two‐dimensional predator‐prey system.

Analytic and numerical study of N-waves governed by the nonplanar burgers equation

In 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].

Singularity Structure of Third‐Order Dynamical Systems. I

A general third-order dynamical system with polynomial right-hand sides of finite degrees in the dependent variables is analyzed to unravel the singularity structure of its solutions about a movable singular point. To that end, the system is first transformed to a second-order Briot–Bouquet system and a third auxiliary equation via a transformation, similar to one used earlier by R. A. Smith in 1973–1974 for a general second-order dynamical system. This transformation imposes some constraints on the coefficients appearing in the general third-order system. The known results for the second-order Briot–Bouquet system are used to explicitly write out Laurent or psi-series solutions of the general third-order system about a movable singularity. The convergence of the relevant series solutions in a deleted neighborhood of the singularity is ensured. The theory developed here is illustrated with the help of the May–Leonard system.

Spherical piston problem in water

In this paper, we study the propagation of a shock wave in water, produced by the expansion of a spherical piston with a finite initial radius. The piston path in the x, t plane is a hyperbola. We have considered the following two cases: (i) the piston accelerates from a zero initial velocity and attains a finite velocity asymptotically as t tends to infinity, and (ii) the piston decelerates, starting from a finite initial velocity. Since an analytic approach to this problem is extremely difficult, we have employed the artificial viscosity method of von Neumann & Richtmyer after examining its applicability in water. For the accelerating piston case, we have studied the effect of different initial radii of the piston, different initial curvatures of the piston path in the x, t plane and the different asymptotic speeds of the piston. The decelerating case exhibits the interesting phenomenon of the formation of a cavity in water when the deceleration of the piston is sufficiently high. We have also studied the motion of the cavity boundary up to 550 cycles.