Phoolan Prasad

Nonlinear Hyperbolic Waves in Multi-Dimensions.

In this book, the authors have discussed the propagation of a curved nonlinear wavefront for the ordinary fluid dynamics(ie without considering magneto-fluids) according to two fundamental physical processes:i) at different points of the wavefront,it travels with different speeds depending on the local amplitude leading to a longitudinal stretching of rays; and ii) a lateral deviation of rays is produced due to non-uniform distribution of the amplitude of the wave front.

Coupled amplitude theory of nonlinear surface acoustic waves

The nonlinear propagation characteristics of surface acoustic waves on an isotropic elastic solid have been studied in this paper. The solution of the harmonic boundary value problem for Rayleigh waves is obtained as a generalized Fourier series whose coefficients are proportional to the slowly varying amplitudes of the various harmonics. The infinite set of coupled equations for the amplitudes when solved exhibit an oscillatory slow variation signifying a continuous transfer of energy back and forth among the various harmonics. A conservation relation is derived among all the harmonic amplitudes.

Transonic flow of a fluid with positive and negative nonlinearity through a nozzle

The one‐dimensional transonic flow of an inviscid fluid, which at large values of the specific heats exhibits both positive (Γ>0) and negative (Γ<0) nonlinearity regions {Γ=(1/ρ)[∂(ρa)/∂ρ]s} and which remains in a single phase, is studied. By assuming that Γ changes its sign in the small neighborhood of the throat of the nozzle where transonic flow exists and introducing a new scaling of the independent variables, an approximate first‐order partial differential equation (PDE) with a nonconvex flux function is derived. It governs both the steady transonic flows and the upstream moving waves near sonic point. The existence of continuous and discontinuous steady transonic flows when the throat area is either a maximum or a minimum is shown. The existence of standing sonic discontinuities and rarefaction shocks in the transonic flow are noted for the first time. Unlike in the classical gas flows, there are two sonic points and continuous transonic flows are possible only through one of them. The numerical...

A mathematical analysis of nonlinear waves in a fluid filled visco-elastic tube

SummaryOur investigations in this paper are centred around the mathematical analysis of a “modal wave” problem. We have considered the axisymmetric flow of an inviscid liquid in a thinwalled viscoelastic tube under certain simplifying assumptions. We have first derived the propagation space equations in the long wave limit and also given a general procedure to derive these equations for arbitrary wave length, when the flow is irrotational. We have used the method of operators of multiple scales to derive the nonlinear Schrödinger equation governing the modulation of periodic waves and we have elaborated on the “long modulated waves” and the “modulated long waves”. We have also examined the existence and stability of Stokes waves in this system. This is followed by a discussion of the progressive wave solutions of the long wave equations. One of the most important results of our paper is that the propagation space equations are no longer partial differential equations but they are in terms of pseudo-differential operators.ZusammenfassungDie vorliegenden Untersuchungen beziehen sich auf die mathematische Behandlung des „Modalwellen”-Problems. Die achsensymmetrische Strömung einer nichtviskosen Flüssigkeit in einem dünnwandigen viskoelastischen Rohr, unter bestimmten vereinfachenden Annahmen, wird betrachtet. Zuerst werden die Gleichungen des Ausbreitungsraumes im Langwellenbereich abgeleitet und eine allgemeine Methode zur Herleitung dieser Gleichungen für beliebige Wellenlängen bei nichtrotierender Strömung angegeben. Eine Operatorenmethode mit multiplem Maßstab wird verwendet zur Herleitung der nichtlinearen Schrödinger-Gleichung für die Modulation der periodischen Wellen, und die „langmodulierten Wellen” sowie die „modulierten Langwellen” werden aufgezeigt. Weiters wird die Existenz und die Stabilität der Stokes-Wellen im System untersucht. Anschließend werden die progressiven Wellenlösungen der Langwellengleichungen diskutiert. Eines der wichtigsten Ergebnisse dieser Arbeit ist, daß die Gleichungen des Ausbreitungsraumes keine partiellen Differentialgleichungen mehr sind, sondern Ausdrücke von Pseudo-Differentialoperatoren.

A new theory of shock dynamics part I: analytic considerations

A new theory for the calculation of the successive positions of a shock is suggested here. For one-dimensional problem, it requires integration of a finite number (say, 4 or 5) of ordinary differential equations derived from an infinite system of compatibility conditions. Numerical results in Section 2 show that error is less than 0.1 percent.

Kinematics of a multi-dimensional shock of arbitrary strength in an ideal gas

SummaryIt has been shown that the kinematics of a shock front in an ideal gas with constant specific heat can be completely described by a first order nonlinear partial differential equation (called here — shock manifold equation or SME) which reduces to the characteristic partial differential equation as the shock strength tends to zero. The condition for the existence of a nontrivial solution of the jump relations across the shock turns out to be the Prandtl relation. Continuing the functions representing the state on the either side of the shock to the other side as infinitely differentiable functions and embedding the shock in a one parameter family of surfaces, it has been shown that the Prandtl relation can be treated as a required shock manifold equation for a function Φ, where Φ=0 is the shock surface. We also show that there are other forms of the SME and prove an important result that they are equivalent. Shock rays are defined to be the characteristic curves of a SME and it has been shown that when the flow on either side of the shock is at rest, the shock rays are orthogonal to the successive positions of the shock surface. Certain results have been derived for a weak shock, in which case the complete history of the curved shock can be determined for a class of problems.

Propagation of a curved weak shock

Propagation of a curved shock is governed by a system of shock ray equations which is coupled to an innite system of transport equations along these rays. For a twodimensional weak shock, it has been suggested that this system can be approximated by a hyperbolic system of four partial dierential equations in a ray coordinate system, which consists of two independent variables

Nonlinear wave propagation on an arbitrary steady transonic flow

(\xi,t)

Numerical simulation of converging nonlinear wavefronts

where the curves t = constant give successive positions of the shock and

A New Theory of Shock Dynamics Part II: Numerical Solution

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