P. L. Bhatnagar

Propagation of an isothermal shock in stellar envelopes

SummaryWe have studied in this paper the propagation of an isothermal shock in the radiative envelopes of the Bosman-Crespin model for a hot star and Boury’s model for a giant star. A spherically symmetric disturbance is supposed to be originated at or outside the surface of the convective core. We have used Whitham’s rule to study the variation in the shock strength and the shock velocity after modifying it for inclusion of pressure, energy and flux of radiation. We find the shock increases in strength as it propagates through the envelopes of decreasing density, pressure and temperature. The velocity of the shock decreases for very weak initial shock strengths, for intermediate initial shock strength it first decreases and then increases, while for large initial shock strength, it always increases. This aspect of the problem throws some light on the stability of the models under consideration.RiassuntoSi è studiata in questo articolo la propagazione di un urto isotermico negli strati radiativi del modello di Bosman-Crespin di una stella calda e del modello di Boury di una stella gigante. Si suppone che una perturbazione a simmetria sferica si origini sulla o fuori della superficie del nucleo convettivo. Si è usata la regola di Whitham per studiare le variazioni dell’intensità dell’urto e della velocità dell’urto, dopo averla modificata per includervi la pressione, l’energia ed il flusso di radiazione. Si trova che l’urto cresce di intensità nel propagarsi attraverso strati di decrescente densità, pressione e temperatura. La velocità dell’urto decresce per debolissime intensità iniziali dell’urto, per intensità iniziali dell’urto intermedie prima decresce e poi cresce, invece per forti intensità iniziali dell’urto cresce sempre. Questo aspetto del problema getta qualche luce sulla stabilità dei modelli presi in considerazione.

Study of self-similar and steady flows near singularities

One-dimensional steady state flow or a self-similar flow is represented by an integral curve of the system of ordinary differential equations and, in many important cases, the integral curve passes through a singular point. Kulikovskii & Slobodkina (1967) have shown that the stability of a steady flow near the singularity can be studied with the help of a simple first-order partial differential equation. In § 2 of this paper we have used their method to study steady transonic flows in radiation-gas-dynamics in the neighbourhood of the sonic point. We find that all possible one-dimensional steady flows in radiation-gas-dynamics are locally stable in the neighbourhood of the sonic point. A continuous disturbance on a steady flow, while decaying and propagating, may develop a surface of discontinuity within it. We have determined the conditions for the appearance of such a discontinuity and also the exact position where it appears. In §3 we have shown that their method can be easily generalized to study the stability of self-similar flows. As an example we have considered the stability of the self-similar flow behind a strong imploding shock. In this case we find that the flow is stable with respect to radially symmetric disturbances.

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.

Transport processes in dense gases

A simple and direct procedure for evaluating the properties of dense gases has been attempted based on the BBGKY hierarchy of equations. The basic idea of Enskog, namely that increasing the density affects the behaviour of the assembly, mainly by reducing the specific volume and by providing acertain amount of shielding to molecular interactions, has been developed at length in this investigation. The decrease in specific volume allows one to approximate the three-particle distribution function in terms of one-particle and two-particle distribution functions. These distribution functions are expanded in terms of generalized Hermite polynomials to study small departures from equilibrium . In the simpleCouette flow and one-dimensional heat flux problems, explicit expressions for viscosity andheat conductivity have been obtained. This enables one to study the variation of these with density and temperature. Numerical results are compared with experimental values for simple gases like argon, neon and helium . The values for the inverse-power-law forces behave monotonically and approach the Enskog curve. The Lennard -Jones potential shows, as density in-creases, an increase of viscosity and heat conductivity that is less rapid than for other power laws. The experimental values agrees well for the force laws studied here, as seen from the figures.

Comparative study of some constitutive equations characterising non-Newtonian fluids

This paper compares, in a general way, the predictions of the constitutive equations given by Rivlin and Ericksen, Oldroyd, and Walters. Whether we consider the rotational problems in cylindrical co-ordinates or in spherical polar co-ordinates, the effect of the non-Newtonicity on the secondary flows is collected in a single parameterα which can be explicitly expressed in terms of the non-Newtonian parameters that occur in each of the above-mentioned constitutive equations. Thus, for a given value ofα, all the three fluids will have identical secondary flows. It is only through the study of appropriate normal stresses that a Rivlin-Ericksen fluid can be distinguished from the other two fluids which are indistinguishable as long as this non-Newtonian parameter has the same value.

Propagation of a spherical shock in an inhomogeneous self-gravitating or nongravitating system

SummaryThe propagation of a shock wave of finite strength due to an explosion into inhomogeneous nongravitating and self-gravitating systems has been considered, using similarity principles, supposing that the density varies as an inverse power of distance from the centre of explosion. A large number of systems, characterised by different density exponents and different adiabatic coefficients of the gas have been considered for different shock strengths. The numerical integration from the shock inward has been continued to the surface of singularity where density tends to infinity and which acts like a piston in the self-gravitating case and to the surface where the velocity gradient tends to infinity in the nongravitating case. The effect of variation of shock strength, density exponent and adiabatic coefficient on the location of these singularities and on the distribution of flow parameters behind the shock has been studied. The initial energy of the system and the manner of release of the explosion energy influence strongly the flow behind the shock. The results have been graphically depicted.RiassuntoSi è studiata la propagazione di un’onda d’urto di intensità tinita dovuta ad una esplosione in sistemi inomogenei non gravitazionali ed autogravitazionali, facendo uso dei principi di similarità e supponendo che la densità vari come l’inverso di una potenza della distanza dal centro della esplosione. Si sono studiati un gran nnmero di sistemi caratterizzati da diversi esponenti della densità e diversi coefficienti adiabatici del gas, per differenti intensità dell’urto. Si è continuata l’integrazione numerica dell’urto verso l’interno fino alia superficie della singolarità dove la densità tende all’infinito e ehe agisce come un pistone nel caso autogravitazionale e verso la superflcie su cui il gradiente della velocità tende all’inflnito nel caso non gravitazionale. Si è stvidiato l’effetto della variazione della intensità dell’urto, dell’esponente della densita e del coefficiente adiabatico sulla posizione di queste singolarita e sulla distribuzione dei parametri del flusso a tergo deil’urto. L’energia iniziale del sistema e il processo di liberazione dell’energia dell’esplosione influenzano fortemente il flusso a tergo dell’urto. I risultati sono espressi in forma di grafico.

Boundary layer on a flat plate with suction

SummaryIn this paper we have discussed the boundary layer on a plate with suction. The problem is solved near the leading edge as well as far downstream. A linear suction law is assumed near the leading edge for simplicity, whereas no restriction is placed on the suction law in the region downstream. An explict expression for boundary layer thickness in terms of suction speed and distance from leading edge is derived. It is found that the thickness of the boundary layer depends on the derivative of the suction speed. The skin friction also has been evaluated. Though near the leading edge a linear law of suction is assumed, the method used in the paper can be easily generalised for any other power law, for example, we may use a power series expansion for the function defining the suction velocity.

On superposable flows

SummaryIn the first section of the present paper we obtain the condition of superposability working through vectors and in particular give explicity the condition of superposability of axi-symmetrical flows. This enables us to make some general remarks on the possibility of superposition of two axi-symmetrical flows. The rest of the paper is devoted to the consideration of the possibility of superposition of general rotational flow on the flows due to a vortex, spiral-vortex and vortex-doublet and of irrotational flow on a radial flow in two-dimensions.In case of the flow due to a vortex, we find that a family of rotational flows of which the isocurls are concentric circles is superposable on it. In the case of the remaining three flows we find that the contemplated types of flows do not exist.