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Showing papers on "Similarity solution published in 1976"


Journal ArticleDOI
TL;DR: In this paper, a fluid dynamical treatment of an ultra-relativistic spherical blast wave enclosed by a strong shock is presented, and a simple similarity solution describing the explosion of a fixed amount of energy in a uniform medium is derived, and generalized to include cases in which power is supplied by a central source and the density of the external medium varies with radius.
Abstract: A fluid dynamical treatment of an ultra‐relativistic spherical blast wave enclosed by a strong shock is presented. A simple similarity solution describing the explosion of a fixed amount of energy in a uniform medium is derived, and this is generalized to include cases in which power is supplied by a central source and the density of the external medium varies with radius. Radiative shocks, in which the escaping photons carry away momentum as well as energy, are also discussed. Formulas that interpolate between the non‐ and ultra‐relativistic limits are proposed.

1,164 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of uniform flow past a flat plate whose surface has a constant velocity λU opposite in direction to that of the mainstream is considered for large values of the Reynolds number R.
Abstract: The problem of uniform flow past a flat plate whose surface has a constant velocity λU opposite in direction to that of the mainstream is considered for large values of the Reynolds number R. In a previous communication (Klemp & Acrivos 1972) it was shown that, if the region of reverse flow which is established next to the plate as a consequence of its motion is O(R−1/2) in thickness, the appropriate laminar boundary-layer equations have a solution provided λ ≤ 0·3541. Here the analysis is extended to the range λ > 0·3541, which cannot be treated using a conventional boundary-layer approach. Specifically, it is found that for λ > 0·3541 the flow consists of three overlapping domains: (a) the external uniform flow; (b) a conventional boundary layer with reverse flow for xs 1.Also, detailed streamline patterns were obtained numerically for various λ's in the range of 0 ≤ λ ≤ 2 using a novel computational scheme which was found to be more efficient than that previously reported. Interestingly enough, the drag first decreased with λ, reached a minimum at λ = 0.3541, and then increased monotonically until, at λ = 2, it was found to have attained essentially the value predicted from the asymptotic λ → ∞ similarity solution available in the literature. Thus it is felt that the present numerical results plus the two similarity solutions for λ = 0 and for λ → ∞ fully describe the high-R steady flow for all non-negative values of λ.

88 citations


Journal ArticleDOI
TL;DR: In this article, a temperature-dependent power-law rheology for mantle plume dynamics under real conditions is presented. But the authors focus on the temperature dependence of both linear Newtonian rheologies and nonlinear olivine rheologists in accounting for narrow mantle flow structures.
Abstract: Stress is placed on the temperature dependence of both a linear Newtonian rheology and a nonlinear olivine rheology in accounting for narrow mantle flow structures. The boundary-layer theory developed incorporates an arbitrary temperature-dependent power-law rheology for the medium, in order to facilitate the study of mantle plume dynamics under real conditions. Thermal, kinematic, and dynamic structures of mantle plumes are modelled by a two-dimensional natural-convection boundary layer rising in a fluid with a temperature-dependent power-law relationship between shear stress and strain rate. An analytic similarity solution is arrived at for upwelling adjacent to a vertical isothermal stress-free plane. Newtonian creep as a deformation mechanism, thermal anomalies resulting from chemical heterogeneity, the behavior of plumes in non-Newtonian (olivine) mantles, and differences in the dynamics of wet and dry olivine are discussed.

80 citations


Journal ArticleDOI
TL;DR: In this paper, a quasi-static Bennett equilibrium, with isothermal distribution of transverse momenta, is shown to be a similarity solution of the Fokker-Planck equation with scale radius increasing in accord with Nordsieck's formula.
Abstract: A Fokker–Planck equation is derived to study the evolution of a stable, low‐current beam propagating in a gas‐plasma medium. Small‐angle scattering of the beam particles by the medium causes diffusion in the phase space projected transverse to the direction of propagation. The projected components of dynamical friction vanish. As a result, there is a continued input of energy into the transverse particle motions, which is taken up in expansion against the pinch field. A quasi‐static Bennett equilibrium, with isothermal distribution of transverse momenta, is shown to be a similarity solution of the Fokker–Planck equation with scale radius increasing in accord with Nordsieck’s formula. An H theorem is proved and the Bennett distribution is shown to minimize both H and −dH/dt; hence, it is the time‐dependent asymptotic state. The predicted current profile and radius are shown to be in fair agreement with experiment.

75 citations


Journal ArticleDOI
TL;DR: In this paper, the process of non-uniform laser-driven DT plasma burning caused by the thermonuclear burn wave produced and propagating in plasma is investigated theoretically, and the energy transfer from the burning plasma region to the remaining cold portion is assumed to be realized either by α-particles or by free electrons.
Abstract: The process of non-uniform laser-driven DT plasma burning caused by the thermonuclear burn wave produced and propagating in plasma is investigated theoretically. The energy transfer from the burning plasma region to the remaining cold portion is assumed to be realized either by α-particles or by free electrons. A similarity solution of this problem has been obtained and, deriving from this solution, the conditions for "firing-up" non-uniform thermonuclear targets are defined.

65 citations


Journal ArticleDOI
TL;DR: In this paper, an asymptotic description of the laminar flow in a symmetric tube is presented, in which there is outward centrifuging of the secondary motion in the inviscid core, supplemented by a faster viscous return motion near the pipewalls, while the downpipe velocity increases steadily across the core.
Abstract: The forcing of fluid through a tube, of uniform cross section $(\sim a)$ and coiled to form an arc of a circle of radius R$(\gg a)$, by the application of a steady constant pressure gradient along the tube is discussed for large values of the Dean number D. An asymptotic description is presented for the fully-developed laminar flow within a general symmetric section, in which there is outward centrifuging of the secondary motion in the inviscid core, supplemented by a faster viscous return motion near the pipewalls, while the downpipe velocity increases steadily across the core. A similarity solution of the viscous flow, with the associated core flow thereby being determined, is calculated for a triangular cross section, but for a rectangular tube analytical and numerical arguments are put forward that point fairly conclusively to the non-existence of an attached laminar motion, near the inside bend at least, at high Dean numbers. For if the pressure-gradient is imagined to vary like the mth power of distance across the section, then computed results indicate that local solutions of the boundary layer problem can be found only for $m>\frac{1}{2}$. The assertion is backed up by an analytical study for small positive values of $(m-\frac{1}{2})$. The dynamical properties in the neighbourhood of a flat outside bend, or of a 'pinched' inside bend, of a tube also need careful consideration, although these may be essentially passive flow regions and some account of the local features there can be given. Near a flat outside bend two possible flow models arise, depending on the local nature of the coreflow.

27 citations


Journal ArticleDOI
TL;DR: In this article, for two opposite orientations of active surface of semi-infinite strip delivering the mass downwards, viz. upward position and the downward one, the corresponding easily applicable correlation equations, in dependence on two characteristic parameters: Sc, Schmidt number and yAo, initial concentration on the plate, have been obtained.

20 citations


Journal ArticleDOI
TL;DR: In this article, the heat transfer rate to the wall in a laser-heated rocket thruster using pure hydrogen, where the average temperature of the hot plasma core is about 14,000 K and the core Reynolds number based on nozzle throat diameter is about 2000.
Abstract: The laminar boundary-layer equations with local similarity approximation are solved in order to estimate the heat transfer rate to the wall in a laser-heated rocket thruster using pure hydrogen, where the average temperature of the hot plasma core is about 14,000 K and the core Reynolds number based on nozzle throat diameter is about 2000. Under these conditions, the density-viscosity product at the wall can be 10 times the freestream value, and the hydrogen is completely dissociated. Hence the boundary layer equations with variable transport properties are solved by a quasi-linearization technique, and the equilibrium properties of hydrogen are used in the calculations. Velocity profiles are obtained, and the wall shear and wall stagnation enthalpy gradient are plotted against the pressure gradient parameter.

9 citations


Journal ArticleDOI
01 Jan 1976
TL;DR: In this article, the temperature distribution in a laminar plane wall jet has been studied and it is found that a similarity solution of the energy equation exists, which is reduced to a hypergeometric equation by a suitable transformation of the similarity variable and the solution, for arbitrary values of the Prandtl number, is obtained.
Abstract: In the present note the temperature distribution in a laminar plane wall jet has been studied. It is found that a similarity solution of the energy equation exists. The resulting ordinary differential equation is reduced to a hypergeometric equation by a suitable transformation of the similarity variable and the solution, for arbitrary values of the Prandtl number, is obtained. It is concluded that the heat transfer at the wall at a given section and the product of volume and heat-flux through any cross-section of the boundary layer increase with the increase in the value of Prandtl number respectively.

7 citations


Journal ArticleDOI
TL;DR: In this paper, an experimental and theoretical investigation of the velocity distribution in the laminar boundary layers at a moving interface generated by a counter-current gas-liquid stratified flow has been made.

6 citations


Journal ArticleDOI
TL;DR: In this article, a singular perturbation analysis is used to determine the shape of a body which will comply with the requirements for a well-known similitude for the cylindrical form of Burgers' equation, and the far field shock profiles are presented.
Abstract: A singular perturbation analysis is utilized for determined the shape of a body which will comply with the requirements for a well‐known similitude for the cylindrical form of Burgers’ equation, and the far‐field shock profiles are presented.

ReportDOI
01 Feb 1976
TL;DR: In this article, a similarity solution for void shape is derived, and assuming growth in a particular manner, the similarity shape is determined by the choice of a/sup 3/v/..cap omega..v* where a is the void half-length, v is the speed of propagation of the void tip, and v* is a temperature dependent material parameter.
Abstract: Intergranular void growth at elevated temperature is studied in cases for which the shape of the void is determined by void surface diffusion and growth occurs by diffusion from the void tip and along the grain boundary. Equations governing the void shape are linearized and this linearization should not lead to gross errors in light of the large differences in the reported values of diffusion coefficients. A similarity solution for void shape is derived, and assuming growth in a particular manner, the similarity shape is shown to be determined by the choice of a/sup 3/v/..cap omega..v* where a is the void half-length, v is the speed of propagation of the void tip, ..cap omega.. is the atomic volume, and v* is a temperature dependent material parameter. An iterative solution for void shape which depends on a/sup 3/v/..cap omega..v* is derived also. By assuming the grain boundary flux to be distributed in such a manner that the grains separate as rigid bodies, the similarity and iterative shapes are linked to the applied stress, and the resulting growth models compared with the limiting cases of equilibrium and crack-like growth. The comparison indicates that void growth can be represented by a two-part solutionmore » where either the linearized equilibrium or the similarity model applies for small a/sup 3/v/..cap omega..v*, and the crack-like model applies for larger a/sup 3/v/..cap omega..v*. An analysis is included of relaxation times associated with a variety of diffusion mechanisms which are useful in determining the dominant mechanisms of matter transport.« less

Book ChapterDOI
F. Obermeier1
01 Jan 1976
TL;DR: In this paper, the singular perturbation methods of strained coordinates and matched asymptotic expansions are applied to the basic equations of a two-dimensional, nonviscous flow describing a transonic motion around slender bodies of finite length.
Abstract: The singular perturbation methods of strained coordinates and matched asymptotic expansions are applied to the basic equations of a two-dimensional, nonviscous flow describing a transonic motion around slender bodies of finite length. The analytical solution determined by these methods is valid in the supersonic region of the flow field, in the near field as well as in the far field. Furthermore, at the surface of the body a first approximation of the solution agrees with a solution by J.R. Spreiter determined with the help of the method of local linearization, while in the far field it is asymptotically equal to a similarity solution found by A.J.A. Morgan. Finally, our solution shows a singular behaviour along the sonic line. This may indicate that in the immediate vicinity of the sonic line the usual transonic equation should be replaced by equations more suitable to the problem.