Showing papers on "Similarity solution published in 1978"
TL;DR: In this paper, an axisymmetric gravity wave, for which each of nonlinearity, dispersion and radial spreading is weak but significant, is determined as a similarity solution with slowly varying amplitude is the radius, d is the depth, and [Sscr ] is the family parameter of the solutions.
Abstract: An axisymmetric gravity wave, for which each of nonlinearity, dispersion and radial spreading is weak but significant, is determined as a similarity solution with slowly varying amplitude is the radius, d is the depth, and [Sscr ] is the family parameter of the solutions. It is shown that the free-surface displacement η(r,t) is either a wave of elevation (η ≥ 0) or a wave of depression (η ≤ 0) and that (|η|/)½ satisfies a Painleve equation that is a nonlinear generalization of Airy's equation. Representative numerical solutions and asymptotic approximations for small and large [Sscr ] are presented. It is shown that the similarity solution conserves energy but not mass, in consequence of which (in order to obtain a complete solution to a well-posed initial-value problem) it must either be accompanied by some other component or components or be driven by a source (or sink) in some interior domain in which the implicit restriction r [Gt ] d is violated. A linear model is developed that is valid for r [lsim ] d and compensates for the mass defect of, and matches, the nonlinear similarity solution for |[Sscr ]| [Lt ] 1.
45 citations
TL;DR: In this article, a flat vertical plate was assumed to be suddenly loaded internally with a constant and uniform flux while immersed in an extensive body of quiescent and unstratified fluid, and the partial differential equations describing the conservation of mass, momentum, and energy were solved in their time-dependent forms, using a finite difference technique.
Abstract: A flat vertical plate, of finite thickness and appreciable thermal capacity, was assumed to be suddenly loaded internally with a constant and uniform flux while immersed in an extensive body of quiescent and unstratified fluid. The partial differential equations describing the conservation of mass, momentum, and energy were solved in their time-dependent forms, using a finite-difference technique. The computed transient velocity and temperature fields are in good agreement with the results of previous integral and numerical analyses and with experimental data. The final steady-state profiles are also in good agreement with the similarity solution for the uniform surface flux condition. For some conditions, with plates of small thermal capacity, the transient temperature and velocity levels locally exceeded the final steady-state distributions.
28 citations
TL;DR: In this article, the authors derived and discussed exact solutions of the full (nonlinear) Boltzmann equation for the velocity relaxation problem, and pointed out the usefulness of similarity solutions in the search for exact solutions.
Abstract: We derive and discuss some exact solutions of the full (nonlinear) Boltzmann equation. One of these is the similarity solution recently found by Krook and Wu for the velocity relaxation problem. Other similarity solutions do exist, and we point out their usefulness in the search for exact solutions of the spatially inhomogeneous Boltzmann equation.
28 citations
TL;DR: In this article, the existence and uniqueness of a solution to a multi-phase one-dimensional Stefan problem are established, provided that a certain reasonable order relation exists among the boundary values, and a nonexistence result is established for the situation when most of the relations are satisfied but one is seriously violated.
Abstract: The similarity solution of a multi-phase one-dimensional Stefan problem gives rise to a system of nonlinear equations for the coefficients of ..sqrt..t which determine the speeds of the several interphase boundaries. The existence and uniqueness of a solution to this system are established, provided that a certain reasonable order relation exists among the boundary values. In addition, a nonexistence result is established for the situation when most of the relations are satisfied but one is seriously violated. This result implies the existence of physically plausible Stefan problems with constant boundary values for which there do not exist similarity solutions. 4 figures.
20 citations
TL;DR: In this article, an exact similarity solution for a spherical magneto-gas dynamic shock wave is obtained in the case when the loss of energy due to radiation escape is significant, and it is shown that the total energy of the wave is not constant but decreases with time.
Abstract: An exact similarity solution for a spherical magneto-gas dynamic shock wave is obtained in the case when the loss of energy due to radiation escape is significant. The total energy of the shock wave is not constant but decreases with time. We have shown that due to the magnetic field, the radiation flux changes considerably.
17 citations
TL;DR: In this article, the two-dimensional laminar fire plume over a vertical burning surface was examined both theoretically and experimentally, using wicks soaked with methanol, ethanol, and 1-propanol.
9 citations
TL;DR: In this paper, a similarity solution is presented for a three-dimensional boundary layer on a heated cone to stimulate the water flow past the forward part of an axisymmetric slender body.
Abstract: Heat transfer and shear stress may be significantly affected by buoyancy-forced and associated free-convection motions in many forced-convection flows. A crossflow is induced when a uniform, horizontal stream passes along a heated, axisymmetric slender body. The crossflow effects on heat transfer and shear stress grow as the fluid flows downstream, and eventually become one of the dominant mechanisms even for a moderate-speed forced-convection flow. Early study of the longitudinal cylinder flow showed that the crossflow may destabilize the boundary layer and degrade the heat transfer over the upper half of the body. On the other hand, heating can be used to stabilize the water boundary layer due to its temperature-dependent viscosity, since (dμ/dT) of water is negative. However, the influences of the pressure gradient on the destabilizing crossflow effect and on the stabilizing variable-viscosity effect have never been studied before. It is important to know the interaction of the buoyancy-force effect and the variable-viscosity effect under the non-zero pressure gradient conditions in stabilizing the boundary layer by heating. In this paper a similarity solution is presented for a three-dimensional boundary layer on a heated cone to stimulate the water flow past the forward part of an axisymmetric slender body. The numerical solutions of the ordinary differential equations reduced by the similarity transformation are presented in the region near the vertex of the cone. The results indicate that the crossflow grows as the fluid flows downstream for the cone of its half angle less than 66.25°. For a cone of its half angle larger than 66.25°, the magnitude of the crossflow is about the same order as that of the axial flow in the neighborhood of the cone vertex and is suppressed by the favorable pressure gradient as the fluid moves downstream. The effect of the temperature-dependent water viscosity has been shown to enhance the favorable pressure-gradient effects and to counterbalance the crossflow effects.
8 citations
TL;DR: In this article, the convective atmospheric vortex model of Kuo (1966) is modifed and a time-dependent two-cell similarity solution is obtained to predict the dissipation of convective vortices.
Abstract: The convective atmospheric vortex model of Kuo (1966) is modifed and a time-dependent two-cell similarity solution is obtained. Predictions of flow fields and thermodynamic property distributions have been made and they compare favorably with available data measured from natural occurrences of tornadoes. The present solution predicts the dissipation of convective atmospheric vortices. DOI: 10.1111/j.2153-3490.1978.tb00853.x
7 citations
TL;DR: Similarity solutions for nonhomogeneous elastic semi-infinite thin rods subjected to suddenly initiated and uniformly sustained stress at the origin were determined in this paper. But the similarity solutions were not applied to the case of the homogeneous rod.
Abstract: Similarity solutions are determined for nonhomogeneous elastic semi‐infinite thin rods subjected to suddenly initiated and uniformly sustained stress at the origin. The system of governing nonhomogeneous partial differential equations along with its initial and boundary conditions is transformed to an ordinary differential equation and the corresponding auxiliary conditions by means of the similarity transformations. The representation so obtained is integrated. From the general solution of the problem, two special cases, one for the homogeneous rod and the other for the nonhomogeneous rod, are solved in closed form. The difference of behavior in two cases is graphically discussed.
5 citations
TL;DR: In this paper, the effect of the disturbances behind a strong shock moving in a medium of nonuniform density is considered and the similarity solution is used to find the strength of the overtaking disturbance which modifies Chisnell's approximation to become an exact theory.
Abstract: The effect of the disturbances behind a strong shock moving in a medium of nonuniform density is considered. The similarity solution is used to find the strength of the overtaking disturbance which modifies Chisnell’s approximation to become an exact theory.
5 citations
TL;DR: In this paper, it was shown that the similarity solutions are possible only for the bodies for which a cross-sectional radius is defined, where the radius is the longitudinal distance from the nose point to the cross-section.
Abstract: The possible existence of similarity solutions for the unsteady three-dimensional boundary layer flows with heat and mass transfer around stationary axisymmetric bodies which are fully immersed in purely viscous moving non-Newtonian fluids has been searched in general by the application of transformations, involving a single linear parameter In particular, the cases involving rotational flows around stationary bodies and rotating bodies have been discussed as corollaries of the main analysis The main analysis shows that the similarity solutions are possible only for the bodies for which
$$\bar r \propto \bar x^n $$
where
$$\bar r$$
is a cross-sectional radius; and
$$\bar x$$
is the longitudinal distance from the nose point to the cross section In case of rotating bodies, similarity solutions exist only for cones and disks The analysis, as an example, has successfully been applied to the Powell-Eyring model It is seen that for the same rate of shear, expenditure of energy for maintaining the rotation of the solid body is comparatively higher for a flow with a higher Eyring number where the Eyring numberEy=1/µBE µ, B, andE are the material functions of the Powell-Eyring fluid
TL;DR: In this paper, the effect of rotation on a magnetosphere containing a conducting fluid is investigated by means of a similarity solution and it is shown that rotation increases the current density in the equatorial region at the expense of that in the polar regions.
07 Aug 1978