Self-similar solution

FIgura 4.1. Overpressure as a function of flame speed for three geometries. The relationships are based on calculations by use of a self-similar solution (Kuhl et al. 1973). 

G. I. Barenblatt and Ya. B. Zel dovich, Self-similar solutions as intermediate asymptotics, Armu. Rev. Fluid Mech. 4, 285 (1972). 

In order to understand better what happens when a nucleation point, say x = Xo, is selected, let us focus on the small time behavior of the nontrivial self-similar solution. Consider a solution (2.5) at time t = At. It is convenient to parametrize the functions w(x,Ar) and v x,At) by x and present them as a curve in the (w,v) plane. It is not hard to see that one then obtains a loop, beginning and ending in a point (Wo,0) (see Fig. 8b) the details of the loop depend, of course, on the fine internal structures of shocks and kinks (see Fig. 8a). 

Since the energy ofthe nucleus is identically zero, the integral impact of this localized contribution to the initial data can be measuredby the corresponding energy density which is finite. For our self-similar solution (2.5) one can equivalently calculate the rate of dissipation R (Dafermos, 1973)... 

Except for some special cases, the presence of a free boundary introduces a nonlinearity, so that only a few exact solutions are known. These are in all cases self-similar solutions, which means that the differential equation and associated initial and boundary conditions can all be expressed in terms of a single independent variable. The problem is thereby reduced... 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

In searching for self-similar solutions, it follows that the left-hand side of Eq. (140) must vanish, and from Eqs. (141) and (142) that (dg/dT ) and 6W must be constant. Equation (140) then reduces to an ordinary differential equation... 

The Self-Similar Solution and the Laws of Conservation of Energy and Momentum... 

However, taken together, these two equations contradict one another We are left with the impression that no self-similar solution can at once satisfy both conditions. Both necessarily follow from the equations of mechanics and, consequently, the problem does not have a self-similar solution at all. 

Below, we shall find a specific self-similar solution of the power type. We will show that the actual exponent is always in the interval 1 < n < 2 for a diatomic gas n = 1.333 4/3. 

We turn to the energy of the gas, calculated according to the self-similar solution as the volume integral of the kinetic and thermal gas energies. The... 

The condition indicates that we take the self-similar solution and find within it the boundary of the region containing the mass mQ (Fig. 2). 

Thus, the concept of zero total momentum, and of the existence of a small region which does not obey the self-similar solution, resolves the paradoxes which arose when the solution was compared with the conservation laws. 

The law of pressure decay /(f/r) is restricted by the obvious condition that the pressure at the piston must decrease faster than the pressure amplitude in the self-similar solution itself, i.e., faster (with a greater absolute value of the negative exponent) than i-4/5. 

We may expect that, beginning from the values M, X at which II — IIS, a greater value of II will occur in subsequent motion. This criterion yields an upper bound of applicability of the self-similar solution from the condition... 

It is well known that self-similar solutions are divided into two sharply differing types. 

This example illustrates that it is necessary to exercise great care and thoroughness in finding self-similar solutions. 

A self-similar solution is found in which the pressure II at the shock wave front propagating in the gas decreases as a power function of the distance traveled, X II X n, where 1 < n < 2. In the general form, for any adiabatic index of the gas, it is proved that 1 < n < 2 for 7 = 7/5 the numerical value of n is 1.333. The law found yields the greatest possible rate of the plane shock wave decay under any circumstances. 

The gas momentum in self-similar motion is identically zero. The energy calculated from the self-similar solution is expressed by a divergent integral. 

The existence of two types of self-similar solutions, explicitly formulated by Ya.B. for the first time, stimulated extensive studies to clarify the general character of the difference between them and to apply the concept of self-similar solutions of the second kind to various problems in mathematical physics. The present state of the problem can be found in a monograph by G. I. Barenblatt.6 We note also the existence of an exact analytic solution with a rational self-similarity exponent when the adiabatic index is equal to 7/5.7... 

The problem considered in this paper of a self-sustaining propagation wave of flame is closely related to self-similar solutions of the second kind, considered by Ya.B. in his paper, Gas Motion Under the Action of Short-Duration Pressure (Impulse) (see article 9 of the present volume) nearly twenty years later. Indeed, we are dealing here with wave-like solutions, for instance,... 

Hunt [37] developed self-similar solutions to the von Smoluchowski equation based on dimensional considerations. Three assumptions that were required are as follows. 

Abstract Generalization of the self-similar solution for ultrarelativistic shock waves (Bland-ford McKee, 1976) is obtained in presence of losses localized on the shock front or distributed in the downstream medium. It is shown that there are two qualitatively different regimes of shock deceleration, corresponding to small and large losses. We present the temperature, pressure and density distributions in the downstream fluid as well as Lorentz factor as a function of distance from the shock front. 

Following the recipe of Blandford and McKee (1976) we generalize their well-known self-similar solutions for relativistic blast waves for the case, where the energy and momentum of the relativistic fluid is carried away by various species of neutral particles. 

To find a self-similar solution we assume T2= t rn and introduce the similarity variable = jEjj > where r is distance from the center and R the current radius of the shock front. We find self-similar solutions in the case of localized losses and in the case of uniform distributions of losses if a a(x)- The velocity, pressure and particle density are found in terms of variables T, %. 

The problem is to determine the velocity distribution in the fluid as a function of time. In this problem, the fluid motion is due entirely to the motion of the boundary - the only pressure gradient is hydrostatic, and this does not affect the velocity parallel to the plate surface. At the initial instant, the velocity profile appears as a step with magnitude Uat the plate surface and magnitude arbitrarily close to zero everywhere else, as sketched in Fig. 3 11. As time increases, however, the effect of the plate motion propagates farther and farther out into the fluid as momentum is transferred normal to the plate by molecular diffusion and a series of velocity profiles is achieved similar to those sketched in Fig. 3-11. In this section, the details of this motion are analyzed, and, in the process, the concept of self-similar solutions that we shall use extensively in later chapters is introduced. 

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