The Self-Similar Solution

Broadly, the self-similar solution identifies invariant domains in the space of the independent variables along which the solution remains the same or contains a part that is the same. Consider, for example, the number density function / (x, t) that may satisfy a population balance equation such as (3.2.8) or (3.3.5). By a self-similar solution of either of these equations we mean one to be of the form 

the cumulative function is itself time-invariant along rj = c so that it has a self-similar form. There is some question as to the existence of the integrals in (5.1.2), which we shall ignore for the present with a promise return to it presently. 

Clearly, the functions g t h t), and i/ ( ) must depend upon the population balance equation. However, it is easy to show that the functions are related to the zeroth moment and the first moment as defined in Section 4.4. Thus 

The concept of a self-similar solution is well known to the student familiar with the development of a boundary layer along a flat plate where the velocity profile remains the same when distance from the wall is scaled with the boundary layer thickness that varies along the direction of flow. Similarly, diffusion profiles in semi-infinite media are known to be self-similar when distance is scaled with respect to the square root of diffusion time. 

If Sp 1, both integrals exist in (5.1.4) so that (5.1.3) is appropriate. Suppose now that 1. Then (5.1.3) is no longer valid, since the integral to the extreme left of (5.1.4) does not exist in this case we seek a new similarity variable rj associated with a higher integral moment / (r) calculated as 

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

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... 

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

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. 

This is precisely the self-similar solution of the Rayleigh problem, (3-134), which was obtained in the previous section. Notice that the length scale d drops completely out of this limiting form of the solution (3-158). This is consistent with our earlier observation that the presence of the upper boundary should have no influence on the velocity field for sufficiently small times 7 <[Pg.151]

By following an approach analogous to the above, Darhuber et al. [6] studied the dynamics of capillary spreading along hydrophobic microstripes. The smooth surface was processed chemically to create narrow hydrophilic stripes on a hydrophobic background. The equation governing the self-similar solutions, in that case, can be described as... 

The self-similar solution of an unsteady rarefaction wave in a gas-vapour mixture with condensation is investigated. If the onset of condensation occurs at the saturation point, the rarefaction wave is divided into two zones, separated by a uniform region. If condensation is delayed until a fixed critical saturation ratio Xc > 1 is reached, a condensation discontinuity of the expansion type is part of the solution. Numerical simulation, using a simple relaxation model, indicates that time has to proceed over more then two decades of characteristic times of condensation before the self-similar solution can be recognized. Experimental results on heterogeneous nucleation and condensation caused by an unsteady rarefaction wave in a mixture of water vapour, nitrogen gas and chromium-K)xide nuclei are presented. The results are fairly well described by the numerical rdaxation model. No plateau formation could be observed. 

In analyzing the self-similar solution of an unsteady rarefaction wave in a gas-vapour mixture with condensation, we will first discuss the situation where the onset of condensation occurs at a saturation ratio of unity. Then, the change in the state of the mixture is continuous. The solution is fully isentropic and follows from the characteristic form of the Euler equations, which for a left running wave is ... 

The self-similar solution is an asymptotic solution that develops when t/r - m, with r as the characteristic time of the condensation process. This development is studied numerically for a mixture of nitrogen gas and water vapour. At t = 0, a piston is accelerated instantaneously to a constant velocity, causing a one-dimensional unsteady rarefaction wave in a nadxture of gas and vapour. The wave is travelling to the left and the wave front runs with x/aQt = -1. A simple relaxation model is assumed ... 

Since we are interested in dynamic analysis starting from some initial conditions which can be arbitrary, it is clear that one cannot associate a self-similar solution from the very beginning of the process except for an initial condition that happens to be compatible with the self-similar solution. Thus, the question of a self-similar solution basically arises when the system has evolved away from the initial condition. There is thus a sense of independence of the self-similar solution from the initial condition. This independence may, however, apply only for a class of initial conditions outside of which no self-similar solution may be attained. Questions in regard to the conditions under which a self-similar solution exists for a population balance equation, and the class of initial conditions for which the solution can approach such a self-similar solution, are indeed mathematically very deep and cannot be answered within the scope of this treatment. On the other hand, numerical solutions can be examined for their approach to self-similarity. 

The corresponding equation in the self-similar solution il/ rj) is readily identified since from (5.1.2) we gave d rj) = rjij/irj) drj, given (5.1.4). Thus we obtain... 

A quick demonstration may be made of the constant aggregation kernel, which allows exact calculation of the self-similar solution. We let a x, x ) = a, and recognize it to be homogeneous as described by (5.2.14) with m = 0 so that the scaling size h t) is proportional to t. Equation (5.2.16) may be solved by Laplace transform, which is left as an exercise to the reader. However, the reader may more readily verify that for this case by... 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

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