MARCH 1971TECHNICAL NOTES 533 the technique used previously by Mitchell, Crocco, and Sirignano 1 and Crocco and Mitchell. 2 The relevant dependent variables u, p, r, C, cr and w are represented as power series in e. For example, p = p 0 + epi + e 2 p 2 + e 3 p 3 + . . . etc., where e = M 112 . The coefficients of e in these expansions are comprised of a steady-state and a time-dependent part. Thus, pi = pi + pi 1 , Pz = PZ + P2 1 etc. Both u\ and rj are taken to be of 0(e). Since, as will be discussed shortly, the analysis is carried out through 0 (e 5 ) , the combustion zone can actually occupy a fairly sizeable fraction of the chamber axial length, and is not of zero length as in the "concentrated combustion" model used by Mitchell, Crocco, and Sirignano. 1 It is necessary to use the simplest kind of coordinate stretching in order to ensure well-behaved periodic solutions even when waveforms are discontinuous. Therefore, the stretched time variable 5 is introduced and P = 2(1 + cPi + e 2 P 2 + ...). P is the nondimensional period of the oscillations and 2 is the wave travel time for an acoustic (zero amplitude) wave. The technique of multiple scales is applied with the introduction of a second axial variable y, which is defined as y = x/rj. Derivatives of u and w with respect to this variable are of 0(1) instead of 0(l/e) as they are with respect to x. All dependent variables are then considered to be functions of the 3 independent variables x, y, and 5. Thus, u = u(x,y,d), p = p(x,y,5), etc. The governing partial differential equations and the droplet vaporization equation are then rewritten in these variables, and the power series representations of the dependent variables are substituted into the equations which result.The system of equations is first solved for the steady state. To the order of approximation necessary for consistent solution of the time-dependent equations, the results are = w 2 = 1 -(1 -?/) 3 = 1, pz = yu 2 ui = Wi = Us = WS = pi = Pi = 00 = 0"! = (72 = 0 r 0 = (I-2/) 1 ' 2 = ui/2C Q Carrying out the analysis of the time-dependent equations through 0(e 3 ) and applying the appropriate boundary conditions leads to the following expressions: U2 1 = /(« -x) -f(d + x) p^1 -2rJ! f' J 8 -Ui l = pi 1 = = (71 1 = Wi l = = 0Here, $ = rji/ui (the steady-state droplet lifetime) and / is an arbitrary function periodic in 2. In order to determine the form of / and therefore of p 2 and u%, it is necessary to continue the analysis through 0 (e 5 ) . Doing this and applying the appropriate order boundary conditions, the following nonlinear integro-differential equation for / is finally derived:(87 -1 -where ^ = 2(7 -!)),The developments that resulted in Eq. (2) are valid for either continuous or discontinuous pressure waves. In order to determine the waveform and amplitude of the pressure oscillations, this equation must be solved for either type of oscillations. Solution of Eq. (2) for small / is easily carried out by linearizing the equation and assuming / = sin?r6. A neutral stability relationship...
Oblique wave formation giving rise to a crosshatched wave pattern in the melt layer of a body in supersonic flow is examined. The three-dimensional disturbance equations of the liquid including the effects of viscosity are formulated. The disturbance motion of the supersonic gas stream is taken into account neglecting viscosity. Three-dimensional disturbances are considered for which the flow is supersonic in the direction normal to the wave fronts and for which energy is fed into the disturbance motion of the liquid from the external gas flow through the pressure variation of the Mach waves at the interface. Numerical results are presented for waves in the liquid for a range of liquid Reynolds numbers. Analytic results in closed form are presented for a highly viscous melt layer. The connection between the dispersion relation presented for thin liquid films and the dispersion relation presented by earlier investigators for liquids of infinite depth is established. It is demonstrated that the inclusion of a finite depth for a viscous liquid leads to a lower cutoff wavenumber. Furthermore, it is shown that this cutoff is present even for initially quiescent liquid films as it is for films in shearing motion. For initially quiescent liquid films the results are correlated in terms of a single parameter.
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