Subsonic Flows in a Multi-Dimensional Nozzle

Abstract

Abstract. In this paper, we study the global subsonic irrotational flows in a multi-dimensional (n ≥ 2) infinitely long nozzle with variable cross sections. The flow is described by the inviscid potential equation, which is a second order quasilinear elliptic equation when the flow is subsonic. First, we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension for sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow. Furthermore, we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely, provided that the incoming mass flux is less than the critical value. This gives a positive answer to the problem of Bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension [5]. Finally, under suitable asymptotic assumptions of the nozzle, we obtain the asymptotic behavior of the subsonic flow in far fields by a blow-up argument. The main ingredients of our analysis are methods of calculus of variations, the Moser iteration techniques for the potential equation and a blow-up argument for infinitely long nozzles.

Three dimensional steady subsonic Euler flows in bounded nozzles

“…Since 1950's, tremendous progress has been made on the study for potential flows. Subsonic potential flows around a body were studied extensively by Shiffman [26], Bers [2,3], Finn, Gilbarg [17,18], and Dong [13], et al The existence of subsonic potential flows in multidimensional infinitely long nozzles was achieved in [29,30,15]. The subsonic-sonic flow as a limit of subsonic flows were studied in [8,29,30,21] via compensated compactness method.…”

On multi-dimensional sonic-subsonic flow

“…Now we give another example of subsonic-sonic limit through an infinite long nozzle. As in [26], denote the multi-dimensional nozzle domain by Ω which satisfies the following regularity assumption: there exists an invertible C 2,α map T :Ω →C : x → y satisfying…”

“…The nozzle approaches to a cylinder in the far fields, i.e, Ω ∩ {x n = k} → S ± as k → ±∞, respectively, (3.3) where K is a uniform constant, C = B(0, 1) × (−∞, +∞) is a unit cylinder in R n , B(0, 1) is unit ball in R n−1 centered at the origin, S ± are n − 1 dimensional simply connected C 2,α , x n is the longitudinal coordinate and x ′ = (x 1 , · · · , x n−1 ) ∈ R n−1 , see Fig 3. The main result of [26] is stated as follows: . Then as m ε 0 → M c , the solution sequence u ε (x) possess a subsequence (still denoted by) u ε (x) that converge strongly a.e.…”

“…Since the equations of uniform subsonic flow possess ellipticity, its solutions have extra-smoothness to those related to transonic flow or supersonic flow. There are a large of literatures on the smooth uniform subsonic solutions, for instance, see [5], [29], [32], [47], [48] for two dimensional flow and [24], [25], [26], [30], [44], [51], [52], [53] for three dimensional flow. Among them, Frankl and Keldysh [32] obtained the first result about the subsonic flow past a two dimensional finite body (or airfoil).…”

“…Dong [24] extended the results of Finn and Gilbarg [30] to maximum Mach number M < 1 and to arbitrary dimensions. Recently, Du, Xin and Yan [26] obtained the smooth uniform subsonic for n-dimensional(n ≥ 2) flow in an infinitely long nozzle. For other related results, we refer to [2,3,4,6,28,31,33,34,35] and references therein.…”