# Compressible Jet Flows Effluxing from a Semi-infinitely Long Nozzle

Project: Research

## Description

The aim of this project is to develop the mathematical theory of compressible jet flows,particularly focusing on the rigorous mathematical analysis of the existence of jet flowseffluxing from a semi-infinite long nozzle and related problems by the variational method.Jet flows can be frequently observed by gas steaming in parallel flow out of an orifice intothe atmosphere. On the basis of observation we suppose that the jet of escaping gas is separatedfrom the quiet air at atmospheric pressure by a boundary consisting of a discontinuitysurface. Moreover, we consider the steady, isentropic, and irrotational flow, and assume thatthe incoming flow at the inlet is subsonic. More description about the steady, isentropic andirrotational jet flow can be found in Courant-Friedrichs’ classic book (cf. [25, p289–290]).Moreover, the jet flow is completely different from the shock wave, since jet flow is characteristic,i.e., the wave generates along the velocity field. Therefore, as a typical prototypeof the elementary waves in the gas dynamics, the study of the well-posedness theory relatedto the compressible jet flows contributes a lot to the mathematical theories of the interfaceand free boundary.It is a long history to study the jet problem by the variational method. For the irrotationalincompressible Euler flows, the variational method was first introduced by P.R. Guardian, H.Lewy and M. Schiffer to deal with the axially symmetric cavitational flows in [31]. Then ithas been significantly developed by Alt-Caffarelli-Friedman in their elegant works in [1, 2, 3],and was applied to deal with jet flows, cavitational flows and jet flows with gravity in [2],and two phase fluids in [3]. On the other hand, for the compressible flow, as far as the P.I.know, there are few results (cf. [4] and [7]) on the global existence of the compressible jetflow. Because compressible jet flow is fundamental and important even though there arevery few results, it is valuable for us to work on the objectives proposed in this project.We have been working on the related problems for several years. Recently, we haveestablished the well-posed theory of the Rethy flows in two dimensions [22] and in theaxially symmetric case [23]; incompressible jet flows for various pressures at the outlet [24];subsonic smooth Euler flows in infinitely long nozzles [12]; and compressible two phase flowsin infinitely long nozzles [19]. In this project, we will mainly focus on a further developmentof the technics of proving the obtained results, to study the mentioned objectives.In [4] and [7], the speed on the boundary of the jet is given a priori to make the flow tobe subsonic, while the incoming flow at the inlet is not given and determined by the speedon the boundary of the jet. There is a nature question on the existence of the jet flows whenthe flow at the inlet is given a priori. Hence the results and techniques developed from thisresearch project will also provide a better understanding of this problem.## Detail(s)

Project number | 9042552 |
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Grant type | GRF |

Status | Finished |

Effective start/end date | 1/01/18 → 8/12/21 |