Wavy Motion of Viscous Bubbly Liquid in Tubes of Orthotropic Material

Wavy Motion of Viscous Bubbly Liquid in Tubes of Orthotropic Material

Rafael Yusif Amenzadeh (Baku State University, Azerbaijan), Akperli Reyhan Sayyad (Baku State University, Azerbaijan) and Faig Bakhman Ogli Naghiyev (Baku State University, Azerbaijan)
DOI: 10.4018/ijcce.2012010104
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Abstract

This article investigates the pulsating flow of a compressible two-phase bubble of viscous fluid contained in an elastic orthotropicle direct axis tube. In this work, one-dimensional linear equations have been used. It is assumed that the tube is rigidly attached to the certain environment. In the case of finite length the pressure is applied at the end of its faces. In the limited process, relations obtained for a very long tube. Such a description, in a sense generalizes and strengthens the work of this type. In the numerical experiment a semi-infinite tube with flowing water containing small amount of air bubbles is considered. The influence of volume fraction of bubbles on wave characteristics is determined.
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Introduction

The problem of wave propagation in deformable shells with flowing fluid in their cavities is quite popular. When considering the problems of this kind it should be encouraged to consider the equations of motion cover, considering the effect of moving the liquid in it. It is assumed that one – dimensional approximation is applicable when the tube length is much greater than its radius. Such approximation describes the basic properties of the “cover-fluid”. To date, the aggregate of such problems is well developed area of fluid dynamics (Lightfoot, 1977; Pedley, 1983). However, the mechanism of the phenomena associated with simultaneous consideration of two-phase fluid in the compartment considering its compressibility, viscosity and orthotropy of the material tube, is not well understood. The interest in problems of wave dynamics of bubbly liquids flowing in deformable tubes is due to the importance of application of research results to problems in calculating the hydraulic systems in aircrafts, oil and gas industry, chemical technology, thermodynamics (Skalak, 1956; Suo & Wylie, 1990). Fluid–structure interaction (FSI) problems arise in industrial piping systems, underwater explosions and turbomachinery (Cole, 1948; Wylie & Streeter, 1993; Brennen, 1994). These flows often involve gas (or vapour) bubbles that alter the dynamics of the fluid dramatically (Brennen, 1995, 2005). Dynamic loading of fluid-filled, deformable tubes have been extensively studied as an FSI model problem (Tijsseling, 1996; Ghidaoui et al., 2005). Liquid-ðlled tubes were ðrst studied by (Korteweg, 1878) and (Joukowsky, 1898), who introduced a linear wave speed that accounts for the compressibility of both the liquid and the structure. The Korteweg–Joukowsky wave speed is also known as the Moens–Korteweg wave speed in a biomedical context concerning pressure pulses through blood vessels (Pedley, 1980). The wave speed in the case of bubbly liquids was later validated by Kobori, Yokoyama, and Miyashiro (1955). For cases without FSI, shock problems in bubbly liquids have also been considered by many researchers. The shock theory has been validated by experiments (Campbell & Pitcher, 1958; Noordzij & van Wijngaarden, 1974; Beylich & Gülhan, 1990; Kameda & Matsumoto, 1996; Kameda et al., 1998). In these experiments, bubbly mixtures were created in a tube, but the shock pressure was small enough to minimize the FSI effect. The detailed shock structure was also confirmed by computations (Kuznetsov et al., 1978; Nigmatulin, Khabeev, & Hai, 1988; Watanabe & Prosperetti, 1994; Kameda & Matsumoto, 1996; Kameda et al., 1998; Delale, Nas, & Tryggvason, 2005; Delale & Tryggvason, 2008). However, to the authors’ knowledge, a (nonlinear) shock theory that includes both structural compressibility and bubbles has not been presented so far (Smereka, 2002; Yates, & Satterfield, 1991; Zhang & Prosperetti, 1994).

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