The study is devoted to modelling of step pressure pulse propagation in an Oldroyd type viscoelastic liquid, confined in a circular thin-walled elastic pipe. The model is developed within quasi-one-dimensional approach; previous results of the authors [1, 2], obtained in a frequency domain for a similar waveguide, were used at the basic formulation. It is well known that the main mechanism influencing final signals propagation in elastic tubes is fluid-structure interaction (FSI); its description is considered usually as an extension of conventional water hammer theory. The most of existing one-dimensional models account for viscous friction in the wave through shear stress at the tube wall, where maximum velocity gradients are achieved [3]. The wall shear is especially important in applications using high-molecular liquids, for instance, in polymer processing technology. Such liquids are characterized by large Newtonian viscosity; however, because of mechanical relaxation, the wave attenuation in a liquid-filled tube is much less than it follows from predictions of the classical theory, based on a pure viscous description of fluid. The main target of the study was to describe pressure signals propagation in a tube with account for the stress relaxation in the wave. Rheological Oldroyd equation in an integral form is used at the hydrodynamic problem formulation, the momentum and mass balance equations for liquid flow in the wave are averaged along the tube cross section, and transient friction at the wall is calculated from solution of non-stationary hydrodynamic problem for the same waveguide but with rigid walls. The resulting system of equations combined with the stress-displacement relation for the tube wall with appropriate boundary conditions, is solved by operational method. The pressure amplitude in the wave was found by numerical inversion of Laplace transform, combined with asymptotic expansion of the Bessel functions, involved in the solution for transient friction at the wall in the s-domain. Results of simulations illustrate pressure profile evolution in the waveguide for different system parameters and demonstrate essentially less wave attenuation along the tube, as compared with a similar pure viscous liquid.
- Levitsky, R. Bergman, O. Levi, J. Haddad, Pressure waves in elastic tube with polymeric solution, J. Appl. Mech. Eng. 4 (1999) 561-574.
- P. Levitsky, R. M. Bergman, J. Haddad, Sound dispersion in deformable tube with polymeric liquid and elastic central rod, J. of Sound and Vibration 275 (2004) 267-281.
- E. Nakoryakov, B. G. Pokusaev, I. R. Shreiber, Wave propagation in gas-liquid media, CRC Press, New-York, 1993.
