The shock wave structure by model equations of capillarity

Abstract

In this paper we investigate the influence of the capillarity terms on the shock wave structure. To this end we compare the shock structures derived from the viscosity-capillarity model and from the Navier-Stokes equations, i.e. t he viscosity model. Let A and ε characterize the values of the capillarity and viscosity effects, respectively. First, we prove that if the ratio A/ε² â‰ª 1 then the viscosity-capillarity and viscosity shock structures differs but only a little. Secondly, if A/ε² â‰« 1 then the viscosity-capillarity shock waves are oscillatory, whereas the viscosity waves are never such. Thirdly, to investigate the intermediate case of A/ε² ≈ 1 we study numerically so-called impending shock splitting. This effect consists in that the shock profile has two inflection points, under suitably chosen data, instead of one, what is usual. Our calculations show that the capillarity, if strong enough, kills this effect totally.