Dynamics of a vapor bubble near a thin elastic plate
چکیده
Numerical and experimental results show that during the collapse phase of a vapor bubble near a rigid boundary, in the absence of strong buoyancy forces, a liquid micro jet is developed on the side of the bubble far from the rigid surface and directed towards it. Numerical and experimental results also show that, in the case of a bubble near a free surface, during the collapse phase of the bubble and in the absence of strong buoyancy forces, the vapor bubble is repelled by the free surface. In this case a liquid micro jet is developed on the closest side of the bubble to the free surface and is directed away from it. The dynamic behavior of a vapor bubble near a free surface leads to the idea that a vapor bubble during its growth and collapse phases near a deformable diaphragm may have a behavior similar to its behavior near a free surface. In this paper dynamics of a vapor bubble during its growth and collapse phases near a thin elastic plate is investigated. It has been shown that the growth and collapse of a vapor bubble generated due to a high local energy input causes considerable deformation on the nearby thin elastic plate. Different thin elastic plates with different thicknesses and different flexural rigidities are assumed and the dynamic behavior of a vapor bubble near each of these plates is investigated. Results show that during the growth and collapse of a vapor bubble near a thin elastic plate with a proper thickness and flexural rigidity, in the absence of strong buoyancy forces, a liquid micro jet may develop on the closest side of the bubble to the thin elastic plate and directed away from it. INTRODUCTION Experimental and numerical investigations on the dynamic behavior of a vapor bubble near deformable surfaces have been carried out by some researchers. These investigations show that during the collapse phase of a vapor bubble near a deformable surface, in the absence of strong buoyancy forces, the minute displacement of the deformable surface may cause the liquid micro jet to be directed away from it [1-7]. In an important study, Duncan and Zhang [8] investigated the dynamics of a collapsing cavity near a compliant wall numerically. They noted that when the wall is rigid, the generated liquid jet during collapse phase is directed towards the boundary but near an elastic membrane the liquid jet may be directed away from the membrane. In their research, they assumed that the vapour bubble, initially, is in its maximum size. This is because of the fact that, in the case of a vapour bubble initially in its minimum size, the pressure distribution on the nearby boundary changes very fast. Consequently, the iteration scheme which is employed by Duncan and Zhang [8] for evaluating of initial pressure distribution on the compliant wall does not work any more. In the present paper, theory of explosion bubble which is developed by Best [9], is employed and in this case it is possible to simulate an explosion bubble growing from its initial minimum volume. THEORY OF THE PROBLEM In this paper dynamics of a vapor bubble generated by a high local energy input near a thin elastic plate is investigated. The used samples are Steel, Aluminum and Magnesium. The physical characteristics of the plates have been shown in the Table.1. The generated vapor bubble is located in the midpoint and below of the thin plate. In the all cases the standoff parameter,γ , is equal to 1. The initial pressure inside the vapor bubble is very high and is related to the initial size of the bubble. Figure 1 shows the initial position of the bubble and the metal plate. It is assumed that the generated vapor bubble contains a mixture of non-condensable and non-chemically reacting gas and saturated vapor. The non-condensable gas inside the vapor bubble is assumed to be an ideal gas. Therefore the pressure inside the bubble is obtained by summation of the partial pressures of the saturated vapor and the ideal gas. It is obvious that the partial pressure of the ideal gas inside the bubble is the dominant pressure and is obtained by the isentropic relation between the pressure inside the bubble and the bubble volume. According to Best [9] in the case of an explosion vapor bubble with an initial small radius, , the bubble generated by a high local energy input contains a mixture of saturated vapor pressure and an ideal gas with a very high partial pressure. The vapor bubble is assumed to be spherical in its initial minimum volume. The equation of purely radial motion of a bubble generated by a high local energy input is given as: 0 R 0 2 3 2 = − + + ∞ ρ b P P R R R & & & (1) Where is the variable pressure inside the bubble. is the pressure in the far field and R is the radius of the bubble, with dots denoting time derivatives b P ∞ P For obtaining governing equation of the hydrodynamic behavior of the liquid domain around the vapor bubble, it is assumed that the liquid is incompressible, inviscid, and irrotational and surface tension is neglected. In this case the flow of the liquid around the vapor bubble is a potential flow. Therefore the Green’s integral formula is the governing equation of the flow around the vapor bubble and is given as: