Analytical Solution of Wave Equations in Viscous Flow with Gas Bubbles and Heat Transfer Considerations

The first purpose of this paper was to obtain non-linear equations for describing pressure waves in a liquid with gas bubbles. For this purpose, non-linear equations of fourth order and some of their special cases for describing pressure waves in a mixture of fluid and gas were considered. Then, considering the influence of heat transfer and viscosity in fluid, a differential relation between pressure and perturbation radius of gas bubbles is obtained. It is shown that the Burgers, the KdV and KdV- Burger’ equations are special cases for describing pressure waves. Also, we introduced a -expansion method, which can obtain exact solutions of non-linear differential equations without requiring the boundary and initial conditions. In this method, by choosing special constants in the obtained solutions, more solutions of these equations can be obtained. These features exhibit the superiority of the introduced approach. compared to the other methods. Furthermore, exact solutions of these equations are obtained using -expansion method, which has many applications in science and engineering.