Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer

Abstract

Nonlinear waves are studied in a mixture of liquid and gas bubbles. Influence of viscosity and heat transfer is taken into consideration on propagation of the pressure waves. Nonlinear evolution equations of the second and the third order for describing nonlinear waves in gas-liquid mixtures are derived. Exact solutions of these nonlinear evolution equations are found. Properties of nonlinear waves in a liquid with gas bubbles are discussed.

Local conservation laws, symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

“…This equation reduces to the KdV equation when b = c = d = e =0, and to the Burgers‐KdV Equation when b = c = e =0. The KS Equation has been derived as a model for describing pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles …”

Lie symmetry analysis and exact explicit solutions of three-dimensional Kudryashov–Sinelshchikov equation

“…In 2010, Kudryashov and Sinelshchikov presented the following nonlinear partial differential equation [1]:…”

“…(1) is called Kudryashov-Sinelshchikov equation and is for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer, it is generalization of the KdV and the BKdV equation and similar but not identical to the Camassa-Holm equation. Equation (1) was studied by many researchers in 2 various methods [2][3][4][5][6][7][8][9]. Further, in the more realistic situations, the following three-dimensional…”

Exact solutions of the Kudryashov-Sinelshchikov equation by modified exp-function method

“…In 2010, Kudryashov and Sinelshchikov [2] obtained a more common nonlinear partial differential equation for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer, that is…”

“…Eq. (1) is called Kudryashov-Sinelshchikov equation, it is generalization of the KdV and the BKdV equation and similar but not identical to the Camassa-Holm (CH) equation, it has been studied by some authors [2,[4][5][6]. Undistorted waves are governed by a corresponding ordinary differential equation which, for special values of some integration constant, is solved analytically in [1].…”