Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution

Abstract

In this article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve in x − y x-y plane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.

Existence of solutions to gas expansion problem through a sharp corner for 2‐D Euler equations with general equation of state

“…Hence, using (16), we obtain a 𝐶 1 norm estimate of the solution to the Goursat problem ( 4) and (31), so by the theory of global classical solution for quasi-linear hyperbolic equations, 29 the lemma can be proved. ■…”

“…[18] for more details. We use the commutator relation (17) on 𝑢 and use (16) to obtain Then one can apply the commutator relation on 𝜌 and use the relations (15) to prove this proposition. ■ Proposition 3.…”

“…In recent years, a lot of significant work has been done for the 2-D compressible Euler system as well as numerous other related models for a variety of initial and boundary value problems; see, namely, Refs. [5][6][7][8][9][10][11][12][13][14][15][16]. In particular, for gas expansion problems through a sharp corner or wedge, we refer the reader to Refs.…”

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

“…Similarly, with $$$ f\left(h,b\right)=g\left(h,b\right)=h $$$, the system () was discussed by Shen et al 34 for classical and nonclassical wave interactions and for construction of non self‐similar solutions by Sun 35 . Recently, Barthwal and Raja Sekhar studied () for a non‐self‐similar Riemann problem and proved the existence of global solutions 36 …”

“…35 Recently, Barthwal and Raja Sekhar studied (1) for a non-self-similar Riemann problem and proved the existence of global solutions. 36 Our objective in this article is to study the system (1) with the initial data of the form…”