Lie group analysis, exact solutions and conservation laws to compressible isentropic Navier–Stokes equation

Abstract

The current study dedicated to the compressible isentropic Navier–Stokes equations in one-dimensional with a general pressure law. The Lie Group method is employed to reduce the compressible Navier–Stokes equations to a system of highly nonlinear ordinary differential equations with suitable similarity transformations. Consequently, with the help of exact solutions of reduced ordinary differential equations, similarity variable and similarity solutions, exact solutions of the main equation are obtained. Finally, using conservation laws multiplier, we find the complete set of local conservation laws of compressible isentropic Navier–Stokes equations for the arbitrary constant coefficients.

Appendix

Theorem 1

Proof

It is easy to verify that these four invariant vector fields form a vector space, of dimension four which is called the Lie algebra.

Theorem 2

Proof

Since each group $\,\,\left( {T_{a}^{i} \,\, = \,\,1,\,\,2,\,\,3,\,\,4} \right)$ is a symmetric group this implies that If $\theta \,\, = \,\,\,f\,\left( {x,\,\,t} \right),\,\,\,\ell \,\,\, = \,\,\,g\,\left( {x,\,\,\,t} \right)$ is a given solution of system (1), so are the $\,\,\theta_{i} ,\,\,\,\ell_{i} \,\,\left( {i\,\, = \,\,1,\,\,2,\,\,3,\,4} \right)$.

Theorem 3

Proof

The categorization of four-dimensional subalgebras of the Lie algebra (14) is obtained with the help of an inductive approach [22]. Here, we start to deal with a general element of the form $A = \alpha_{1} A_{1} + \alpha_{2} A_{2} + \alpha_{3} A_{3} + \alpha_{4} A_{4}$ and subject it to different adjoint transformations to make simpler it as much as simple. As a result, the one-parameter optimal system of isentropic Navier- Stokes equations is obtained which is spanned by (18).

Theorem 4

Proof

The general solutions Eq. (19) involves three constants. Now with respect to one arbitrary constant, we obtained the similarity variable (new independent variable). The other two constants play the role of group invariant solution (new dependent variables).