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# The Analytical Solutions for the N‐Dimensional Damped Compressible Euler Equations

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Employing matrix formulation and decomposition technique, we theoretically provide essential necessary and sufficient conditions for the existence of general analytical solutions for N‐dimensional damped compressible Euler equations arising in fluid mechanics. We also investigate the effect of damping on the solutions, in terms of density and pressure. There are two merits of this approach: First, this kind of solutions can be expressed by an explicit formula $\mathbit{u}=\mathbit{b}\left(t\right)+A\left(t\right)\mathbit{x}$ and no additional constraint on the dimension of the damped compressible Euler equations is needed. Second, we transform analytically the process of solving the Euler equations into algebraic construction of an appropriate matrix $A\left(t\right)$. Once the required matrix $A\left(t\right)$ is chosen, the solution $\mathbit{u}$ is obtained directly. Here, we overcome the difficulty of solving matrix differential equations by utilizing decomposition and reduction techniques. In particular, we find two important solvable relations between the dimension of the Euler equations and the pressure parameter: $\gamma =1-2/N$ in the damped case and $\gamma =1+2/N$ for no damping. These two cases constitute a full range of solvable parameter $0\le \gamma <+\infty$. Special cases of our results also include several interesting conclusions: (1) If the velocity field $\mathbit{u}$ is a linear transformation on the Euclidean spatial vector $\mathbit{x}\in {R}^{N}$, then the pressure p is a quadratic form of $\mathbit{x}$. (2) The damped compressible Euler equations admit the Cartesian solutions if $A\left(t\right)$ is an antisymmetric matrix. (3) The pressure p possesses radially symmetric forms if $A\left(t\right)$ is an antisymmetrical orthogonal matrix.
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Document Type: Research Article

Publication date: April 1, 2017

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