Solution of a linear model of a single-piston pump by means of methods for differential equations in Hilbert spaces.

*(English)*Zbl 0644.76081A mathematical model of a fluid flow in a single-piston pump is formulated and solved. Variation of pressure and rate of flow in suction and delivery piping respectively is described by linearized Euler equations for barotropic fluid. A new phenomenon is introduced by a boundary condition with discontinuous coefficient describing function of a valve. The system of Euler equations is converted to a second order equation in the space \(L^ 2(0,l)\), where l is length of the pipe. The existence, unicity and stability of the solution of the Cauchy problem and the periodic solution is proved under explicit assumptions.

##### MSC:

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

35B35 | Stability in context of PDEs |

35B10 | Periodic solutions to PDEs |

##### Keywords:

telegraph equation; time-dependent boundary condition; single-piston pump; linearized Euler equations; barotropic fluid; boundary condition with discontinuous coefficient; existence; Cauchy problem; periodic solution
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##### References:

[1] | V. Kolarčík: Linear model of a piston pump. Communication during the cooperation of Mathematical Institute of Czechoslovak Academy of Sciences and Research Institute of Concern Sigma Olomouc in 1984. Also to appear in Acta Technica ČSAV 1987-8. |

[2] | V. Lovicar: Private communication. · Zbl 0793.34040 |

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