A Panorama of Pure Mathematics (Pure and Applied Mathematics by Jean A. Dieudonne

By Jean A. Dieudonne

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G. 6. open From the above equations, the reduced open system force term f¯ , which is responsible for the rocket thrust, can be identified as the force caused by the difference of the velocity of the ejection v¯ with respect to the rocket head velocity v. 6) 21 3 Balance equations Note that in the literature, the propulsive term [ v¯ − v ] R is sometimes also referred to as ’irreversible’ contribution while the extra force v R generated by the ejection leaving the system at the same velocity as the remaining rocket head is then denoted as ’reversible’ contribution.

52) t whereby bτclosed contributes to the external, the internal and the kinetic contributions open bτext , bτint and ∂ϕ Kτ while b¯ τ contributes exclusively to the external sources bτext . 53) and thus ¯ int ¯ tD ) + v · b¯ ext ¯t ρ0 Dt K = Div (v · Π 0 − m0 K − Π : D t F + v · b 0 . 55) 0 37 3 Balance equations int keeping in mind that the internal force term b¯ 0 = 0 vanishes identically for the spatial motion case and has only been included to stress the duality with the definition based on the material motion problem.

Yet, it proves significant to discuss it in detail since it will help to introduce work conjugate pairs of stress and strain. Moreover, the balance of kinetic energy will be used to identify the external and internal mechanical power which are essential for our further thermodynamical considerations. 1 Spatial motion problem The balance of momentum, which can be understood as the continuum version of Newton’s axiom for a system of discrete particles, balances the rate of change of the spatial momentum density pτ with the spatial or rather physical forces generated by a change of the actual spatial placement of physical particles.

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