tailieunhanh - Hydrodynamics Advanced Topics Part 3

Tham khảo tài liệu 'hydrodynamics advanced topics part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 46 Hydrodynamics - Advanced Topics accordance with its integrals of motion mean mass displacement vector u velocity of this displacement v du dt is determined by integrals of motion v P M rotation angle p angular velocity of rotation Q p is determined by integrals of motion Q M 1 where I - inertia moment and heat displacement uT determining variation of temperature and related with integral of energy E . In accordance with the set of independent field variables we can represent the kinetic K and the free F energies as corresponding quadratic forms 2K pu2 Ip2 41 2F Ằ 2p Vu 2 p Vu 2 2ỗp Vu ơp 2 V p 2 g V p 2 42 Taking into account that the dissipation dealt only with field of micro rotations and omitting for shortness dissipation of mean displacement field described by heat conductivity we can write the dissipation function in the following form 2D YP2 43 Equations of motion derived from GVP without temperature terms have the forms d dK V dF V dF dD dt du dVU d VU du d dK dK dF r dF dD --V - V - - dt dp 9p dV p d Vp 9p 44a 45a Without dissipation p 0 the motion equations obtained with use of quadratic forms 41 43 correspond to the ones for Cosserat continuum Kunin 1975 Novatsky 1975 Erofeev 1998 . Indeed for this case the equations 44 have forms d - . _ _ __ _ pd-u - Ằ 2p V Vu V Vu - V p 0 dt 44b d Id- P - V V p g V V p ơp Vu 0 45b dt The explicit form of these equations confirms that they are indeed the Cosserat continuum. If one sets formally ỗ 0 then equations 44b and 45b are split and the equation 44b reduces to ordinal equation of the elasticity theory and the equation 45b represents the wave equation for angular momentum. When dissipation exists the system of equations 44 - 45 contains additional terms responsible for this dissipation pu - Ằ 2p V Vu p V VP - J V P 0 44c IP - V V p g V V p ơp J Vu -yp 45c For the case 0 g 0 and I 0 the second equation 45c reduces to the pure relaxation form Generalized Variational Principle for Dissipative .

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