Geodesic
The solution of equations of ideal gas that describes Galileo invariant motion with helical level lines, with the collapse in~the helix
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 4, pp. 797-808

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We consider the equations of ideal gas dynamics in a cylindrical coordinate system with the arbitrary equation of state and one two-dimensional subalgebra from the optimum system of an 11-dimensional Lie algebra of differentiation operators of the first order. The basis of the subalgebra operators consists of the operator of Galilean transfer and the operator of movement on spiral lines. Invariants of operators set representation: type of speed, density and entropy. After substitution of the solution representation into the equations of gas dynamics the assumption of the linear relation of a radial component of speed and spatial coordinate is entered. Transformations of equivalence which are allowed by a set of equations of gas dynamics after substitution of the solution representation are written down. For the state equation of polytropic gas all four solutions depending on an isentropic exponent are found. For each case the equations of world lines of gas particles motion are written down. The transition Jacobian from Eulerian variables to Lagrangian is found. The instants of collapse of gas particles are determined by value of the Jacobian. As a result the solutions describe movement on straight lines from a helicoid surface. Movements of the particles on equiangular spirals lying on a paraboloid and on hyperbolic spirals, lying on a cone.
Keywords: gas dynamics, rank two submodel, linear velocity field, polytropic gas, collapse surface, helicoid. 
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     author = {Yu. V. Yulmukhametova},
     title = {The solution of equations of ideal gas that describes {Galileo} invariant motion with helical level lines, with the collapse in~the helix},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {797--808},
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Yu. V. Yulmukhametova. The solution of equations of ideal gas that describes Galileo invariant motion with helical level lines, with the collapse in~the helix. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 4, pp. 797-808. http://geodesic.mathdoc.fr/item/VSGTU_2019_23_4_a13/