A method for replicating exact solutions of the Euler equations for incompressible Beltrami flows

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In the paper, Beltrami flows or helical flows are flows in which the vorticity and velocity vectors are collinear, and the proportionality coefficient between these vectors is nonzero and is the same at all points of the flow. A method is proposed that allows using known helical solutions to obtain new helical solutions of the Euler equations for an incompressible fluid. Some of these new solutions cannot be obtained by the known methods of replicating solutions by shifting and rotating the coordinate system, symmetry, scaling, cyclic permutation of the velocity and coordinate components, vector summation. The new replication method is applied to such parametric families of exact solutions in which the proportionality coefficient between velocity and vorticity remains unchanged for different values of the parameter. The essence of the method is that for such families the derivative of the velocity with respect to the parameter is also the helical velocity. The sequential differentiation of the speed of a new solution with respect to a parameter gives an endless chain of new exact solutions.

About the authors

Grigory Borisovich Sizykh

Moscow Aviation Institute (State Technical University)

Email: o1o2o3@yandex.ru

Candidate of physico-mathematical sciences, Associate professor


  1. Prosviryakov E. Yu., "Exact solutions to generalized plane Beltrami-Trkal and Ballabh flows", Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 24:2 (2020), 319-330
  2. Хорин А. Н., Конюхова А. А., "Течение Куэтта горячего вязкого газа", Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 24:2 (2020), 365-378
  3. Kuzmina K., Marchevsky I., Ryatina E., "Exact solutions of boundary integral equation arising in vortex methods for incompressible flow simulation around elliptical and Zhukovsky airfoils", J. Phys.: Conf. Ser., 1348 (2019), 012099
  4. Голубкин В. Н., Сизых Г. Б., "Течение вязкого газа между вертикальными стенками", ПММ, 82:5 (2018), 657-667
  5. Bosnyakov S. M., Mikhaylov S. V., Podaruev V. Yu., Troshin A. I., "Unsteady discontinuous Galerkin method of a high order of accuracy for modeling turbulent flows", Math. Models Comput. Simul., 11:1 (2019), 22-34
  6. Dergachev S. A., Marchevsky I. K., Shcheglov G. A., "Flow simulation around 3D bodies by using Lagrangian vortex loops method with boundary condition satisfaction with respect to tangential velocity components", Aerospace Science and Technology, 94 (2019), 105374
  7. Trkal V., "A note on the hydrodynamics of viscous fluids", Czech. J. Phys., 44 (1994), 97-106
  8. Громека И. С., Некоторые случаи движения несжимаемой жидкости (докторская диссертация), Казань, 1881, 107 с.
  9. Beltrami E., "Considerazioni idrodinamiche", Nuovo Cim., 25 (1889), 212-222
  10. Васильев О. Ф., Основы механики винтовых и циркуляционных потоков, Госэнергоиздат, Л.; М., 1958, 144 с.
  11. Ковалев В. П., Просвиряков Е. Ю., Сизых Г. Б., "Получение примеров точных решений уравнений Навье-Стокса для винтовых течений методом суммирования скоростей", Труды Московского физико-технического института, 9:1 (2017), 71-88
  12. Arnold V. I., "Sur la topologie des ecoulements stationnaires des fluides parfaits", C. R. Acad. Sci. Paris, 261 (1965), 17-20
  13. Childress S., "New solutions of the kinematic dynamo problem", J. Math. Phys., 11:10 (1970), 3063-3076
  14. Berker R., "Integration des equations du mouvement d'un fluide visqueux incompressible", Encyclopedia of Physics, v. 8/2, Springer-Verlag, Berlin, 1963, 1-384



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