Flow of a thin fluid film over a plane with a periodic system of jumps in the case of time-dependent viscosity and external force.

*(English. Russian original)*Zbl 1337.76002
Dokl. Phys. 55, No. 1, 23-27 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 430, No. 2, 179-184 (2010).

Summary: A class of flows of viscous incompressible fluids with varying viscosity (for example, due to varying the temperature or the presence of polymerization processes) in the field of time-dependent external forces (for example, in the non-inertial coordinate system associated with the streamlined body) is considered. Concrete expressions of these dependences are found at which the set of the Navier-Stokes equations, via
change of variables, can be transformed to equations with time-independent coefficients or to a modified set of Navier-Stokes equations. The latter allows one
to consider time-dependent analogs of the steady-state solutions of the Navier-Stokes equations.

As an example, the problem of the flow of a layer of fluid with a variable viscosity over the plane under the effect of variable external force is considered. An exact solution of this problem is used to obtain the set of equations describing the flow in a thin liquid layer in a shallow-water approximation. It is shown that, along with the time-dependent uniform solution, a nonuniform time-dependent solution with a periodical system of jumps exists (an analog of the steady-state solution constructed in [the authors, Dokl. Phys. 54, No. 5, 248–251 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk, 426, No. 3, 321–324 (2009; Zbl 1337.76007)]).

As an example, the problem of the flow of a layer of fluid with a variable viscosity over the plane under the effect of variable external force is considered. An exact solution of this problem is used to obtain the set of equations describing the flow in a thin liquid layer in a shallow-water approximation. It is shown that, along with the time-dependent uniform solution, a nonuniform time-dependent solution with a periodical system of jumps exists (an analog of the steady-state solution constructed in [the authors, Dokl. Phys. 54, No. 5, 248–251 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk, 426, No. 3, 321–324 (2009; Zbl 1337.76007)]).

##### MSC:

76A20 | Thin fluid films |

35Q35 | PDEs in connection with fluid mechanics |

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |

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\textit{V. A. Buchin} and \textit{G. A. Shaposhnikova}, Dokl. Phys. 55, No. 1, 23--27 (2010; Zbl 1337.76002); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 430, No. 2, 179--184 (2010)

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##### References:

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[9] | V. A. Buchin and G. A. Shaposhnikova, Dokl. Akad. Nauk 404(6), 762 (2005). |

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