Richard Andrasik, Rostislav Vodak
Abstract:
Problems with three-dimensional models lie very often in their large
complexity leading to impossibility to find an analytical solution.
Numerical solutions are sometimes an option, but they can be unduly
complicated in the case of three-dimensional models. Frequently,
researchers investigate models where one or even two dimensions are
almost negligible and nothing important is occurring in them.
These models can be simplified and turned into one- or two-dimensional
models, which is very helpful, because their solutions are easier than
solutions of the original three-dimensional models. Since nonsteady
Navier-Stokes equations for compressible nonlinearly viscous fluids
in a three-dimensional domain belongs to the class of models which need
a simplification, when possible, to be effectively solved, we performed
a dimension reduction for this model. We studied the dynamics of a
compressible fluid in thin domains where only one dimension is dominant.
We present a rigorous derivation of a one-dimensional model from the
three-dimensional Navier-Stokes equations.
Submitted January 23, 2018. Published May 14, 2018.
Math Subject Classifications: 35Q30, 35Q35, 76D05.
Key Words: Navier-Stokes equations; compressible fluids; nonlinear viscosity;
dimension reduction; asymptotic analysis.
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Richard Andrasik Department of Mathematical Analysis and Applications of Mathematics Faculty of Science, Palacky University Olomouc tr. 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic email: andrasik.richard@gmail.com |
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Rostislav Vodak Department of Mathematical Analysis and Applications of Mathematics Faculty of Science, Palacky University Olomouc tr. 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic email: rostislav.vodak@upol.cz |
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