https://doi.org/10.1140/epjp/i2017-11659-5
Regular Article
Plane shearing waves of arbitrary form: Exact solutions of the Navier-Stokes equations
1
Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691, Stockholm, Sweden
2
Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, D-37077, Göttingen, Germany
3
Raman Research Institute, 560 080, Bangalore, India
* e-mail: singh@mps.mpg.de
Received:
20
July
2017
Accepted:
11
August
2017
Published online:
28
September
2017
We present exact solutions of the incompressible Navier-Stokes equations in a background linear shear flow. The method of construction is based on Kelvin’s investigations into linearized disturbances in an unbounded Couette flow. We obtain explicit formulae for all three components of a Kelvin mode in terms of elementary functions. We then prove that Kelvin modes with parallel (though time-dependent) wave vectors can be superposed to construct the most general plane transverse shearing wave. An explicit solution is given, with any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions.
© The Author(s), 2017
