Some unsteady pseudo-plane motions for the Navier-Stokes equations

Abstract

Some unsteady motions of a viscous incompressible fluid are determined in which all streamlines lie in the planes z = constant. These include motions in which the fluid in each plane z = constant moves as in a rigid body rotating with angular speed ω about a point x = ζ (z, t), y = η(z) of stagnation points as solutions of linear ordinary differential equations.

For unsteady motions ordinary solutions have been published.

Since the Navier-Stokes equations are satisfied for any unsteady motion with ζ = ζ (z, t) = F(z, t)cos ωt + G(z, t) sin ωt + m(t), η = η(z, t) = − F(z, t)sin ωt + G(z, t) cos ωt + n(t), where ω is a constant, m(t) and n(t) are arbitrary functions of t, while F and G are independent solutions of the heat equation θ t = zz , with v the kinematic viscosity, we can analyse flows including timeharmonic flow fields and the decaying modes within an orthogonal rheometer. Other, more general pseudo-plane flows are described and the use of Lie groups to generate explicit flow fields is outlined.

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Correspondence to D. F. Parker.

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Németh, S.Z., Parker, D.F. Some unsteady pseudo-plane motions for the Navier-Stokes equations. ARI 50, 161–168 (1998). https://doi.org/10.1007/s007770050010

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Key words

  • Navier-Stokes equations
  • Unsteady solutions
  • Decaying modes
  • Pseudo-plane motions
  • Orthogonal rheometer
  • Generalised stagnation point flows