The Navier-Stokes equations : a classification of flows and by P G Drazin; N Riley; London Mathematical Society
By P G Drazin; N Riley; London Mathematical Society
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Additional resources for The Navier-Stokes equations : a classification of flows and exact solutions
Example text
Berker (1963) gives a comprehensive bibliography and discusses some of these in detail. 2, steady Beltrami flows for which v ∧ ω = 0 can only exist in a viscous fluid when sustained by a non-conservative body force. 15). For axisymmetric flows we have v = (vr , 0, vz ) and ω = (0, ωθ , 0). 13) and ∂ ∂r 1 ∂(r ωθ ) ∂ 2 ωθ + = 0. 13) has the solution ωθ = r f (ψ) showing that ωθ /r is a constant along streamlines. However, Marris and Aswani (1977) have determined that the only possibility for these axisymmetric generalised Beltrami flows is that f = constant, or ωθ = αr .
8 Tangential velocity profile for a two-fluid stagnation-point flow. 29). There, as remarked, Crane (1970) interpreted it as the flow due to a stretching plate. Whilst this might be difficult to envisage physically, in the present context the ‘stretching plate’ is simply the free surface at which the velocity does increase linearly with distance. 51). 22841. 8. Wang (1987) has also considered the impingement of two stagnation-point flows at a two-fluid interface, whilst Tilley and Weidman (1998) have extended Wang’s analysis to oblique stagnation-point flow in a two-fluid system.
As |R| → ∞ a boundary layer forms, either at y = 1 with thickness O(R −1 ) for R > 0, or at y = 0 with thickness O(|R|−1/2 ) for R < 0. 4 Channel flows 29 for R > 12 but it was Robinson (1976) who uncovered the flow structure in greater detail. Thus, with R > 0 there are Type I solutions for which the velocity maximum is on the centre-line y = 0, which are the solutions obtained by Terrill. 165 two further solutions emerge. 119 there are Type II solutions for which f (y) > 0, with the maximum velocity located between the centre-line and solid boundary.