• P N Shankar

      Articles written in Pramana – Journal of Physics

    • On a class of three-dimensional corner eddies

      P N Shankar

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      Consider Stokes flow in a viscous fluid filling a corner, of angle 2α, bounded by two infinite plane walls. Assume that the flow is symmetrical about some plane which is normal to the walls bounding the corner. Since superposition is valid we may consider flows that are symmetrical about the plane bisecting the comer and those that are antisymmetrical about this plane. In either case it is shown that for a class of corner eddies, the corner flow is made up of an infinite sequence of eddies asr → 0, wherer is the radial distance from the corner. Moreover, the eigenvalues λ which determine the structure of the corner eddy fields satisfy the same equation, sin λα = ± λ sin 2α, that arises in the corresponding plane case. The three-dimensional velocity fields are, however, quite different from those seen in the plane case. In particular, in the symmetric case the streamlines are not closed and foci, rather than elliptic stagnation points, are the centres of the eddies in the plane of symmetry. These results represent, in this special context, a generalization to three-dimensions of Moffatt’s classical result for planar corner eddies.

    • The constancy of the contact angle in viscous liquid motions with pinned contact lines

      P N Shankar R Kidambi

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      Consider motion initiated in a viscous liquid in a smooth walled container. The liquid is initially at rest under uniform pressure from an inert gas of negligible inertia. We show that if the contact line is pinned and the interface is single valued, the contact angle has to remain constant throughout the motion. This is true even for motions of finite amplitude. Some implications of the result are discussed.

    • The shape of an axisymmetric bubble in uniform motion

      P N Shankar

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      We consider in a frame fixed to a bubble translating with steady speedU, the inviscid, axisymmetric, irrotational motion of the liquid past it. If all speeds are normalized byU and lengths by {ie437-1}, whereT is the surface tension of the liquid-bubble interface, it can be shown that the unknown bubble shape and field depend on a single parameter {ie437-2} alone, where the pressures are the ones in the bubble and far away respectively. WhenΓ is very large the bubble is almost spherical in shape while for Γ<- Γ* ≈ -0.315, bubbles whose exteriors are simply connected do not exist. We solve the non-linear, free boundary problem for the whole range Γ* < Γ < ∞ by the use of an analytical representation for the bubble shape, a surface singularity method to compute potential flows and a generalized Newton’s method to continue inΓ. Apart from providing explicit representations for bubble shapes and detailed numerical values for the bubble parameters, we show that the classical linearized solution for largeΓ is a very good approximation, surprisingly, to as low values of Γ as 2. We also show that Miksiset al [1] is inaccurate over the whole range and in serious error for large and smallΓ. These have been corrected.

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