On source-sink flows in a rotating fluid

Abstract

An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the (x, y)-plane, the m rigid side walls and the two rigid end walls being in contact with the fluid. Here (x, y, z) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container at Q cm³/sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocity u = −∇ϕ − ∇ × A, is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects on u of uniform rotation of the whole system with angular velocity Ω about the z-axis when the normal component of u on the side walls is independent of z.

In the simplest conceivable case, D ≡ z_u − z_l is infinite (but D/Q remains finite). End effects are then negligible and u is everywhere independent of z. The solenoidal component of u, − ∇ × A, corresponds to j gyres, one for each of the j irreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within the m-ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇²ϕ = 0, depends on Q but not on Ω, j and v (the coefficient of kinematic viscosity), ∇ × A depends on all these quantities but vanishes identically when jΩ = 0. When jΩ ≠ 0 but v → 0, ∇²A + 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form.

End effects cannot be ignored when D is finite. When D is independent of x and y and equal to D₀ (say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3(v/Ω)^½ (δ [Lt ] D₀), the end effects that then arise are only confined to these boundary layers when j = 0. When j ≠ 0 boundary-layer suction influences the flow everywhere; thus ∇²A and ∇ϕ (but not ∇ × A) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇²A being the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δ_s (typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δ_s. When D is not everywhere independent of x and y, non-viscous end effects arise which produce relative vorticity in the main body of the fluid even when j = 0.

Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.