Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L²+²r²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p₀, p pressures of O(1) and O() - Re Reynolds number=aW₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r₀ radius of the projection of C - t unit tangent vector to C - V_r, V, V_s velocity components along the nonorthogonal directions - V_r, V, V_s nondimensional velocity components along - W₀ average velocity in a straight annulusGreek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V_r, V, V_s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulusAfter this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].