© 1982 by Oxford University Press
DUAL SOLUTIONS FOR STEADY LAMINAR FLOW THROUGH A CURVED TUBE
( Department of Applied Mathematics, University of Western Ontario London, Ontario, Canada )
Numerical solutions of the Navier-Stokes equations are presented for steady laminar flow through a slightly curved tube of constant circular cross-section. The motion is due to the action of a constant axial pressure gradient and depends upon a similarity parameter which is designated the Dean number D. The solutions are obtained by a method in which the basic two-dimensional partial differential equations governing the motion are reduced to infinite sets of ordinary differential equations for the dependent variables using substitutions of Fourier series. The infinite sets of equations are then reduced to finite sets by truncating the series and the finite sets are solved numerically to give an approximation to the flow. The solutions cover the range D = 96 to 5000. For D<956 only one solution can be found for a given value of D. The secondary flow consists of a symmetrical pair of counter-rotating vortices of Taylor-Goertler type. The same type of solution can be found for D>956, but in addition a second family of solutions can be obtained in this case in which the secondary flow has a four-vortex pattern consisting of two symmetrical vortex pairs. The properties of the two-vortex family of solutions agree well with those of previous calculations; the four-vortex solutions have not previously been obtained for a tube of circular cross-section.