Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

Abstract

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications. Keywords: divergent channel, viscous fluid, Jeffery-Hamel problem, steady-state flow, bifurcation. 0015-4628/05/4003-0359  2005 Springer Science+Business Media, Inc. z = y − 1, z(0) = 0; y(1) = z(1) = 0 (2.2)

Bifurcation of multimode flows of a viscous fluid in a plane diverging channel

“…Furthermore, the dependence of the solution on the angle ␤ is also non-regular. [16][17][18][19] These computational difficulties are aggravated by the insufficient accuracy (10 −4 -10 −6 , whereas an accuracy of 10 −8 -10 −10 and higher is required) of the tabulated data for elliptic functions and integrals. Hence, the procedure of reduction to a system of transcendental equations is unnecessary since it hinders a constructive solution of the boundary value problem.…”

“…[18][19][20] It follows from an analysis of the last dimensionless factor of expression (2.3) that the magnitude of the pressure p increases without limit when the parameter |b| is reduced.…”

“…As in the case of flows in a converging channel, 17 it is convenient to subdivide the range of values of the semi-aperture angle of the diverging channel ␤ into several ranges corresponding the different qualitative behaviour of the variables ␥(b) and (b) in the procedure of continuation with respect to the parameter b < 0, |b| 1: The flow pattern is quite complicated 18 and requires a careful step-by-step presentation. In essence (and traditionally), it is assumed to be natural to begin it from the low values of n = 1, 2 which corresponds to the basic 19 and following 18 ranges of values of ␤ and Re. The bifurcation pattern is shown in Fig.…”

“…However, it has been explicitly established by the authors (this property was noted earlier, see Refs. 2, 3, etc) that, in the case of flows in a converging channel Q < 0 16,17 and a diverging channel Q > 0 18,19 , this problem, together with symmetric odd-mode solutions, has even-mode solutions which are asymmetric about the axis. This property is a consequence of the axial symmetry of the problem and it testifies to the channel walls (boundaries) "having equal rights" We note that a recent, detailed and constructive analysis of the effect of symmetry properties in the Navier-Stokes equations is available.…”

“…This property is a consequence of the axial symmetry of the problem and it testifies to the channel walls (boundaries) "having equal rights" We note that a recent, detailed and constructive analysis of the effect of symmetry properties in the Navier-Stokes equations is available. 4 The pattern of bifurcations of single-mode, purely radial flows for a diverging channel has been constructed by the authors 19 for comparatively small values of Re␣(2␤Re ≤ 6) and a wide range of variation in the semi-aperture angle ␤(0 < ␤ ≤ /2). The announcement 18 of investigations for a small semi-aperture angle 0 < ␤ ≤ 10 • suggests that, when Re is increased, only multimode flows, containing domains of forward and backward flows, are possible.…”