The Stability of Two-Dimensional Linear Flows

Citation

Lagnado, Ronald Robert (1985) The Stability of Two-Dimensional Linear Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DDJC-PE17. https://resolver.caltech.edu/CaltechETD:etd-03272008-105253

Abstract

This thesis presents the results of a theoretical and experimental investigation concerned with the hydrodynamic stability of extensional flows. In particular, model extensional flows in the class of two-dimensional linear flows are considered. These flows may be classified by a parameter λ ranging from λ = 0 for simple shear flow to λ = 1 for pure extensional flow.

In Chapter I, a linear stability analysis is given for an unbounded Newtonian fluid undergoing two-dimensional linear flows. The linearized velocity disturbance equations are analyzed to yield the large-time asymptotic behavior of spatially periodic initial disturbances. The results confirm the established fact that simple shear flow (λ = 0) is linearly stable. However, it is found that unbounded extensional flows in the range 0 < λ ≤ 1 are unconditionally unstable. Spatially periodic initial disturbances which have lines of constant phase parallel to the inlet streamline of the basic flow and have sufficiently small wavenumbers in the direction normal to the plane of the basic flow must grow exponentially in time. A complete analytical solution of the vorticity disturbance equation is obtained for the case of pure extensional flow (λ = 1).

Chapter II presents a linear stability analysis for an Oldroyd-type fluid undergoing two-dimensional linear flows throughout an unbounded region. The effects of fluid elasticity on extensional-flow stability are considered. The time derivatives in the constitutive equation can be varied continously from co-rotational to co-deformational as a parameter β varies from 0 to 1. It is again found that unbounded flows in the range 0 < λ ≤ 1 are unconditionally unstable with respect to spatially periodic initial disturbances that have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. For small values of the Weissenberg number, only disturbances with sufficiently small wavenumbers α_s in the direction normal to the plane of the basic flow give rise to instability. However, for certain values of β, there exist critical values of the Weissenberg number above which flows are unstable for all values of the wavenumber α_s.

The results of an experimental investigation of the flow of a Newtonian fluid in a four-roll mill are found in Chapter III. The four-roll mill may be used to generate an approximation to two-dimensional linear flow in a central region between the rollers. A photographic flow-visualization technique was employed to study the stability of a pure extensional flow (λ = 1). Two four-roll mills with different ratios of roller length to gap width between adjacent rollers (namely, L/d = 3.39 and 12.73) were used in order to study end effects on flow stability. At sufficiently small Reynolds numbers the flow in both devices is essentially two- dimensional throughout most of the region between the rollers, except near the top and bottom bounding surfaces where three-dimensional flow involving four symmetrically positioned vortices appears. The vertical extent of this two- dimensional flow gradually diminishes and the vortices grow in size and strength as the Reynolds number is increased up to a quasi-critical range. An increase in Reynolds number through this quasi-critical range results in an abrupt transition to a steady three-dimensional flow throughout the entire region between the rollers. The three-dimensionality is significantly less pronounced in the device with L/d = 12.73, however. At sufficiently high Reynolds numbers beyond the quasi-critical range, the flow becomes unsteady in time and eventually turbulent.

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