Citation
Lagnado, Ronald Robert (1985) The Stability of TwoDimensional Linear Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DDJCPE17. https://resolver.caltech.edu/CaltechETD:etd03272008105253
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 twodimensional 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 twodimensional linear flows. The linearized velocity disturbance equations are analyzed to yield the largetime 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 Oldroydtype fluid undergoing twodimensional linear flows throughout an unbounded region. The effects of fluid elasticity on extensionalflow stability are considered. The time derivatives in the constitutive equation can be varied continously from corotational to codeformational 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 fourroll mill are found in Chapter III. The fourroll mill may be used to generate an approximation to twodimensional linear flow in a central region between the rollers. A photographic flowvisualization technique was employed to study the stability of a pure extensional flow (λ = 1). Two fourroll 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 threedimensional 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 quasicritical range. An increase in Reynolds number through this quasicritical range results in an abrupt transition to a steady threedimensional flow throughout the entire region between the rollers. The threedimensionality is significantly less pronounced in the device with L/d = 12.73, however. At sufficiently high Reynolds numbers beyond the quasicritical range, the flow becomes unsteady in time and eventually turbulent.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Chemical Engineering  
Degree Grantor:  California Institute of Technology  
Division:  Chemistry and Chemical Engineering  
Major Option:  Chemical Engineering  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  9 January 1985  
Record Number:  CaltechETD:etd03272008105253  
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd03272008105253  
DOI:  10.7907/DDJCPE17  
Related URLs: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  1183  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  08 Apr 2008  
Last Modified:  20 Dec 2019 20:01 
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