Flow behavior of viscoelastic fluids in the inlet region of a channel

Abstract

The behavior of viscoelastic fluids in the inlet region of a channel formed by two parallel flat plates has been analyzed with a Rivlin-Ericksen approximation to describe the fluid properties. I t was found that viscoelasticity changed the entry length; both the magnitude and direction of this change were predicted to depend upon the detailed fluid properties. These predictions of the change in the entry length were seen to find some qualitative support in available experimental evidence.The directions for further analysis, based primarily on the important observation that these materials may exhibit solidlike behavior in the immediate vicinity of the entrance, are indicated. This solidlike behavior is seen to be closely related to "melt fracture" and similar instability phenomena common to these materials; i t serves to explain directly the otherwise perplexing observations concerning the similarity of such fluid fracture phenomena in materials as divergent in their properties as viscoelastic polymeric systems and inelastic dilatant slurries.A currently important engineering problem is that of the analysis of the flow behavior of viscoelastic fluids in complex geometries. While a number of problems of interest in the polymer processing industry may be analyzed a t least approximately in terms of laminar shearing motions (12, 16, 42, 5 8 ) , most problems of engineering interest are far more complex. Examples of such complex motions are found in the turbulent flow of dilute polymer solutions in a pipe (28, 4 6 ) ; the behavior of bulk polymers in extruders, mills, and mixers ( 2 4 ) ; and in fiber spinning operations (56). I n this paper, we shall investigate one such fluid dynamic problem: the flow of a viscoelastic material in the entrance region of a conduit. PREVIOUS WORKThe entrance length problem for the flow of Newtonian fluids into conduits has attracted the efforts of a number of researchers during the past half century, many of whom have made use of the Prandtl-von Karman boundary-layer theory ( 4 8 ) and have applied it in computing the velocity and pressure fields. Schiller (47) used the von Karman-Pohlhausen integral momentum procedure, while Schlichting (49) and Atkinson and Goldstein (19) matched a Blasius type of solution with a reverseward perturbation of the fully developed velocity profile. Tomita ( 5 3 ) introduced a variational principle to solve the entrance region boundary-layer equation, while Bodoia and Osterle ( 5 ) used finite difference methods. On the other hand, some authors have avoided boundary-layer methods. Langhaar ( 2 1 ) obtained a solution to the capillary tube problem by linearizing the Navier-Stokes equation and by presuming a Bessel function solution. Wang and Longwell ( 5 5 ) carried out a full numerical solution of the complete two-dimensional Navier-Stokes equation for several special cases chosen to ascertain the limitations introduced by boundary-layer approximations.A boundary-layer theory for purely viscous power law fluids was developed by a number of researchers...

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