Journal of Non -Newtonian Fluid Mechanics, 12 (1983) 383-386 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

Short Communication

383

FLOWS IN PRESSURE HOLES

VIVIAN O’BRIEN

Milton S. Eisenhower Research Center, Applied Physics Laboratory, Johns Hopkins University, Laurel, Maryland 20707 (U.S.A.)

(Received November 29, 1982)

In a recent article [l] there was a request for a theoretical criterion for onset of “secondary flow” in a pressure hole. The answer is, of course, no such criterion exists; whatever the Reynolds number there is always a recirculation pattern in the sharp-edged hole whenever a transverse shear flow over the open face drives it [2,3,4]. The real question is whether the asymmetric secondary flows lead to .erroneous viscoelastic pressure measure- ments. In the article, their Fig. ‘22 indicates opposite asymmetries due to non-zero Reynolds number (in Newtonian flow) and to elastic effects. This is not surprising, for second-order fluid two-dimensional asymmetry is related

to [WI

028 = 0,

where Re is the Reynolds number (U./v) and Ws is Weissenberg number (Uh/L), 1c, is the Stokes streamfunction and vorticity G = - v ‘4. [Here L, U, X are characteristic length (gap height), velocity (driving plate velocity), and elastic relaxation time (Maxwell model), respectively, and v is kinematic viscosity.] Although this is a particular fluid model, it probably indicates correctly that near pressure holes the two types of nonlinearity are often in competition.

If Fig. la, the 2-D hole (slot), is long and thin, and provided the open span 1 is no greater than the viscometric shear flow gap, C, we can estimate the pressure error P, due to finite length h (from the Newtonian solution). Even if the shear flow Reynolds number is high (based on gap height C and the velocity of the smooth driving boundary in a Couette flow arrangement), that corresponding to the highest velocity on the dividing streamline $ = 0 that bounds the cavity flow is much smaller. V.elocities are even lower in the hole. So in the cavity the polar Stokes theory [7] is valid to a good approximation for Newtonian fluid. Since the main dividing streamline is

0377-0257/82/$03.00 0 1983 Elsevier Science Publishers B.V.

-0.5.

v/c

--l.O-

-1.5-

5.3 x 1’0-2

-2.o-

0 0.5 0 I.”

X/C

a) Deep narrow cavity b) Wide square cavity

Fig. 1. balculated streamlines for flat Stokes Couette flow (U + 0) over rectangular cavities

[2] (one half of symmetric flow field shown).

located just below the top of the hole, within the hole the absolute velocity magnitude between vortices falls off as ( P,)~’ where 1 f pr = Re (A,). Here X, is the first complex eigenvahte for the even I/J solution which depends on (2a), the angle between the sides of the cavity. (For a rectangular cavity, 2ar = 0.) Whatever the odd variation of a/ along the dividing streamline, odd 4 falls off faster with depth than even #, because its first eigenvalue is greater than A,. Essentially the relative magnitude of 4 at the bottom of the hole is nearly zero when h/l > 2p, (see [3,8]) and the fluid is virtually static. Then the local effect of the elastic non-linearity for moderate Ws must also be negligible. As h/l -+ 00 the proportionality between bottom measured p,,

and (p,,-~~~) in the shear flow approaches the Tanner-Ripkin constant, l/4

[91.

385

On the other hand, if the slot is square and wide relative to the shear gap, Fig. lb, things are different. Under a flat Couette flow, the Stokes dividing streamline for Newtonian fluid may be located partially in the gap [2]. Corresponding, Fig. 2 shows a recently calculated circular shear flow exam- ple after Pritchard [IO]. The velocity along I,!J = 0 will be higher and the inertial and elastic non-linearities will be more apparent in asymmetric effects. For the same depth h as the thin cavity, there will be considerable activity near the bottom and the asymmetry may still be quite pronounced [ll]. Then the,proportionality between pH and (p,,-~~~) will not be con- stant; it will depend on Re and elastic parameters as well as the depth of the hole [6].

If the shear flow is pressure-driven (Poiseuille), instead of a Couette flow there will be a slightly different dividing streamline and cavity flow pattern, even for Stokes Newtonian flow [2,3,12]. For non-zero Re, Ws there will be inertial contributions and elastic ones to flow asymmetries in the wide holes [5,6]. If the fluid is shear-thinning as well as elastic, this will also contribute to the asymmetry of the flow as non-linearities take effect. All these changes will be reflected in the proportionality between p_, and (p,,-~~~). So Han’s reluctance to accept a universal constant for all fluids is well-taken for geometries resembling Fig. lb or Fig. 2.

b) Centerline vorticity

a) Flow field

Fig. 2. Calculated flow features for a circplar Stokes Couette flow over a sector cavity [4] (Q, is the undisturbed shear flow vorticity).

386

It is obvious that analogs to two-dimensional flow are found in axi-sym- metry flow. The Couette recirculation patterns shown here will have their counterparts in streamlines on median planes. (The axi-symmetric pressure slots must be transverse to an annulus where flow is driven by sliding one cylinder axially.) A circular pressure hole in one plate of a flat Couette flow may bear some resemblance to the two-dimensional situation, but the hole flow patterns are really three-dimensional and have not yet been described for Stokes Newtonian flow, much less the nonlinear ones.

Acknowledgement

The support for this work came from the U.S. Army through ARRADCOM, Aberdeen Proving Ground, Maryland.

References

1 C.D. Han and KJ. Yoo, J. Rheol., 24 (1980) 55-79. 2 V. O’Brien, Phys. Fl., 15 (1972) 2089-2097. 3 S.A. Trogdon and D.D. Joseph, Matched eigenfunction expansions for slow flow over a

slot. J. Non-Newt. Fl. Mech., 10.(1982) 185-213. 4 V. O’Brien, Viscous flow in an annulus with a sector cavity. Trans. ASh4E.J. Fl. Eng. 104

(1982) 500-504. 5 V. O’Brien and L.W. Ehrlich, Planar entry flow of viscoe!astic fluid. Presented at Brown

University Workshop on Numerical Simulation, 1979. 6 P. Townsend, Rheol. Acta, 19 (1980) 1 - 11. 7 H.K. Moffatt, J. Fl. Mech., 18 (1964) l-18. 8 J. Sanders, V. O’Brien, and D.D. Joseph, Trans. ASME (J. Appl. Mech.), 102 (1980)

482-484. 9 R.I. Tanner and A.C. Pipkin, Trans. Sot. Rheol., 13 (1969) l-9.

10 W.G. Pritchard, Phil. Trans. Roy. Sot., A270 (1971) 507-555. 11 F. Pan and A. Acrivos, J. Fl. Mech., 28 (1967) 643-655. 12 D.S. Malkus, Proceedings of the VIIth International Congress on Rheology, Gothenberg,

Sweden, 1976.