time required for a particle, starting from any point in the field, t o reach the sink is finite and is smaller the larger the stretching at the starting point. The steady sink flow is unsteady in a Lagrangian sense and is increasingly accelerated as the sink is approached. The above discussion shows that much caution has to be exercised in extrapolating rheological results, even qualitatively, between different, although in some respects similar, types of flow.
D
= =
Q
= = = = = =
R
= = =
stretching tensor, see-’ pressure, dynes/cm2 sink strength, cm3/sec or cm2/sec value of r in reference configuration, cm polar coordinate, cm physical components of stress, dynes/cm2 time lapse, sec time, see velocity, cm/sec value of z in reference configuration, ern cylindrical coordinate, em
GREEKLETTERS “IC(S)
=
P
=
Y
= =
r
r
= =
e e u , UI, u2
= = = = =
7
=
h I.(
P , mi
=
9
=
value of @ in the reference configuration, dimensionless spherical coordinate, dimensionless
SPECIAL SYMBOLS m
5
s=o
,
m
m
F1, F~ = material functionals, dynes/cm2
s=o
s=o
6/6t
=
contravariant convected derivative, sec-1
Acknowledgement
One of the authors (R.E.M.) gratefully acknowledges support from N.S.F. and Project Themis.
Nomenclature
P,P o
@
history of i t h principal stretch ratio, dimensionless material constant, dynes sec2/cm2 principal stretching, see-’ material constant, dynes sec2/cm2 dummy integration variable value of e in reference configuration, dimensionless polar coordinate, dimensionless relaxation time, see viscosity, dynes sec/cm2 density, dynes sec2/cm4 material functions, dynes/cm2 extrastress tensor, dynes/cm2
literature Cited
Astarita, G., IND. ENG.CHEM.FENDAM. 6, 257 (1967) Carroll. 11.M.. Int. J . Ena. Sci. 5 . 515 11967). Coleman, B. D:, Proc. Roy: Soc. London A306, 449 (1968). Coleman, B. D., Noll, W., Phys. Fluids 5 , 840 (1962). Fosdick, R . L., Arch. Rat. Mech. Anal. 29, 272 (1968). Giesekus, H., Rheol. Acta 7, 127 (1968). Kaloni, P. N., J . Phys. SOC.Japan 20, 610 (1965). Lodge. A. S..“Elastic Liauids.” Academic Press, New York. 1964. RlaAovitz, H., Coleman; B. D., Advan. Appl. Mech. 8 , 69 (1964). RIetzner, A. B., Uebler, E. A., Chan Man Fong, C. F., i n press, 1970. JIurch, R . E., Ph D. thesis, University of Delaware, Newark, Del., 1970. Ramacharyulu, N.Ch.P., ZARlbl47, 9 (1967). Uebler, E. A,, Ph.D. thesis, University of Delaware, Newark, Del., 1966. GIUSEPPE MARRUCCI‘
R. E. MURCH University of Delaware Newark, Del.
Present address, Istituto di Principi d’Ingegneria Chimica, Universita di Napoli, Kaples, Italy RECEIVED for review June 23, 1969 ACCEPTEDApril 16, 1970
Pressure Drop in Packed Beds of Spheres Different pressure drop equations applicable over limited ranges of the Reynolds number are well repRe/(l - E) 60,000. This implies that the resented b y a single nonlinear equation in the range 300 viscous and kinetic parameters in the Ergun equation are not true constants.
