Summary
Relations have been derived for the frictional resistance of finite discs and cones rotating in Ostwald-de Waele (power-law) type non-Newtonian fluids. The obtained equations can be formulated as dimensionless relations between the dimensionless moment coefficient and the generalized Reynolds number; the flow-behaviour index n enters the equations as a parameter. The relations derived for cones contain the apex angle 2α0 as an additional parameter in the form of A=sin α0. The validity of the theoretically derived relations has been verified by measurements of the torque of discs and cones for a number of pseudoplastic power-law fluids.
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Abbreviations
- A :
-
sin α 0 parameter
- b :
-
exponent in regression equation (16)
- C :
-
coefficient in regression equation (16)
- c Mi :
-
dimensionless moment coefficient, for bodies wetted on one side (i=1) and for completely wetted bodies (i=2), equations (8) and (9b)
- d :
-
diameter of turntable
- F, G :
-
velocity functions of exact solution, equation (4)
- K :
-
consistency coefficient of non-Newtonian fluids
- M Ki :
-
torque of rotating bodies, i=1 for bodies wetted on one side, i=2 for completely wetted bodies
- n :
-
flow-behaviour index of non-Newtonian fluids
- N=K/ρ :
-
kinematic consistency coefficient
- P :
-
tangential force
- r(y) :
-
perpendicular distance of point on cone surface from axis
- R :
-
radius of disc or of base of cone
- ℛ :
-
modified Reynolds number defined by equation (14)
- Re ow :
-
generalized Reynolds number defined by equation (10)
- S, S′ :
-
area
- u, v :
-
components of velocity vector
- x, y, z :
-
coordinates according to fig. 1
- α 0 :
-
half the apex angle of cone
- β :
-
coefficient of frictional resistance defined by equation (11)
- δ :
-
thickness of boundary layer
- ζ :
-
independent variable in exact solution, defined by equation (5)
- ρ :
-
density of fluid
- τ zx, τzy :
-
tangential stresses
- ω :
-
angular velocity of rotation
- T :
-
theoretical value
- E :
-
experimental value
- 0:
-
refers to surface of rotating body
References
Mitschka, P. and J. Ulbrecht, Coll. Czech. Chem. Comm. 30 (1965) 2511, comm. IV.
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Karmán, Th., Z. angew. Math. Mech. 1 (1921) 244.
Cochran, W. G., Proc. Cambr. Phil. Soc. 30 (1934) 365.
Theodorsen, Th. and A. Regier, NACA Report No 793 (1944).
Dorfman, L. A., Gidrodinamicheskoe soprotivlenie i teplootdacha vraschayuschikhsia tel, Moscow 1960.
Schlichting, H., Grenzschichttheorie, Karlsruhe 1958.
Tomita, Y., Bull. Jap. Soc. Mech. Eng. 4 (1961) 671.
Mitschka, P., Coll. Czech. Chem. Comm. 29 (1964) 2892, comm. II.
Standart, G., Coll. Czech. Chem. Comm. 23 (1958) 1163.
Mitschka, P., Theses, Czechoslovak Acad. of Sciences, Prague 1964.
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Mitschka, P., Ulbrecht, J. Non-Newtonian fluids v frictional resistance of discs and cones rotating in power-law non-Newtonian fluids. Appl. sci. Res. 15, 345–358 (1966). https://doi.org/10.1007/BF00411568
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DOI: https://doi.org/10.1007/BF00411568