Abstract
The object of this paper is to present accurate numerical data concerning the creeping flow in curved annular channels with rectangular cross sections of which the outer wall is rotating with constant angular velocity. Dimensionless expressions for velocity profiles, flow rates and friction factors are obtained analytically for both the “drag” and “pressure” flow contributions. Numerical data were obtained on a digital computer and are presented in tabular and graphical form. The results of the theoretical analysis are also expressed in terms of the flow rate correction factors widely used in calculating the pumping efficiency of screw-pumps, agitators and extruders. This enables to estimate the effect of flight curvature on the pumping capacity.
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Abbreviations
- A(n):
-
constant, defined in (28)
- A 1(n):
-
constant, defined in (37)
- A 2(n):
-
constant, defined in (42)
- B(n):
-
constant, defined in (29)
- B 1(n):
-
constant, defined in (38)
- B 2(n):
-
constant, defined in (43)
- C(n):
-
constant, defined in (30)
- D H :
-
equivalent diameter
- f :
-
Fanning friction factor
- F(n):
-
constant, defined in (37)
- F d :
-
flow rate correction factor for drag flow
- F p :
-
flow rate correction factor for pressure flow
- G :
-
geometric shape function defined in (46)
- H :
-
channel height
- I 0, I 1 :
-
modified Bessel functions, first kind, zeroth and first order
- K 0, K 1 :
-
modified Bessel functions, second kind, zeroth and first order
- p :
-
hydraulic pressure
- p⋆:
-
dimensionless pressure defined in (11)
- p⋆⋆:
-
pressure defined in (44)
- r :
-
radial coordinate
- R :
-
outer radius of the channel
- Re:
-
Reynolds number
- s :
-
arc length measured along the outer wall of the channel
- t :
-
time
- u :
-
velocity vector
- u θ :
-
velocity component in the θ direction
- 〈u〉:
-
volumetric mean velocity
- U :
-
peripheral velocity of the barrel
- W :
-
channel width
- \(\dot V\) :
-
volumetric flow rate
- z :
-
axial coordinate
- x :
-
argument in (54), (55)
- κ :
-
ratio of the inner to outer radii
- λ :
-
friction factor defined in (49)
- л :
-
dynamic viscosity
- η :
-
dimensionless radial variable in (8)
- ø :
-
dimensionless velocity
- \(\bar \phi _s\) :
-
Fourier sine transform of ø
- θ :
-
azimuthal coordinate
- ϕ :
-
shape factor defined in (22)
- ϕ⋆:
-
shape factor defined in (12)
- ρ :
-
density of the fluid in (1), (2), (49) and (50) otherwise dimensionless radial variable defined in (21)
- ξ :
-
dimensionless axial variable (9)
- \(\dot \omega\) :
-
angular velocity
- ∇ :
-
nabla operator
- p.f.:
-
pressure flow
- d.f.:
-
drag flow
References
Cheng, K. C. and M. Akiyama, Int. J. Heat Mass Transfer 13 (1970) 471.
Bird, R. B., W. E. Stewart and E. N. Lightfoot, Transport Phenomena, J. Wiley New York 1960.
Tranter, C. J., Integral Transforms in Mathematical Physics, J. Wiley New York 1950.
Mc Lachlan, N. W., Bessel Functions for Engineers, Clarendon Press Oxford 1955.
Rieger, F., Thesis, Faculty of Mechanical Engineering, Czech Technical University, Prague 1970.
Majkapar, G. I., I. A. N. USSR, Mechanics, Mechanical Engineering 2 (1964) 175.
Bernhardt, E. C., Editor, Processing of Thermoplastic Materials, Reinhold New York 1959.
Booy, L. M., Thesis, Technische Hogeschool te Delft 1964.
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Rieger, F., Šesták, J. Creeping flow of Newtonian fluids in curved rectangular channels. Appl. Sci. Res. 28, 89–106 (1973). https://doi.org/10.1007/BF00413059
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DOI: https://doi.org/10.1007/BF00413059