Summary
This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters τ1 and τ2. Maximum values ξ1 and ξ2 of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=τ1+τ2)]. The velocities at ξ1 and ξ2 as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.
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Abbreviations
- γ, θ, z :
-
coordinates in a cylindrical polar system
- z 0 :
-
distance between rotor and stator (gap length)
- ξ :
-
=γ/z 0, dimensionless radial distance
- ζ :
-
=z/z 0, dimensionless axial distance
- ξ s :
-
=γ s/z0, dimensionless disc radius
- V :
-
=(u, v, w), velocity vector
- Ū\(\bar V,\bar W\) :
-
dimensionless velocity components
- ω :
-
uniform angular velocity of the rotor
- ρ, p :
-
fluid density and pressure
- P :
-
=p/(ρΩ2 z 202 , dimensionless pressure
- ν 1, ν2, ν3 :
-
kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively
- τ 1, τ2 :
-
ν 2/z 20 , resp. ν 3/z 20 , dimensionless parameters representing the ratio of second-order and inertial effects
- m :
-
=\(2\pi \rho \int\limits_0^{z_0 } {ru{\text{ d}}z} \), mass rate of symmetrical radial outflow
- l :
-
a number associated with induced circulatory flow
- Rm :
-
=m/(πρz 0ν1), Reynolds number of radial outflow
- R l :
-
=l/(πρz 0ν1), Reynolds number of induced circulatory flow
- Rz :
-
=ωz 20 /ν1, Reynolds number based on the gap
- ξ 1, ξ 2 :
-
maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively
- ξ 1(T), ξ 2(T):
-
ξ 1 and ξ 2 for different T
- U (+)ξ1(T) :
-
=\(\bar U\sqrt {Rz/Rm} \) dimensionless radial velocity, Rm>0
- V (+)ξ1(T) :
-
=\(\bar V\sqrt {Rz/Rm} \), dimensionless transverse velocity, Rm>0
- U (−)ξ2(T) :
-
=\(\bar U\sqrt {Rz/Rn} \), dimensionless radial velocity, Rm=−Rn<0, m=−n
- V (−)ξ2(T) :
-
=\(\bar V\sqrt {Rz/Rn} \), dimensionless transverse velocity, Rm<0
- C m :
-
moment coefficient
References
Soo, S. L., Trans. A.S.M.E., Journal of Basic Engineering 80 (2) (1958) 287.
Sharma, S. K., Proc. of 8th Congress on Theoretical and Applied Mechanics, India, 1963.
Sharma, S. K. and R. K. Gupta, Proc. of 9th Congress on Theoretical and Applied Mechanics, India, 1964.
Coleman, B. D. and W. Noll, Arch. Rat'l. Mech. Anal. 6 (1960) 355.
Noll, W., Ibid 2 (1958) 197.
Markowitz, H., Refer to Srivastava, A. C., J. Fluid Mech. 17 (1963) 171.
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Sharma, S.K., Sharma, H.G. Flow of second-order fluids over an enclosed rotating disc. Appl. sci. Res. 15, 272–288 (1966). https://doi.org/10.1007/BF00411563
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DOI: https://doi.org/10.1007/BF00411563