Unsteady flow in an annulus between two concentric rotating spheres
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Abstract
An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of inertia is introduced to account for the local inertia of the fluid.
Keywords
Torque Secondary Flow Unsteady Flow Outer Sphere Infinite MediumNomenclature
- Ri
radius of the inner sphere
- Ro
radius of the outer sphere
- \(\bar r\)
radial coordinate
- r
dimensionless radial coordinate,\(\bar r/R_i \)
- ϑ
meridional coordinate
- φ
azimuthal coordinate
- \(\bar t\)
time
- t
dimensionless time,\(\bar tv/R_i^2 \)
- Rei
instantaneous Reynolds number of the inner sphere,ω i R i /2 /ν
- Reo
instantaneous Reynolds number of the outer sphere,ω o R o /2 /ν
- \(\bar V_r \)
radial velocity component
- Vr
dimensionless radial velocity component,\(\bar V_r R_i /v\)
- \(\bar V_\theta \)
meridional velocity component
- Vθ
dimensionless meridional velocity component,\(\bar V_\theta R_i /v\)
- \(\bar V_\phi \)
azimuthal velocity component
- Vφ
dimensionless azimuthal velocity component,\(\bar V_\phi R_i /v\)
- \(\bar T\)
viscous torque
- T
dimensionless viscous torque,\(3\bar T/8\pi \mu \nu R_i \)
- \(\bar T_i \)
viscous torque at surface of inner sphere
- Ti
dimensionless viscous torque at surface of inner sphere,\(3\bar T_i /8\pi \mu \nu R_i \)
- \(\bar T_o \)
viscous torque at surface of outer sphere
- To
dimensionless viscous torque at surface of outer sphere,\(3\bar T_o /8\pi \mu \nu R_i \)
- \(\bar T_{p,i} \)
externally applied torque on inner sphere
- Tp,i
dimensionless applied torque on inner sphere,\(3\bar T_{p,i} /8\pi \mu \nu R_i \)
- \(\bar Z_i \)
moment of inertia of inner sphere
- Zi
dimensionless moment of inertia of inner sphere,\(3\bar Z_i /8\pi \rho R_i^5 \)
- \(\bar Z_{i,v} \)
virtual moment of inertia of inner sphere
- Zi,v
dimensionless virtual moment of inertia of inner sphere,\(3\bar Z_{i,v} /8\pi \rho R_i^5 \)
- \(\bar Z_{o,v} \)
virtual moment of inertia of outer sphere
- ωi
instantaneous angular velocity of the inner sphere
- ωo
instantaneous angular velocity of the outer sphere
- ρ
density of fluid
- μ
viscosity of fluid
- ν
kinematic viscosity of fluid,μ/ρ
- λ
radius ratio,R i /R o
- \(\bar \Omega \)
swirl function,\(\bar V_\phi \bar r\sin \theta \)
- Ω
dimensionless swirl function,\(\bar \Omega /v\)
- \(\bar \psi \)
stream function
- ψ
dimensionless stream function,\(\bar \psi /R_i v\)
- γi
acceleration parameter for the inner sphere,\((R_i^4 /\nu ^2 )d\omega _i /d\bar t = d/dt(Re_i )\)
- γo
acceleration parameter for the outer sphere,\((R_o^4 /\nu ^2 )d\omega _o /d\bar t\)
- \(\bar \tau _{r\phi } \)
shear stress
- τrφ
dimensionless shear stress,\(\bar \tau _{r\phi } R_i^2 /\mu \nu \)
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