Slow Rotation of Concentric Spheres with Source at Their Centre in a Viscous Fluid

Abstract

The problem of concentric pervious spheres carrying a fluid source at their centre and rotating slowly with different uniform angular velocities ${\Omega }_1$, ${\Omega }_2$ about a diameter has been studied. The analysis reveals that only azimuthal component of velocity exists, and the couple, rate of dissipated energy is found analytically in the present situation. The expression of couple on inner sphere rotating slowly with uniform angular velocity ${\Omega }_1$, while outer sphere also rotates slowly with uniform angular velocity ${\Omega }_2$, is evaluated. The special cases, like (i) inner sphere is fixed (i.e., ${\Omega }_1=0$), while outer sphere rotates with uniform angular velocity ${\Omega }_2$, (ii) outer sphere is fixed (i.e., ${\Omega }_2=0$), while inner sphere rotates with uniform angular velocity ${\Omega }_1$, and (iii) inner sphere rotates with uniform angular velocity ${\Omega }_1$, while outer sphere rotates at infinity with angular velocity ${\Omega }_2$, have been deduced.