Axisymmetric Radial Stagnation-Point Flow of a Viscous Fluid on a Rotating Cylinder with Time-Dependent Angular Velocity

The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinitely long rotating circular cylinder is investigated, when the angular velocity varies arbitrarily with time. The free stream is steady and with a constant strain rate of k. An exact solution of the Navier-Stokes equations is derived in this problem. The general self-similar solution is obtained when the angular velocity of the cylinder varies as certain functions. The cylinder may perform different types of motion: It may rotate with constant speed, with exponentially increasing/decreasing angular velocity, with harmonically varying rotation speed or with accelerating/decelerating oscillatory angular speed. For completeness, some sample semi-similar solutions of the unsteady Navier-Stokes equations have been obtained numerically using a finite-difference scheme. These solutions are presented for special cases when the time-dependent is step-function, linear and non-linear, with respect to time. All the solutions above are presented for Reynolds numbers, Re=ka^2/2\upsilon, ranging from 0.1 to 1000, where a is the cylinder radius and \upsilon is the kinematic viscosity of the fluid. Shear stresses corresponding to all cases increase with the Reynolds number. The maximum value of the shear-stress increases with an increase in oscillation frequency and amplitude. An interesting result is obtained, in which a cylinder, spun up from rest with a certain angular velocity function and at a particular value of Reynolds number, is azimuthally stress-free.