Aixsymmetric Stagnation Point Flow of a Viscous Fluid on a Moving Cylinder with Time Dependent Axial Velocity

The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinite moving cylinder with time-dependent axial velocity is investigated. The impinging free stream is steady with a strain rate k. An exact solution of the Navier-Stokes equations is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations. The general self-similar solution is obtained when the axial velocity of the cylinder varies as specified time-dependent functions. In particular, the cylinder may move with different velocity patterns. For completeness, 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 axial velocity of the cylinder is a step-function, a ramp, and a non-linear function. All the solutions above are presented for Reynolds numbers, , ranging from 0.1 to 100 where a is cylinder radius and is kinematic viscosity of the fluid. Shear stresses corresponding to all the cases increase with the Reynolds number. The maximum value of the shear stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder moving with certain axial velocity function and at particular value of Reynolds number is axially stress-free.