The unsteady viscous flow and heat transfer in the vicinity of an unaxisymmetric stagnation-point flow of an infinite moving cylinder with time-dependent axial velocity and non-uniform normal transpiration are investigated. The impinging free stream is steady with a strain- rate . A reduction of the Navier-Stokes and energy equations is obtained by use of appropriate similarity transformations. The semi-similar solution of the Navier-Stokes equations and energy equation has been obtained numerically using an implicit finite-difference scheme when the axial velocity of the cylinder and its wall temperature or its wall heat flux varies as specified time-dependent functions. In particular, the cylinder may move with different velocity patterns. 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 for different values of Prandtl number and for selected values of transpiration rate function, where a is cylinder radius and υ is kinematic viscosity of the fluid. Dimensionless shear stresses corresponding to all the cases increase with the increase of Reynolds number and transpiration rate function. 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. The heat transfer coefficient increases with the increases transpiration rate function, Reynolds number and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration rate function. It is shown that a cylinder with certain type of exponential wall temperature exposed to a temperature difference has not heat transfer.

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