The study of run-up flows is gaining importance due to its wide applications in different technologies. This phenomenon arises in petrochemical engineering, lubrication technology, irrigation systems, water supply and bio-fluid mechanics where the pressure gradient is suddenly withdrawn from the steady state flow which henceforth gains unsteadiness due to extraneous influence. In some technological problems, the fluid flow experiences phenomenon viz. run-up which arises due to sudden withdrawal of the pressure gradient causing the flow while its boundaries instantaneously move from rest. Under this phenomenon, steady flow in the unperturbed state gains unsteadiness later. Many research workers have paid attention to words the study of non-Newtonian fluids. In lubrication theory and in many physical situations where we come across slip flows, there arises a class of problems referred to as "run up and spin up flows". The growing importance of the use of non-Newtonian fluids in modern technology and industries has led various researchers to attempt diverse flow problems related to several non-Newtonian fluids.