On the energetic balance for the flow of an Oldroyd-B fluid due to a flat plate subject to a time-dependent shear stress

Exact and approximate expressions for the power due to the shear stress at the wall LL, the dissipation ΦΦ and the boundary layer thickness δδ are established for the unsteady flow of an Oldroyd-B fluid driven by the transverse motion of an infinite plate subject to a time-dependent shear stress. The change of the kinetic energy with time is also obtained from the energetic balance. Similar expressions for Newtonian, Maxwell and second-grade fluids are obtained as limiting cases of general results. Series solutions for the velocity and shear stress are also obtained for small values of the dimensionless relaxations and retardation times. Graphical illustrations corresponding to the exact expressions for LL, ΦΦ and δδ agree with the associated asymptotic approximations. Usually for many industrial applications the velocity of the wall is given and what is required is the energy that is necessary to keep the wall running with the prescribed value. The problem discussed by us now is that where, on the contrary, the wall shear stress is given but the velocity and the energy of the medium are required.