Inertial swimming in an Oldroyd-B fluid

The effects of fluid inertia on a self-propelling inextensible waving sheet in an Oldroyd-B fluid are examined. The swimming velocity of the sheet is calculated in the limit in which the amplitude of the waves propagating along the sheet is small relative to the wavelength of the waves. The rate of work done by the sheet is also calculated. It is found that the swimming speed decreases monotonically approaching a limiting value with increasing Reynolds number (R) for a Newtonian fluid. For an Oldroyd-B fluid, the swimming speed increases to a maximum and then decreases asymptotically to a limiting value with increasing R. In contrast, it increases monotonically to a limiting value with increasing R for a Maxwell fluid. The limiting value is highest for the Maxwell fluid and lowest for the Oldroyd-B fluid. The corresponding value for the Newtonian fluid lies in between. The rate of work done by the sheet increases with increasing Reynolds number for all Deborah numbers. However, the energy consumed at a fixed swimming speed is lesser for an Oldroyd-B fluid than that of a Newtonian fluid. These results suggest that contrary to the Newtonian case, the fluid inertia supports the swimming sheet motion in a complex fluid. At a particular Deborah number, the oscillation frequency of the sheet could be adjusted to achieve the maximum speed. Similarly, at a particular frequency of oscillation, the Deborah numbers could be adjusted to achieve the maximum speed. These observations are in sharp contrast with the previous results reported for Newtonian and second-order fluids.