Transient swimming of an undulating sheet in a second-order fluid

Abstract

The motion of a wavy sheet with time-dependent frequency is discussed in an unbounded non-Newtonian fluid. The rheological behavior of non-Newtonian fluid is captured through the constitutive equation of a second-order fluid. The waves start propagating down the sheet surface with a frequency that achieves a steady-state as an arbitrary function of time. The equation governing the flow is derived under the low Reynolds number approximation. Regular perturbation expansion is employed to develop equations and boundary conditions for stream function at leading and second-order in sheet amplitude. These equations are then solved in Laplace domain to yield expressions of stream functions as arbitrary functions of the frequency of the sheet. Further analysis is carried out for two scenarios. In the first scenario, the sheet is not moving and its undulations produces a net flow. The average velocity of this flow in the horizontal direction is obtained in the Laplace domain. In the second scenario, the sheet is free to move. By employing a force balance at the sheet in the horizontal direction, the swimming velocity of the sheet is also obtained in the Laplace domain. Numerical inversion for some specific choices of sheet frequency is carried out in both scenarios and obtained results are discussed in detail. It is shown that well-behaved pumping and swimming velocities (which are free of jump discontinuity at the initial starting time) for the case in which sheet frequency evolves like a unit-step function are possible in a second-order fluid provided that the amplitudes of longitudinal and transverse waves propagating down the sheet surface satisfy a specific equation.