Infinite swimming sheet of amplitude b beneath a membrane at average height H, which separates two fluids with viscosities . The membrane has surface tension γ and bending rigidity B. The coordinates (x S, y S) and (x M, y M) describe the swimmer and interface, respectively.
Amplitudes and phases of the transverse (left panel) and longitudinal (middle panel) waves of the membrane. The insets for the longitudinal waves show the maximum values. In the right panel, membrane shapes are shown for various heights, with b = 0.3, and the shape of the swimmer (gray dashed line) is superimposed. The directions of the swimmer wave speed, swimmer swimming speed, and membrane swimming speed are also indicated with arrows. Here Ca → ∞ and μ r = 1/2. The color scheme is a function of the Machin number Ma. The dashed vertical red lines (see vertical arrows) indicate the height, H ≈ 0.91, at which the membrane reverses its direction. The other vertical lines in the left and middle panels correspond to (i) H = 1/2, (ii) H = 1, (iii) H = 2, and (iv) H = 3.
Speeds normalized by the Taylor result in the laboratory frame, where V Taylor = cb 2 k 2/2. These results are in the limit where Ca → ∞, for three different values of viscosity ratio ((a) μ r = 1/2, (b) μ r = 1, and (c) μ r = 3/2) and the range of Ma ∈ [0, 50]. The insets in the bottom row show the behavior of as a function of H. The vertical dashed lines (see vertical arrows) in the bottom row and in the insets indicate the separation H at which the membrane speed reverses.
Speeds normalized by the Taylor result in the laboratory frame, where V Taylor = cb 2 k 2/2. These results are in the limit where Ma → 0 and Ca → ∞. In (a) μ r → 0 and (b) μ r ≈ 0.215.
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