In comparison to the flow in a rigid channel, there is a multifold reduction in the transition Reynolds number for the flow in a microchannel when one of the walls is made sufficiently soft, due to a dynamical instability induced by the fluid–wall coupling, as shown by Verma & Kumaran (J. Fluid Mech., vol. 727, 2013, pp. 407–455). The flow after transition is characterised using particle image velocimetry in the $x{-}y$ plane, where $x$ is the streamwise direction and $y$ is the cross-stream coordinate along the small dimension of the channel of height 0.2–0.3 mm. The flow after transition is characterised by a mean velocity profile that is flatter at the centre and steeper at the walls in comparison to that for a laminar flow. The root mean square of the streamwise fluctuating velocity shows a characteristic sharp increase away from the wall and a maximum close to the wall, as observed in turbulent flows in rigid-walled channels. However, the profile is asymmetric, with a significantly higher maximum close to the soft wall in comparison to that close to the hard wall, and the Reynolds stress is found to be non-zero at the soft wall, indicating that there is a stress exerted by fluid velocity fluctuations on the wall. The maximum of the root mean square of the velocity fluctuations and the Reynolds stress (divided by the fluid density) in the soft-walled microchannel for Reynolds numbers in the range 250–400, when scaled by suitable powers of the maximum velocity, are comparable to those in a rigid channel at Reynolds numbers in the range 5000–20 000. The near-wall velocity profile shows no evidence of a viscous sublayer for $(yv_{\ast }/{\it\nu})$ as low as two, but there is a logarithmic layer for $(yv_{\ast }/{\it\nu})$ up to approximately 30, where the von Karman constants are very different from those for a rigid-walled channel. Here, $v_{\ast }$ is the friction velocity, ${\it\nu}$ is the kinematic viscosity and $y$ is the distance from the soft surface. The surface of the soft wall in contact with the fluid is marked with dye spots to monitor the deformation and motion along the fluid–wall interface. Low-frequency oscillations in the displacement of the surface are observed after transition in both the streamwise and spanwise directions, indicating that the velocity fluctuations are dynamically coupled to motion in the solid.
Fig 5. Velocity along the drill-string. A. Maximum velocity. B. Root mean square velocity v1 (protocols.io.ke6cthe)
