Session E28: Viscous Flows I: Flow Past Interferences

Oscillatory Stokes Flow Past a Slip Cylinder

Two-dimensional transient slow viscous flow past a circular cylinder with Navier slip boundary conditions is considered in the limit of low-Reynolds number. The oscillatory Stokes flow problem around a cylinder is solved using the stream function method leading to an analytic solution in terms of modified Bessel functions of the second kind. The corresponding steady-state behavior yields the familiar paradoxical result first detected by Stokes. It is noted that the two {\it key} parameters, viz., the frequency $\lambda$, and the slip coefficient $\xi$ have a significant impact on the flow field in the vicinity of the cylinder contour. In the limit of very low frequency, the flow is dominated by a term containing a well-known biharmonic function found by Stokes that has a singular behavior at infinity. Local streamlines for small times show interesting flow patterns. Attached eddies due to flow separation - observed in the no-slip case - either get detached or pushed away from the cylinder surface as $\xi$ is varied. Computed asymptotic results predict that the flow exhibits inviscid behavior far away from the cylinder in the frequency range $0 < \lambda \ll 1$. Although the frequency of oscillations is finite, our exact solutions reveal fairly rapid transitions in the flow domain.   

Viscous power-law flow past a finite flat plate

Viscous flow past a finite flat plate is studied numerically, using a high order implicit finite difference scheme. The plate moves in direction normal to itself with velocity $V_{\infty}=t^p$. We present the dependence of the vorticity evolution, streamlines and streaklines on $p \in [0, 2]$ and on Reynolds number $Re \in [250, 2000]$, and compare with experimental results of Pullin \& Perry (1980). We observe that, unlike in the p=0 case, for $p\ne 0$ the vortex core position oscillates as it moves away from the plate.   

Experimental and modeling study of global circulation by bent rod precession in low Reynolds number flows

The precessing motion of a bent rod over a plane in viscous dominated regimes can generate global fluid flow structures in the form of recirculating tori. Such motion can play an important role in the development of multicellular organisms, where primary cilia are the main agent for the embryonic forms of nutrient circulation. Results from an experimental investigation using PIV techniques to analyze the flow field will be presented and compared with a first principle theory based on slender body approximations. While good qualitative agreement can be achieved with Blake images enforcing the no-slip condition at the plane, quantitative agreement requires a more sophisticated approach, which will be outlined.   

The Oscillatory Motion of a Sphere in a Stokes Flow

We report results of an experimental investigation into the dynamic response of a single sphere to magnetic forcing and the resultant motion of the surrounding viscous fluid. Permanent magnets embedded into the surface of a neutrally buoyant sphere enable actuation of torsional oscillations of the sphere through the application of an alternating magnetic field. The applied field induces a torque on the embedded magnets, and the torsional response of the sphere to magnetic forcing has been systematically characterized as a function of the dimensionless forcing parameter F$=$8$\pi \mu $ a$^{\mathrm{3}}\omega $. Excellent agreement is found between the experimentally observed and numerically computed behavior of the sphere. Furthermore, the flow generated by the rotary motion of a sphere has visualized using Particle Image Velocimetry and good agreement is also found between the observed and the analytic solution for the fluid velocity as a function of radial distance.