On the flow past ellipses in a Hele-Shaw cell

Abstract

In this work we investigate the effect of vertical confinement and inertia on the flow past thin ellipses in a Hele-Shaw cell (with centre line velocity $U_c$ and height 2 $h$ ) with different aspect ratios for symmetrical flows and at an angle of attack, using asymptotic methods and numerical simulations. A Stokes region is identified at the ellipse vertices which results in flow different to flow past bluff bodies. Comparison with asymptotic analysis indicates close agreement over the ‘flat’ portion of the ellipse, for $\delta =(b/a)=0.05$ , where $a$ and $b$ are the semi-major and -minor ellipse axes, respectively. Two flow conditions are investigated for ellipses at an angle of attack of 10 $^\circ$ for a fixed $\delta =0.05$ . Firstly, for $\varLambda =(U_ca/\nu )(h/a)^2 \ll 1$ , the effect of increasing the vertical confinement of the Hele-Shaw cell results in the rear stagnation point (RSP) moving from close to the potential-flow prediction when $\epsilon =h/a$ is very small to the two-dimensional Stokes-flow prediction when $\epsilon$ is large. Secondly, for a fixed $\epsilon \ll 1$ , when inertia is increased past $\varLambda ={O}(\epsilon )$ the RSP moves towards the trailing edge and is located there for $\varLambda ={O}(1)$ . Under these conditions an attached exponentially decaying shear layer or ‘viscous tail’ is formed. A modified Bernoulli equation of the depth-averaged flow, together with the Kutta–Joukowski theorem is used to predict the drag and lift coefficients on the ellipse, which include a linear and a nonlinear contribution, corresponding to a Hele-Shaw and circulation component, respectively. Close agreement is found up to $\varLambda ={O}(1)$ .