Anti-Inertial Lift in Foams: A Signature of the Elasticity of Complex Fluids

To understand the mechanics of a complex fluid such as a foam we propose a model experiment (a bidimensional flow around an obstacle) for which an external sollicitation is applied, and a local response is measured, simultaneously. We observe that an asymmetric obstacle (cambered airfoil profile) experiences a downwards lift, opposite to the lift usually known (in a different context) in aerodynamics. Correlations of velocity, deformations, and pressure fields yield a clear explanation of this inverse lift, involving the elasticity of the foam. We argue that such an inverse lift is likely common to complex fluids with elasticity.

Figure 1

Top view of the airfoil with a small part of the flowing foam. The superimposed white line indicates its boundary. It is a Joukovski profile [17], which equation of shape writes $x(t)=\{1.56+[(0.168+\cos t{)}^2+(0.25+\sin t{)}^2{]}^{-1}\}(0.168+\cos t)$ and $y(t)=\{1.56-[(0.168+\cos t{)}^2+(0.25+\sin t{)}^2{]}^{-1}\}(0.25+\sin t)$, with lengths in centimeters, the angle $t$ ranging from $-\pi$ to $\pi$. Its length from leading to trailing edge is 5.2 cm. The arrows indicate the direction of drag and lift (flow from left to right), and the white dot marks the contact point between the airfoil and the fiber. The dotted lines show where the quantities displayed in Fig. 4 are evaluated. A movie is available online [29].

Figure 2

Forces exerted by the flowing foam on the airfoil versus flow rate: drag (□) and lift (●). Inset: leading angle (○) spontaneously selected by the airfoil versus flow rate ($\mathrm{ml}{\min }^{-1}$); the standard deviation of its time fluctuations are plotted as error bars.

Figure 3

Experimental measurements of velocity, area, and deformation fields around the airfoil across the whole channel (width: 10 cm). Each measurement results from an average over a representative volume element (square box of side 1.1 cm) and over time (750 successive images in steady regime). Arrows: velocity of bubble centers of mass. Background gray levels: bubble area, with 12% relative variation between the most (dark gray) and less (white) compressed bubbles; the mean area corresponds to the contour line between the two gray levels at the left side of the figure. Ellipses: texture tensor, the major axis representing the direction and magnitude of maximal bubble elongation (an isotropic region would be represented by a circle).

Figure 4

Experimental measurements of velocity, area, and deformation fields (a) 1 cm above and (b) 1 cm below the airfoil. Same data as Fig. 3. The streamwise component $v_x$ of the velocity (□), the bubble area (●) and the $yy$ component of the texture tensor, $M_{yy}$ ($△$) are adimensioned by their value at the channel entrance ($2.7\mathrm{mm}{\mathrm{s}}^{-1}$, $16{\mathrm{mm}}^2$, and $3.5{\mathrm{mm}}^2$, respectively), and we represent their variations relative to these values versus the streamwise coordinate $x$ relative to the leading edge. Vertical dots indicate the leading and trailing edges. The inversion below the airfoil occurs 1.8 cm after the leading edge.