The effect of a hole on aerodynamic forces acting on a rectangular plate in Stokes flow is shown. The effect does not depend strongly on the area or location of the hole, because the hydrodynamic forces near the hole are increased and because the effect of the hole on the hydrodynamic forces acting on the plate is decreased. This result can be used to decrease the mass of a cantilever in a micro-flow sensor without affecting its characteristics. When this micro-flow sensor is attached to a flapping wing, the decrease in sensor mass is important because inertia acting on the sensor causes deformation of the flapping wing. This result is important in understanding the flight of insects with bristled wings.Key Words: Viscous Flows, Stokes Flow, Micro Flow Sensor Micro-flow sensors composed of a cantilever have been developed.1,2) Theoretical analysis to estimate hydrodynamic forces acting on a cantilever with thickness, (a rectangular solid) have clarified 3) the characteristics of such micro-flow sensors. At low Reynolds numbers (Re % 1), the difference between aerodynamic forces acting on a bristled wing and on a membranous wing with the same outline is small. 4) If aerodynamic forces acting on a cantilever in a micro-flow sensor are independent of hole size, we can decrease the mass of a micro-flow sensor without affecting its characteristics. When this sensor is attached to a flapping wing, 2) the decrease in sensor mass is important because inertia acting on the sensor causes deformation of the flapping wing. This paper shows how a hole in a plate without thickness affects the aerodynamic forces acting on the plate in Stokes flow. Moreover, the results clarify our understanding of the flight of an insect with bristled wings. 5,6)
Numerical MethodAs shown in Fig. 1, the rectangular plateThe centers of the plate and hole are located at (0,0) and (X h ,Y h ), respectively. The plate is considered in uniform flows U X , U Y , or U Z . Note that the hydrodynamic forces in the X, Y, and Z directions acting on an object in Stokes flow are determined by the uniform flow velocity in each direction. The Reynolds number is definedThe Stokes flow around a rectangular plate with a hole in uniform flow was investigated by boundary-element analysis. The details of the analytical method are explained elsewhere. 7,8) Here, the method is explained simply. The velocity u i ðxÞ at point x on the plate is satisfied by the following equation:where,H ij ðx; yÞ ¼ À 3 4%R 5 r i r j ðr i n i Þ;y, À , p, ij , and n are a point on the plate, the plate surface area, the pressure, the Kronecker delta, and a unit outward normal on the plate surface, respectively. From the boundary condition, when the total flow velocity is 0 on the plate surface, u i ðxÞ is given by U X , U Y , and U Z . The plate surface is divided into many rectangular elements, and solving Eq. (1) gives the values of p and @u i @x 3 (i ¼ 1, 2) for each element. Moreover, the aerodynamic forces acting on the plate are calculated from the values of p and2) on each element....