A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows

Abstract

In this paper we describe the design and operating characteristics of a computercontrolled four-roll mill for investigations of particle and drop dynamics in twodimensional linear flows. The control system is based upon the use of: a video camera to visualize the instantaneous position of the drop or particle ; a PDP 11/23 computer, with a pipeline processor acting as an interface between the camera and computer, to calculate the position and implement a control strategy, and d.c. stepping motors to convert an electronic signal to angular velocities of the four rollers. The control objective is to keep the centre of mass of the drop/particle at the centre of the region between the rollers where there is a stagnation point in the undisturbed flow, while maintaining the shear-rate and the ratio of vorticity to strain rate in the flow at fixed values. The resulting system is suitable for studies of: the rotational motions of single solid particles; the deformation and burst of single droplets; or the hydrodynamic interactions of two particles or drops, one of which is held with its centre-of-mass fixed at the stagnation point of the undisturbed flow. In all cases, the flow can be varied from pure rotation to pure strain, and the shear rate can be either steady or changing as a prescribed function of time.

Accurate control of hyperbolic trajectories in any dimension

“…Thus, by choosing the velocity field in the form (5) subject to (10), (12), and (13), one can ensure that a hyperbolic trajectory of (2) follows the path x(t,ε), correct to O(ε 3 ). By continuing this process, the control velocity can be determined to as high an order of ε as required; we stop here for brevity.…”

“…A classical example from fluid mechanics is the concept of a fluidic device which achieves droplet manipulation through positioning at the hyperbolic trajectory location [11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In the paradigmatic two-dimensional device with a hyperbolic point with attached one-dimensional stable and unstable manifolds, droplet elongation in the unstable direction is achieved by this process.…”

Controlling the Unsteady Analogue of Saddle Stagnation Points

“…A drop positioned there will be deformed according to the stable and unstable manifolds, and will thus lengthen in the y-direction while shortening in the x-direction. By varying the roller speeds, it is possible to obtain a combination of such purely straining flow with rotational motion at the origin, providing a mechanism for different forms of drop deformation [6,31,52,49,26,23,47,22,43]. A major difficulty in such four-roll mills is maintaining a particle precisely at the point of interest, particularly under unsteady rotation protocols.…”

“…Their stable and unstable manifolds separate the flow regime into regions of fluid which do not mix. Analysis of saddle stagnation points is well established in fluid applications ranging from groundwater modeling [29,46], macro-and micromixing devices [17,2,6,50,42,21], and oceanographic flows [48,3,36,11,9,40]. The analogous entity in unsteady flows is that of a hyperbolic trajectory, a specific type of time-varying fluid parcel trajectory which possesses time-varying stable and unstable manifolds, whose locations govern fluid transport; cf.…”

An experimental study of transient effects in the breakup of viscous drops

“…The experiments reported in this paper were performed using the computercontrolled four-roll mill described in Bentley & Leal (1986a). Here, we discuss only those features of the device that are important to the present work.…”