Kelvin-Helmhotz Instability for Parallel Flow in Porous Media: A Linear Theory

Two fluid layers in fully-saturated porous media are considered. The lighter fluid is above the heavier one so that in the absence of motion the arrangement is stable and the interface is flat. It is shown that when the fluids are moving parallel to each other at different velocities, the interface may become unstable (the Kelvin-Helmholtz instability). The corresponding conditions for marginal stability are derived for Darcian and non-Darcian flows. In both cases, the velocities should exceed some critical values in order for the instability to manifest itself. In the case of Darcy\u27s flow, however, an additional condition, involving the fluids\u27 viscosity and density ratios, is required

Feedback Control Stabilization of the No-Motion State of a Fluid Confined in a Horizontal Porous Layer Heated From Below

We consider a horizontal three-dimensional saturated porous layer, confined in an upright cubic box, heated from below and cooled from above. In the absence of a controller, the fluid maintains a no-motion state for subcritical Rayleigh numbers R \u3c Rc, where Rc, depends on the box’s aspect ratio. Once this critical number is exceeded, fluid motion ensues. We demonstrate that, with the use of feedback control strategies which suppress flow instabilities, one can maintain a stable no-motion state for Rayleigh numbers far exceeding the classical critical one for the onset of convection. To preserve the equilibrium no-motion state of the classical problem, the controller alters the system’s dynamics so as to stabilize an otherwise non-stable state

Theoretical Investigation of Electroosmotic Flows and Chaotic Stirring in Rectangular Cavities

Two dimensional, time-independent and time-dependent electro-osmotic flows driven by a uniform electric field in a closed rectangular cavity with uniform and nonuniform zeta potential distributions along the cavity’s walls are investigated theoretically. First, we derive an expression for the one-dimensional velocity and pressure profiles for a flow in a slender cavity with uniform (albeit possibly different) zeta potentials at its top and bottom walls. Subsequently, using the method of superposition, we compute the flow in a finite length cavity whose upper and lower walls are subjected to non-uniform zeta potentials. Although the solutions are in the form of infinite series, with appropriate modifications, the series converge rapidly, allowing one to compute the flow fields accurately while maintaining only a few terms in the series. Finally, we demonstrate that by time-wise periodic modulation of the zeta potential, one can induce chaotic advection in the cavity. Such chaotic flows can be used to stir and mix fluids. Since devices operating on this principle do not require any moving parts, they may be particularly suitable for microfluidic devices

Numerical investigation of the stabilization of the no-motion state of a fluid layer heated from below and cooled from above

The feasibility of controlling flow patterns of Rayleigh–Bénard convection in a fluid layer confined in a circular cylinder heated from below and cooled from above (the Rayleigh-Bénard problem) is investigated numerically. It is demonstrated that, through the use of feedback control, it is possible to stabilize the no-motion (conductive) state, thereby postponing the transition from a no-motion state to cellular convection. The control system utilizes multiple sensors and actuators. The actuators consist of individually controlled heaters positioned on the bottom surface of the cylinder. The sensors are installed at the fluid\u27s midheight. The sensors monitor the deviation of the fluid\u27s temperatures from preset desired values and direct the actuators to act in such a way so as to eliminate these deviations. The numerical predictions are critically compared with experimental observations

Complex magnetohydrodynamic low-Reynolds-number flows

The interaction between electric currents and a magnetic field is used to produce body (Lorentz) forces in electrolyte solutions. By appropriate patterning of the electrodes, one can conveniently control the direction and magnitude of the electric currents and induce spatially and temporally complicated flow patterns. This capability is useful, not only for fundamental flow studies, but also for inducing fluid flow and stirring in minute devices in which the incorporation of moving components may be difficult. This paper focuses on a theoretical and experimental study of magnetohydrodynamic flows in a conduit with a rectangular cross section. The conduit is equipped with individually controlled electrodes uniformly spaced at a pitch L. The electrodes are aligned transversely to the conduit\u27s axis. The entire device is subjected to a uniform magnetic field. The electrodes are divided into two groups A and C in such a way that there is an electrode of group C between any two electrodes of group A. We denote the various A and C electrodes with subscripts, i.e., Ai and Ci , where i = 0, ±1, ±2, ... . When positive and negative potentials are, respectively, applied to the even and odd numbered A electrodes, opposing electric currents are induced on the right and left hand sides of each A electrode. These currents generate transverse forces that drive cellular convection in the conduit. We refer to the resulting flow pattern as A. When electrodes of group C are activated, a similar flow pattern results, albeit shifted in space. We refer to this flow pattern as C. By alternating periodically between patterns A and C, one induces Lagrangian chaos. Such chaotic advection may be beneficial for stirring fluids, particularly in microfluidic devices. Since the flow patterns A and C are shifted in space, they also provide a mechanism for Lagrangian drift that allows net migration of passive tracers along the conduit\u27s length

Stabilization of the no-motion state in Rayleigh-Bénard convection through the use of feedback control

It is demonstrated theoretically that the critical Rayleigh number for transition from the no-motion (conduction) to the motion state in the Rayleigh-Bénard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of feedback control strategies effecting small perturbations in the boundary data

A mathematical model of lateral flow bioreactions applied to sandwich assays

Lateral flow (LF) bio-detectors facilitate low-cost, rapid identification of various analytes at the point of care. The LF cell consists of a porous membrane containing immobilized ligands at various locations. Through the action of capillary forces, samples and reporter particles are transported to the ligand sites. The LF membrane is then scanned or probed, and the concentration of reporter particles is measured. A mathematical model for sandwich assays is constructed and used to study the performance of the LF device under various operating conditions. The model provides insights into certain experimental observations including the reduction in the level of the detected signal at high target analyte concentrations. Furthermore, the model can be used to test rapidly and inexpensively various operating conditions, assist in the device\u27s design, and optimize the performance of the LF device

The kinematics of bend-induced mixing in micro-conduits

The absence of turbulence and the difficulty associated with introducing moving components into microfluidic systems make the mixing problem in microdevices challenging. We studied steady, laminar, incompressible flow through a sequence of conduits with rectangular cross-sections aligned to form 90o with each other. The feasibility of taking advantage of bend-induced vortices to stir the fluid and enhance the mixing process was evaluated theoretically and experimentally. Since at very low Reynolds numbers the bend-induced vortices decay rapidly, it was necessary to utilize a large number of bends to achieve the desired effects. Since it is not practical to directly simulate the flow through a large number of bends, we borrowed Jones et. al.\u27s (1989) idea of constructing a two-dimensional map to project fluid particles from a cross-section upstream of the bend to a cross-section downstream of the bend. This map was then applied repetitively to trace particle trajectories in various bend arrangements. Under certain conditions, chaotic advection was predicted. A prototype of a stirrer was fabricated with low temperature co-fired ceramic tapes