Ian Sobey

On flow through furrowed channels. Part 1. Calculated flow patterns

Bellhouse et al. (1973) have developed a high-efficiency membrane oxygenator which utilizes pulsatile flow through furrowed channels to achieve high mass transfer rates. We present numerical solutions of the time-dependent two-dimensional Navier–Stokes equations in order to show the structure of the flow. Experimental observations which support this work are presented in a companion paper (Stephanoff, Sobey & Bellhouse 1980). Steady flow through a furrowed channel will separate provided the Reynolds number is sufficiently large. The effect of varying the Reynolds number and the geometric parameters is given and comparisons with solutions calculated using the modern boundary-layer theory of Smith (1976) show excellent agreement. Unsteady flow solutions are given as the physical and geometric parameters are varied. The structure of the flow patterns leads to an explanation of the high efficiency of the devices of Bellhouse.

BIFURCATIONS OF TWO-DIMENSIONAL CHANNEL FLOWS

In this paper we study some instabilities and bifurcations of two-dimensional channel flows. We use analytical, numerical and experimental methods. We start by recapitulating some basic results in linear and nonlinear stability and drawing a connection with bifurcation theory. We then examine Jeffery–Hamel flows and discover new results about the stability of such flows. Next we consider two-dimensional indented channels and their symmetric and asymmetric flows. We demonstrate that the unique symmetric flow which exists at small Reynolds number is not stable at larger Reynolds number, there being a pitchfork bifurcation so that two stable asymmetric steady flows occur. At larger Reynolds number we find as many as eight asymmetric stable steady solutions, and infer the existence of another seven unstable solutions. When the Reynolds number is sufficiently large we find time-periodic solutions and deduce the existence of a Hopf bifurcation. These results show a rich and unexpected structure to solutions of the Navier–Stokes equations at Reynolds numbers of less than a few hundred.

Observation of waves during oscillatory channel flow

We have observed steady and oscillatory flow through a two-dimensional channel expansion. The experimental results are supported by numerical solutions of the unsteady Navier–Stokes equations. This work was prompted by the recent discovery of vortex waves during steady flow past a moving indentation in a channel wall. Our work deals with both asymmetric channels, in which we show that vortex waves are observed during oscillatory flow with rigid walls, and with symmetric channels, in which a vortex street is observed. We believe that the vortex street is not a vortex wave, but the result of a shear-layer instability.

On flow through furrowed channels. Part 2. Observed flow patterns

Observations of flow in furrowed channels support the calculations of part 1 (Sobey 1980). If the mainstream flow is steady there is a critical Reynolds number below which separation does not occur. Above that Reynolds number vortices form and fill the furrow. When the mainstream is oscillatory, the flow may separate during the acceleration to form strong vortices. During the deceleration the vortices grow to fill the furrow and channel. As the mainstream reverses the vortices are ejected from the furrows as the fluid flows between the wall and the vortex. Photographs show that this pattern occurs for sinusoidally varying walls, furrows that are arcs of circles and rectangular hollows.

On the stability of Poiseuille flow of a Bingham fluid

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

A hydroelastic model of hydrocephalus

We combine elements of poroelasticity and of fluid mechanics to construct a mathematical model of the human brain and ventricular system. The model is used to study hydrocephalus, a pathological condition in which the normal flow of the cerebrospinal fluid is disturbed, causing the brain to become deformed. Our model extends recent work in this area by including flow through the aqueduct, by incorporating boundary conditions which we believe more accurately represent the anatomy of the brain and by including time dependence. This enables us to construct a quantitative model of the onset, development and treatment of this condition. We formulate and solve the governing equations and boundary conditions for this model and give results which are relevant to clinical observations.

Oscillatory flows at intermediate Strouhal number in asymmetric channels

We consider separated oscillatory flows in asymmetric channels when the Strouhal number is sufficiently great for the flow not to be quasi-steady, but where the flow is not dominated by viscous effects. In this range, we show that numerical calculations predict the expansion of vortices during a deceleration and we investigate these flows in several different geometries. These calculations are supported by experimental observations of the motion of small particles immersed in water.

Dispersion caused by separation during oscillatory flow through a furrowed channel

Abstract In this paper we use numerical solutions of the unsteady Navier-Stokes equations to study the occurrence of convective dispersion in the absence of diffusion. The calculations are for oscillatory flow in furrowed channels and illustrate how unsteady separation can augment dispersion. The calculations are carried out for symmetric and asymmetric geometries. The results shed new light on the vortex mixing cycle and demonstrate the powerful benefits to mixing which can be obtained when unsteady separation occurs.

Enhancement of plasma filtration using the concept of the vortex wave

Abstract We have investigated a novel microfiltration process for the separation of plasma from whole blood to act as the primary stage of a membrane-based affinity system. The combination of oscillatory flow, with a small mean flow component and flow deflectors have been employed to generate a vortex wave in a flat membrane channel. The vortex wave has been analysed under predominantly laminar flow conditions such that the design is appropriate for shear-sensitive fluids. The performance has been assessed in terms of long term plasma filtration rates, the level of blood damage and total protein recovery. The vortex wave produces an enhancement factor of 3.5 relative to a flat unobstructed channel at Re =123. Membranes of 0.2 μm and 0.45 μm pore sizes gave very similar trends in terms of plasma flux, haemolysis rate and total protein sieving coefficients. Detailed explanations are given for the influence of Reynolds number, Strouhal number, deflector spacing and mean flow rate on plasma flux. At plasma filtration rates as high as J =0.1 cm/min blood damage was found to be negligible.

Analytic solution during an infusion test of the linear unsteady poroelastic equations in a spherically symmetric model of the brain

This work determines the spatial and temporal distribution of cerebrospinal fluid (CSF) pressure and brain displacement during an infusion test in a spherically symmetric model of the brain. The response of CSF pressure and parenchymal displacement to blood pressure pulsations is determined in the solution. We use a spherically symmetric, three-component poroelastic model of the brain, differentiating between the solid elastic matrix, the CSF and the arterial blood compartments. The governing equations are linearized with quasi-constant poroelastic parameters. The solution does reproduce the average intracranial pressure increase during the test as well as the rise in CSF pressure pulsation amplitude due to transmission of blood pressure oscillations. In addition, the CSF flux into and out of the parenchyma is shown over time.