Plasma motions in narrow capillary flow

Plasma motions in the gaps between successive red cells in narrow-capillary blood flow are obtained in an idealized model, using a series of eigenfunctions to represent the disturbance to a basic Poiseuille flow created by the cells. The flow is matched, in the narrow entry and exit regions, to the lubrication flow in the constricted zone around the red cell (Fitz-Gerald 1969). Basically, the circulating toroidal motion predicted by Prothero & Burton (1961) is obtained in a reference frame in which the cells are considered stationary. Small secondary circulations are also found near the axis and close to the red cells, whose intensity is controlled by the amount of leakback past the cells. Zones of high shear are found along the capillary wall and in some cases on part of the red-cell face; implications of this for mass transport are discussed (see §4). Because of the unusual behaviour of the slowest-decaying dominant eigenfunction circulation and wall shear increase as the cell spacing decreases, contrary to expectation, until the spacing becomes very small indeed

A note on a problem of Apollonius

Degenerate cases of the problem of Apollonius, to construct a circle tangent to each of three given circles, are discussed and exhaustively classified for proper circles (finite and non-zero radius). Singular cases are considered, and an outline of the extension of the problem to higher dimensions given. Amusing alternative interpretations of the results are obtained

Non-Newtonian secretion flow in tubes

A model is developed for the phenomenon of non-Newtonian secretion in tubes. The motivation for this study is the problem of glandular secretion, particularly in the pancreas. Power-law fluids are considered in some detail, as being biologically appropriate. It is found that for a power-law fluid whose exponent is less than unity (the biological case) two types of flow occur. For a sufficiently high secretion pressure, all of the tube is used for secretion, and a nonlinear pressure profile results. Numerical solutions are obtained for the pressure and rate of efflux. When the secretion pressure parameter falls below a certain critical value, the upper end of the tube begins to be choked off, only part of the tube being used for secretion. This phenomenon does not occur for exponents greater than or equal to unity. Physiological implications are considered, and a qualitative discussion given for the case of non-power-law fluids

Flow mechanics of red cell trains in very narrow capillaries. I. Trains of uniform cells

From a brief historical survey we extract the basic requirements considered essential for any satisfactory capillary flow model. Within this framework we construct a model incorporating a revised version of the lubricated-cell model matched with improved bolus flow calculations. Pressure-flow relations are discussed for uniform trains of cells and plasma in very narrow capillaries (diameter less than 7.5 μm). Resistances obtained compare favourably with recently published experimental results

Transport of O2 along a model pathway through the respiratory region of the lung

A pathway through the system of branching in the respiratory region of the lung is modelled by a circular cylinder, closed at one end, with partitions which define the component respiratory units. In this model the transport of O2 during inspiration, generated by diffusion is compared with that produced by diffusion together with convection and the importance of convection in the respiratory region in promoting oxygen uptake at the alveolar wall is discussed. For this discussion it is only necessary to consider inspiration. The equations are solved numerically for flow rates of 10, 85 and 200 liters/min. O2 uptake at the wall and curves of constant O2 concentration are shown to illustrate the influence of convection. It is found that after a 2 sec inspiration from an O2 tension of 98 mm Hg and a lung volume of 2300 ml, convection is about 12 per cent as important as diffusion at a flow rate of 85 liters/min, whereas at 10 liters/min convection is only about 0.4 per cent as important as diffusion

Flow patterns in models of small airway units of the lung

Quasi-steady creeping flow in models of small airway units of the lung is investigated. A respiratory unit of the lung is modelled by a sphere, an oblate and a prolate ellipsoid of revolution, and a circular cylinder of finite length. The solution of the Stokes equations for each of these geometries is indicated for general axi-symmetric boundary conditions. For particular cases consistent with the models, streamlines are plotted and some velocity profiles are shown. It is suggested that bulk 00w in the ha1 generations of the lung is significant for gas transport even though diffusion is the predominant mechanism there

Manufacturing sequences for the Economic Lot Scheduling problem

In the basic Economic Lot Scheduling problem, a production schedule is required to manufacture sequentially a number of products on a single machine, with the schedule chosen to minimize set-up and inventory costs. The products suffer continuous demand, and no shortfall is allowed. A recent approach involves repetitions of a production cycle (such as ABCBC for three products A, B and C, with manufacturing times chosen to prevent shortage occurring); an exhaustive search is performed over a large set of possible cycles to discover the optimal schedule. This paper discusses the question “How many such sycles need to be examined?”, Since the answer is very relevant to practical application of the method. The case of three products is considered. Complete information is obtained for cycles up to length 12 (that is, 12 production switch overs), and partial results for longer ones. An estimate, apparently reasonable, is obtained for cycles of any length. The major trend to emerge is that surprisingly few cycles are involved

Mathematical theory of electrochemical machining 2. Anodic shaping

Further solutions to the potential equation with moving boundary conditions, applicable to the electrochemical machining process, are presented. A theory is proposed for the machining of an arbitrary distribution of irregularities on to an anode for the cases where the cathode profile is specified and the resultant anode shape must be found, and where a cathode profile must be designed to give a required anode shape. Limitations on the application of this theory are discussed. The effects of overpotential are also discussed; overpotential only at the cathode is shown to reduce the limiting amplitude which can be obtained on the anode, and to increase the machining time required to achieve it. Overpotential only at the anode has no effect on this limiting amplitude, but again increases the necessary machining time