G. Jayaraman

Casson fluid flow in a pipe filled with a homogeneous porous medium

Abstract The flow characteristics of a Casson fluid in a tube filled with a homogeneous porous medium is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. This analysis can model the pathological situation of blood flow when fatty plaques of cholesterol and artery-clogging blood clots are formed in the lumen of the coronary artery. Two cases of permeability of the porous medium are considered, namely (i) permeability has a constant value K o , and (ii) permeability varies in radial direction according to K ( r ) = K o (1− r )/ r . The generalized equation of motion, which is an integral equation for shear stress, is solved iteratively and is coupled with the Casson constitutive equation to find the velocity distribution. For the case of constant permeability, the analytical solution is found for the shear stress distribution in terms of modified Bessel functions of order 0 and 1. Finally, the effect of permeability factor K o and yield stress θ of the fluid on shear stress distribution, wall shear stress, plug flow radius, flow rates and frictional resistance are examined.

Flow in a catheterized curved artery with stenosis

The fluid mechanics of blood flow in a catheterized curved artery with stenosis is studied through a mathematical analysis. Blood is modelled as an incompressible Newtonian fluid and the flow is assumed to be steady and laminar. An approximate analytic solution to the problem is obtained through a double series perturbation analysis for the case of small curvature and mild stenosis. The effect of catheterization on various physiologically important flow characteristics (i.e. the pressure drop, impedance and the wall shear stress) is studied for different values of the catheter size and Reynolds number of the flow. It is found that all these flow characteristics vary markedly across a stenotic lesion. Also, increase in the catheter size leads to a considerable increase in their magnitudes. These results are used to obtain the estimates of increased pressure drop across an arterial stenosis when a catheter is inserted into it. Our calculations, based on the geometry and flow conditions existing in coronary arteries, suggest that, in the presence of curvature and stenosis, and depending on the value of k (ratio of catheter size to vessel size) ranging from 0.1 to 0.4, the pressure drop increases by a factor ranging from 1.60 to 5.16. But, in the absence of curvature and stenosis, with the same range of catheter size, this increased factor is about 1.74-4.89. These estimates for the increased pressure drop can be used to correct the error involved in the measured pressure gradients using catheters. The combined effects of stenosis and curvature on flow characteristics are also studied in detail. It is found that the effect of stenosis is more dominant than that of the curvature. Due to the combined effect of stenosis, curvature and catheterization, the secondary streamlines are modified in a cross-sectional plane. The insertion of a catheter into the artery leads to the formation of increased number of secondary vortices.

ESTIMATION OF INCREASED FLOW RESISTANCE IN A NARROW CATHETERIZED ARTERY -- A THEORETICAL MODEL

The changed flow pattern in a narrow catheterized artery is studied and an estimate of the increased flow resistance is made. The anomalous behaviour of blood in small blood vessels has been taken into account by modelling blood as a Casson fluid possessing some finite yield stress. Both the cases of steady and pulsatile flow situations are studied. The pulsatile flow is analysed by considering the pressure gradient as a periodic function of time with small inertial effects. The resulting quasi-steady non-linear coupled implicit system of differential equations governing the flow are solved using a perturbation analysis, where it is assumed that the Womersley frequently parameter is small (alpha < 1) which is reasonable for physiological situations in small blood vessels as well as in coronary arteries. The effect of pulsatility, catheter radius and yield stress of the fluid on the yield plane locations, velocity distribution, flow rate, shear stress and frictional resistance are investigated. Because of the yield stress theta, two yield surfaces are found to be located in the flow field. Depending on the ration kappa (catheter size/vessel size) ranging from 0.3 to 0.7 (which is widely used in coronary angioplasty procedures), the frictional resistance to flow in large blood vessels, where the effect of yield stress can be neglected (i.e. theta = 0), increases by a factory ranging from 3 to 33. In small blood vessels with the same range of catheter size and an unit pressure gradient, frictional resistance increase was by a factor of 7-21 when theta = 0.05 and 11-294 when theta = 0.1. For small values of kappa and theta, the frictional resistance increased to several hundred times thus implying that the combined effect of increased catheter radius and yield stress is to obstruct the fluid movement considerably.

Shear Augmented Dispersion of a Solute in a Casson Fluid Flowing in a Conduit

AbstractThe unsteady dispersion of a solute in a Casson fluid flowing in a conduit (pipe/channel) is studied using the generalized dispersion model of Gill and Sankarasubramanian. With this approach, the entire dispersion process is described appropriately in terms of a simple diffusion process with the effective diffusion coefficient as a function of time, in addition to its dependence on the yield stress of the fluid. The results are accurate up to a first approximation for small times, but verified with Sharp to be exact for large times. The model brings out mainly the effect of yield stress, or equivalently, the plug flow region on the overall dispersion process. It is found that the rate of dispersion is reduced (i.e., the effective diffusivity decreases) due to the yield stress of the fluid, or equivalently, the plug flow region in the conduit. Also, the effective diffusivity increases with time, but eventually attains its steady state value below a critical time [0.48(a2/Dm) for dispersion in a pipe and 0.55(a2/Dm) for dispersion in a channel—the critical transient time for a Newtonian fluid—where “a” is the radius of the pipe and Dm is the molecular diffusivity]. At steady state, for dispersion in a pipe with the plug flow radius one tenth of the radius of the pipe, the effective diffusivity is reduced to about 0.78 times of the corresponding value for a Newtonian fluid at equivalent flow rates; for dispersion in a channel, the reduction factor is about 0.73 confirming the earlier result of Sharp. Further, the location of the center of mass of a passive species over a cross section is found to remain unperturbed during the course of dispersion and for different values of the plug flow parameter (i.e., the yield stress of the fluid). The study can be used as a starting first approximate solution for studying the dispersion in the cardiovascular system or blood oxygenators.

Flow in catheterised curved artery.

CATHETERS ARE used extensively in clinical applications, despite the considerable errors that can arise in measurements using them. Transducers attached to a catheter are often used in routine clinical work and in animal experiments. Pressure transducers are used with long, fine catheters so that pressure measurements over a large part of the arterial tree can be taken from a single insertion in a peripheral a~tery. Roos and Lykoudis obtained a urometrogram through the insertion of a catheter inside the ureter (Roos and LYKOUDIS, 1970).

Pulsatile blood flow in a stenosed artery—a theoretical model

To understand the special flow conditions which may be produced by the presence of stenosis in arteries, an analytical solution is obtained for pulsatile laminar flow in an elliptic tube. Blood is approximated by a Newtonian model and the geometry of the stenosis is introduced by specifying the change in area of cross-section of the stenosed artery with axial distance. The results for velocity, pressure, shear stress and impedance are presented. These are compared with the steady flow results as well as with those of the flow in a stenosed tube of circular cross-section. The study indicates that the fluid dynamic characteristics of the flow are affected by the percentage of stenosis as well as the geometry of the stenosis. The frequency of oscillation is also found to influence shearing stress and the impedance.

Effect of Boundary Absorption in Dispersion in Casson Fluid Flow in a Tube

The combined effect of yield stress and irreversible boundary reaction on dispersion process in a Casson fluid flowing in a conduit (pipe/channel) is studied using the generalized dispersion model proposed by Sankarasubramanian and Gill (Sankarasubramanian, R., and W. N. Gill. Proc. R. Soc. London, Ser. A 333:115–132, 1973). The study describes the development of dispersive transport following the injection of a tracer in terms of the three effective transport coefficients, viz., exchange, convection, and dispersion coefficients. The exchange coefficient does not depend on yield stress but the convection and dispersion coefficients depend on yield stress or equivalently plug flow region. For large times, when the plug flow radius is one-tenth of pipe radius, the convective coefficient is reduced by 0.41 times of the corresponding value for a Newtonian fluid at equivalent wall absorption parameter; in channel case the reduction is by 39%. It is seen that the asymptotic dispersion coefficient decreases with increase in wall absorption parameter and yield stress of the fluid. When the plug radius in pipe (channel) is 0.1, depending upon the values of wall absorption parameter, say (0.01–100) the reduction factor in dispersion coefficient is in the range (0.1–0.3) in comparison to the values of the Newtonian case. The results reduce to those of Sankarasubramanian and Gill (Sankarasubramanian, R., and W. N. Gill. Proc. R. Soc. London, Ser. A 333:115–132, 1973) when there is no yield stress for the pipe flow analysis and to those of Dash et al. (Dash, R. K., G. Jayaraman, and K. N. Mehta. Ann. Biomed. Eng. 28:373–385, 2000) when there is no interphase mass transfer. The study can be used as a starting first approximation solution for studying the dispersion in the cardiovascular system.

Numerical scheme for modelling oxygen transfer in tubular oxygenators

The diffusion equation for oxygen transfer in tubular membrane oxygenators has been solved numerically by using the Crank-Nicolson method. The iterative procedure takes care of the nonlinear nature of the equations used in the model. The usual hypotheses have been used for the establishment of the nonlinear partial differential equation. Velocity profile (Newtonian, Cassonian fluid) and membrane resistance have been taken into account. Theoretical results have been compared with those obtained by the advanced front theory.Experimental results with several types of device are presented using either blood or saline. Boundary conditions are analysed. Comparisons between theory and results of experiments are presented.

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.

Flow in a curved tube with constriction—an application to the arterial system

The flow of blood in a curved stenosed artery has been mathematically modelled. Using a system of toroidal co-ordinates and perturbation in terms of two small parameters, complete analytical solutions were obtained. The physiologically important quantities, impedance and shear stress on the wall, have been discussed. In a uniform curved tube the point of maximum shear at a cross-section changes over from the inner bend to the outer bend after a distance of 1·9 times the radius from the entrance, and continues to be so later. However, the presence of a stenosis introduces an additional curvature and hence the point of maximum shear varies with the cross-section concerned. The possible points of separation in a curved occluded artery are tabulated The results obtained in this study provide valuable information for the study of mass transfer between blood and the arterial wall.