N. Padmanabhan

Mathematical model of arterial stenosis

A mathematical model for pulsatile flow in a partially occluded tube is presented. The problem has applications in studying the effects of blood flow characteristics on atherosclerotic development. The model brings out the importance of the pulsatility of blood flow on separation and the stress distribution. The results obtained show fairly good agreement with the available experimental results.

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.

Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section

We analyse the steady streaming generated in an infinite elliptical tube containing a viscous, incompressible fluid when the boundary oscillates in such a way that the area and ellipticity of the cross-section vary with time but remain independent of the longitudinal coordinate. The parameters α −1 = (ν/Ω a 0 2 ) ½ and e = U 0 / a 0 Ω, where ν is the kinematic viscosity, Ω is the oscillation frequency, a 0 is the undisturbed semi-major axis and U 0 is a typical wall velocity, are taken to be small, so that the Stokes layer is thin and the interaction which leads to the steady streaming can be analysed as a small perturbation. Coupled axial and transverse velocities, both oscillatory and steady, are generated. A complication is the need to specify the tangential as well as the normal velocity component on the tube wall, which requires an assumption concerning its elastic properties. We have considered two cases: (i) constant major axis, in which all boundary points move parallel to the minor axis, and (ii) an inextensible wall. The three-dimensional steady streaming in the core of the tube is computed only in the limit of small steady-streaming Reynolds number, R s = e 2 α 2 .

Entry flow in heated curved pipes

Abstract The flow in the entrance region of heated curved pipes is analysed. Two cases of heating—a constant temperature at the wall, and a constant flux of heat at the wall—are considered. Using boundary layer approximations and the method of matched asymptotic expansions, the combined effects of curvature, entrance region and the buoyancy is studied. It is found that buoyancy disturbs the symmetric secondary motion induced by curvature, the deviation depending on the type of thermal input at the wall. It is also found that the oscillatory nature of the Nusselt number in the constant temperature case decreases as the Peclet number is increased.

Mathematical model of an arterial stenosis, allowing for tethering

Closed-form solutions are presented for approximate equations governing the pulsatile flow of blood through models of mild axisymmetric arterial stenosis, taking into account the effect of arterial distensibility. Results indicate the existence of back-flow regions and the phenomenon of flow-reversal in the cross-sections. The effects of pulsatility of flow and elasticity of vessel wall for arterial blood flow through stenosed vessels are determined.

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.

Entry flow into a circular tube of slowly varying cross-section

The entry flow into a circular tube of slowly varying cross-section is studied by considering the flow field to consist of two regions—a core region and a boundary layer region. As in classical boundary layer problem, the equations in the two regions are solved separately using a matching condition at the edge of the boundary layer. For different wall geometries, the velocity, shearing stress and vorticity are computed and the phenomenon of boundary layer separation is discussed. It is found that the shape of the wall geometry plays an important role on the fluid dynamic characteristics. The model is quite versatile and can be applied to a variety of engineering problems involving slowly varying, converging or diverging tubes.

REVERSING FLOW IN THE AORTA: A THEORETICAL MODEL

The viscous boundary layer in a reversing flow in a curved tube is analysed with a view to study the reversing flow in the aorta at the beginning of the diastole. The velocity in the core is taken to vary with time as in a dogs aorta. The flow is considered to be quasi-steady long before the time of reversal. Near the time of reversal, the flow is governed by the diffusion equation--a balance between the unsteady inertia terms and viscous terms. The solution which ensures the continuity of the displacement thickness at the time of take over from the quasi-steady solution to the diffusive solution, is obtained. The knowledge of the distribution of the shearing stress in the circulatory system is essential due to its relevance with regard to atheroma--a disease leading to the hardening of the arteries. With this in mind, the wall shear rate is obtained as a function of time at every cross-section of the tube. The shearing stresses at the inner bend and the outer bend are also plotted. The results are compared with those for the straight tube.