The Variational Method in Engineering

Abstract

N R ~ = Reynolds number = SRup/p, dimensionless p = pressure, dyne/sq.cm. r = radius, cm. R = radius of fall tube, cm. ur, urO, u , uZo = axisymmetric cylindrical velocity component , cm./sec. tin = theoretical terminal velocity without end effects, cm./sec. u1 = theoretical terminal velocity with end effects, cm./ sec. u2 = experimental terminal velocity, cm./sec. x = distance, cm. Greek Letters B0 p2 8,. K , J p u a,,' Q,, = viscometer constant, theoretical, sq.cm. = viscometer constant, experimental, sq.cm. = viscometer constant, reduced, dimensionless = radius of falling cylinder/radius of fall tube, di-= viscosity, g./ (cm.) (sec.) = density of oil, g./cu.cm. = density of cylinder, g./cu.cm. = axisymmetric velocity dissipation function, l/sec.2 = theoretical end effects coefficient, dimensionless mensionless This paper presents on analysis and computed results of the flow of deformoble droplets of one Newtonian fluid suspended in another Newtonian fluid. Surface tension is assumed to act at the interface of the two fluids. Non-Newtonian characteristics of the suspension are demonstrated and are quolitatively similor to flow charocteristics of blood in capillaries. The results show that the deformation of the drops may result in a significant reduction in the pressure gradient compared with that necessary for o suspension of rigid spheres or spherical (nondeformable) liquid drops. The decrease in the pressure gradient is velocity dependent. This constitutes a mechanism of non-Newtonian behavior of the suspension as a whole, and is attributable to the flexibility of the suspended elements. The resultant shapes of the liquid drops ore similar to the shapes of red blood cells that hove been observed in narrow glass capillaries as well as in blood vessels. This study considers the axisymmetric flow of a suspension of deformable liquid drops through a rigid cylindrical tube (Figure 1). The liquid in each drop and the suspending fluid are both assumed to be Newtonian and incompres-W. A. Hyman is at the Department of Mechanical Engineering, Massa-sible. A surface tension is assumed to act at the interface. The analysis allows for an arbitrary ratio of viscosity of the liquid drops to that of the suspending fluid. The motion is assumed to be sufEciently slow so that inertial terms in the equation of motion may be neglected. This system is considered to be a possible model of blood