Vanishing Reynolds Number Research Articles

Experimental studies of the onset of oscillatory instability in viscoelastic Taylor-Couette flow

Laser Doppler velocimetry (LDV) measurements of the flow of a wellcharacterized viscoelastic fluid between concentric cylinders are presented. The results show that the viscoelastic Taylor-Couette instability at vanishing Reynolds number is oscillatory for a Boger fluid; the velocity undergoes a transition from steady to time periodic at a critical Deborah number, De c. The value of De c measured by LDV is in good agreement with earlier mechanical measurements and with the calculations of Shaqfeh et al. for an Oldroyd-B fluid. The frequency of oscillation of the velocity field, however, is considerably higher than the Oldroyd-B prediction. In addition, higher harmonics of the oscillation frequency appear in the power spectrum of the velocity just above the critical condition. This observation is consistent with the calculations of Northey et al. who predict that the flatness of the neutral stability curve with respect to changes in axial wavenumber will lead to nonlinear interactions between families of time-periodic states that are closely spaced in De. Finally, the temporal evolution of the local flow field in the present LDV studies appears qualitatively different from that observed in global flow-visualization studies. This suggests that the flow may depend on the history of the experiment as well as the Deborah number.

Sausage instability of a thread in a matrix; linear theory for viscoelastic fluids and interface

Surface tension makes the shape of a fluid cylinder unstable with respect to periodic corrugations of wavelength greater than the cylinder circumference, and the thread eventually breaks up into a series of droplets separated by the wavelength of the fastest growing instability. We derive in this paper the relation between the wavelength and the growth rate of the sausage instability of a thread embedded in a matrix, in the limit of vanishing Reynolds number and for incompressible fluids. The theory applies to general viscoelastic fluids, and takes into account the dynamic properties of the interfacial tension due to adsorbed contaminants or interfacial agents, as well as the capability of the interface to resist a shear deformation. It includes the special cases treated by Tomotika and by Chin and Han, and provides a valuable tool for measuring the static and dynamic properties of the interfacial tension.

Two-Dimensional Viscous Flow Accompanied by a Stagnation Point on a Plane Wall

The two-dimensional flow, induced by a point source at a finite distance from an infinite plane wall, is analyzed on the basis of the Navier-Stokes equation in the following two cases. In the first case, the source strength is considered to oscillate sinusoidally with the vanishing Reynolds number (unsteady Stokes flow). The wall shear stress can then be evaluated for arbitrary values of the frequency \(\varOmega \) and is found to increase in proportion to \(\sqrt{\varOmega }\) as \(\varOmega \) becomes large. In the second case the steady flow with not-zero but small Reynolds number is dealt with by the perturbation method. It is found that the first-order perturbation to the steady Stokes solution is fairly small but that the back flow may occur far away from the stagnation point.

Field descriptions for a steady, developing tube flow of vanishing Reynolds number

Numerical solutions for the stream function, vorticity, velocity, and pressure fields are presented for the case of a steady, laminar, isothermal, Newtonian flow developing from an initial slug flow in a circular cylinder of infinite length at zero Reynolds number.

Laminar Flows Past Oblate Spheroids of Various Thicknesses

The effect of high surface curvature of bodies on a surrounding laminar flow field is studied by means of steady axisymmetric flows past oblate spheroids of various thicknesses. For vanishing Reynolds number the Oberbeck solution is discussed. For nonzero Reynolds numbers Re ≤ 100 a numerical program is used which was recently developed by Rimon. The numerical stability of this finite-difference scheme deteriorates rapidly as the oblate spheroids become very thin. For Reynolds numbers 0 and 10 two different types of flows are observed, depending on the thickness of the body: for spherelike oblate spheroids the surface pressure has a maximum at the stagnation point, whereas for disklike bodies this maximum moves toward the edge with decreasing thickness. However, for Re = 100 the maximum pressure is always at the stagnation point, no matter how thin the body is. In the limiting case of the infinitely thin disk the edge becomes a singular body point. Although this limit could not be examined for nonzero Reynolds numbers with the numerical scheme used, the trend in the behavior of the solutions as the thickness decreases is discussed. From a previous analysis of the nature of possible solutions near a singular point, it appears from the numerical results that for Re = 100 the flow is regular at the edge. Since for Re = 0 the solution is weakly singular (pressure and vorticity are infinite at the singular body point), a change in the flow characteristics must occur between Re = 0 and Re = 100.

The boundary layer on a paraboloid of revolution

AbstractAn analytical study is made of the boundary-layer approximation for the steady laminar flow of a viscous incompressible fluid past a paraboloid of revolution. The functional form of the vorticity distribution is found and shown to involve exponential decay with distance from the surface of the body, and inequalities are established concerning the displacement thickness.The exact results are compared with various linearizations. The Oseen linearization (12) turns out to be incorrect except in the limit of vanishing Reynolds number; this result is connected with the infinite drag of any paraboloid of non-zero thickness. An alternate linearization, related to a method of Burgers (3) and allowing for the displacement effect of the body and of its boundary layer, is shown to give the correct functional form for these quantities. Numerical results are included which bear out these conclusions.