The linear stability of variable viscosity, miscible core–annular flows is investigated. Consistent with pipe flow of a single fluid, the flow is stable at any Reynolds number when the magnitude of the viscosity ratio is less than a critical value. This is in contrast to the immiscible case without interfacial tension, which is unstable at any viscosity ratio. Beyond the critical value of the viscosity ratio, the flow can be unstable even when the more viscous fluid is in the core. This is in contrast to plane channel flows with finite interface thickness, which are always stabilized relative to single fluid flow when the less viscous fluid is in contact with the wall. If the more viscous fluid occupies the core, the axisymmetric mode usually dominates over the corkscrew mode. It is demonstrated that, for a less viscous core, the corkscrew mode is inviscidly unstable, whereas the axisymmetric mode is unstable for small Reynolds numbers at high Schmidt numbers. For the parameters under consideration, the switchover occurs at an intermediate Schmidt number of about 500. The occurrence of inviscid instability for the corkscrew mode is shown to be consistent with the Rayleigh criterion for pipe flows. In some parameter ranges, the miscible flow is seen to be more unstable than its immiscible counterpart, and the physical reasons for this behaviour are discussed.

A detailed parametric study shows that increasing the interface thickness has a uniformly stabilizing effect. The flow is least stable when the interface between the two fluids is located at approximately 0.6 times the tube radius. Unlike for channel flow, there is no sudden change in the stability with radial location of the interface. The instability originates mainly in the less viscous fluid, close to the interface.

A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Poiseuille–Rayleigh–Bénard and Tollmien–Schlichting modes for Grashof numbers up to ∼ 25 000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary instability, destabilizing significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.

During an attempt to work on a stratified flow problem envisaged as a sequel of the paper by Sameen & Govindarajan (2007), it was found that the original paper contained errors in §§ 3.4 and 4.3 due to a factor of iα, which was inadvertently missed in two places in the code (i) in the buoyancy term due to the use of vertical velocity and streamfunction interchangeably, and (ii) in the apportionment between kinetic and potential energy in the Gmax calculation. Because of this, there were significant differences in the effect of Grashof number on stability. Figure 1 is the modified figure 9 of the original paper, for Pr =7 and ΔT = 25 K. The Poiseuille–Rayleigh–Bénard mode appears at Gr = 39.12 and is seen not to merge with the Poiseuille mode, unlike the conclusion made earlier. This modification applies at any Prandtl number from 10−2 to 102. The corrected versions of figures 17 and 21, showing Gmax contours for different Pr at Gr = 0 and different Gr for Pr = 1, are plotted in figures 2 and 3, respectively. The large growth reported at β = 0 was thus erroneous. The other main conclusions of the paper, that Prandtl number changes transient growth qualitatively, but not the least stable eigenmode, whereas viscosity stratification, which has a huge impact on exponential growth/decay, does not change transient growth much, remain the same. The secondary instabilities also remain unchanged. The stability equations (3.2) to (3.4) in the paper should read (for explanation, please refer to Sameen & Govindarajan 2007) (1)(2)(3)

The stability of a mixing layer made up of two miscible fluids, with a viscosity-stratified layer between them, is studied. The two fluids are of the same density. It is shown that unlike other viscosity-stratified shear flows, where species diffusivity is a dominant factor determining stability, species diffusivity variations over orders of magnitude do not change the answer to any noticeable degree in this case. Viscosity stratification, however, does matter, and can stabilize or destabilize the flow, depending on whether the layer of varying velocity is located within the less or more viscous fluid. By making an inviscid model flow with a slope change across the ‘viscosity’ interface, we show that viscous and inviscid results are in qualitative agreement. The absolute instability of the flow can also be significantly altered by viscosity stratification.

We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100–1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien–Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable.