A study of optimal temporal and spatial disturbance growths is carried out for three-dimensional viscous incompressible fluid flows with slippery walls. The non-modal temporal stability analysis is performed under the framework of normal velocity and normal vorticity formulations. A Chebyshev spectral collocation method is used to solve the governing equations numerically. For a free surface flow over a slippery inclined plane, the maximum temporal energy amplification intensifies with the effect of wall slip for the spanwise perturbation, but it attenuates with the wall slip when perturbation considers both streamwise and spanwise wavenumbers. It is found that the boundary for the regime of transient growth appears far ahead of the boundary for the regime of exponential growth, which raises a question on the critical Reynolds number for the shear mode predicted from the eigenvalue analysis. Furthermore, the eigenvalue analysis or the modal stability analysis reveals that the unstable region for the shear mode decays rapidly in the presence of wall slip, which is followed by the successive amplification of the critical Reynolds number for the shear mode and ensures the stabilizing effect of slip length on the shear mode. On the other hand, for a channel flow with slippery bounding walls, the maximum spatial energy amplification intensifies with the effect of wall slip in the absence of angular frequency, but it reduces with the wall slip if the angular frequency is present in the disturbance. Furthermore, the maximum spatial energy disturbance growth can be achieved if the disturbance excludes the angular frequency. Furthermore, it is observed that the angular frequency plays an essential role in the pattern formation of optimal response. In addition, the pseudo-resonance phenomenon occurs due to external temporal and spatially harmonic forcings, where the pseudo-resonance peak is much higher than the resonance peak.

Non-modal stability analysis in viscous fluid flows with slippery walls

This paper is concerned with the question of the beginning of convective motion in a fluid saturated porous layer, containing a salt in solution, heated below and salted above and below. This model has a single temperature and employs the Darcy theory in the micropores, the Brinkman theory, however, being utilized in the macropores. The effect of slip boundary conditions on the stability of the model is also studied. General boundary conditions regarding temperature and salt are also taken into account. It will be shown that the linear instability threshold is the same as that of nonlinear stability if the layer is salted from above, indicating that the linear theory entirely captures the physics of the onset of thermal convection. In the case of salting from below, the behavior of the transition from stationary to oscillatory convection is investigated in detail, as the boundary conditions change from prescribed temperature and salt concentration toward those of prescribed heat flux and salt flux. The nonlinear stability threshold does not coincide with that of linear instability; thus, regions of possible subcritical instability are still present. We believe that the problem presented in this paper has not been addressed before and that its study will have great scientific value and impact.

Bidispersive double diffusive convection with relatively large macropores and generalized boundary conditions

In this work, we present a systematic numerical investigation of the 1:4 planar expansion creeping flow under the influence of slip boundary conditions for Newtonian and viscoelastic fluids, the latter modeled by the simplified Phan–Thien–Tanner constitutive model. The linear and nonlinear Navier slip laws were considered with the dimensionless slip coefficient kl* varying in the range 0, 4500 and the slip exponents m = 0.5, 1, and 2. The simulations were carried out for a low Reynolds number, Re = 0.001, and for Deborah numbers (De) between 0 and 100. Convergence could not be achieved for higher values of the Deborah number and large values of the slip coefficient due to the large stress gradients near the singularity point (reentrant corner). The results obtained allow us to conclude that for all De, the increase in slip velocity leads to vortex suppression. The flow characteristics are described in detail for low values of the Deborah number, De ≤ 5, while for higher De the main features are only shown for specific values of the slip coefficient. These results find application in polymer processing, where the use of lubricants that migrate to the wall is common, which promotes slip.

Newtonian and viscoelastic fluid flows through an abrupt 1:4 expansion with slip boundary conditions

Linear stability and energy stability of plane Poiseuille flow with isotropic and anisotropic slip boundary conditions

In a fluid-saturated bidisperse porous medium at a single temperature, the issue of thermal convection when the Darcy theory is used in the micropores, particularly the effects of slip boundary conditions on the model stability, was examined (whereas the Brinkman theory is used in the macropores). In addition, the effect of general temperature boundary conditions was also addressed. The governing equations of motion are provided, followed by the way in which the related equations of perturbation were derived. In addition, the linear instability and nonlinear stability analyses of the system were introduced, with the numerical approach used to approximate the eigenvalue system resulting from our analysis. The threshold for linear instability was proved to be the same as the one for nonlinear stability, showing that the linear theory accurately captures the mechanism of the onset of thermal convection. The numerical results for stability/instability thresholds were also introduced. The researchers assumed that this was the first time a mixed Darcy–Brinkman model had been used in bidisperse flow with slip boundary conditions.

Bidispersive thermal convection with relatively large macropores and generalized velocity and temperature boundary conditions

In this work, we revisit the temporal stability of slip channel flow. Lauga & Cossu (Phys. Fluids 17, 088106 (2005)) and Min & Kim (Phys. Fluids 17, 108106 (2005)) have investigated both modal stability and non-normality of slip channel flow and concluded that the velocity slip greatly suppresses linear instability and only modestly affects the non-normality. Here we study the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did. We find that, with sufficiently large slip, both streamwise and spanwise slip trigger three-dimensional leading instabilities. Overall, the critical Reynolds number is only slightly increased by streamwise slip, whereas it can be greatly decreased by spanwise slip. Streamwise slip suppresses the non-modal transient growth, whereas spanwise slip enlarges the non-modal growth although it does not affect the base flow. Interestingly, as the spanwise slip length increases, the optimal perturbations exhibit flow structures different from the well-known streamwise rolls. However, in the presence of equal slip in both directions, the three-dimensional leading instabilities disappear and the flow is greatly stabilized. The results suggest that earlier instability and larger transient growth can be triggered by introducing anisotropy in the velocity slip.

Chunshuo Chai

Papers

Stability of slip channel flow revisited

Stability of slip channel flow revisited