On the influence of boundary condition on stability of Hagen–Poiseuille flow

Abstract

a b s t r a c tWe analyze the influence of choice of boundary condition (no-slip and Navier's slip boundary conditions) on linear stability of Hagen-Poiseuille flow. Several heuristic arguments based on detailed analysis of spectrum of the Stokes operator are given, and it is concluded that Navier's slip boundary condition should have a destabilizing effect on the flow. Finally the linear stability problem is solved by numerical means, and quantitative results confirming the heuristic prediction are obtained. It is shown that the destabilization is not strong enough to maintain an unstable disturbance, and that the significant destabilization effects of Navier's slip boundary condition are restricted to small values of the Reynolds number. As a byproduct we obtain explicit formulas for eigenfunctions of the Stokes operator subject to Navier's slip boundary condition.

Linear stability of slip pipe flow

“…In a word, in the pure streamwise slip case, we did not observe any linear instability in the large ranges of and that we considered, and based on the data shown in figure 5, we propose that streamwise slip destabilizes the flow but does not cause linear instability, regardless of the slip length and Reynolds number. A similar destabilizing effect was reported by Průša (2009) for the isotropic slip case.…”

“…Our results show that the leading eigenvalue increases with streamwise slip length but remains negative, i.e. streamwise slip renders the flow less stable but does not cause linear instability, similar to the effect of isotropic slip length on the flow (Průša 2009). Interestingly, our results suggest that the leading eigenvalue is independent of , or equivalently, the slowest decay rate of disturbances scales as (note that time is scaled by in our formulation).…”

“…Lauga & Cossu (2005) and Chai & Song (2019) considered the same boundary conditions for slip channel flow. Note that in the isotropic slip case considered by Průša (2009), and are related as , which gives for small slip lengths. With boundary condition (2.9 a , b ), given that we impose the same volume flux as in the no-slip case, i.e.…”

“…), far fewer studies were dedicated to the linear stability of single-phase pipe flow with slip boundary condition. Průša (2009) investigated this problem and showed that, subject to Navier slip boundary condition, pipe flow becomes less stable compared with the no-slip case, however, the destabilization effect is constrained to small Reynolds numbers and is not sufficient to render the flow linearly unstable. Their results indicated that the stability property of pipe flow is not qualitatively affected by the slip boundary condition, regardless of the slip length.…”

“…The slip boundary condition can be thought of as a perturbation to the linear operator with no-slip boundary condition. However, Průša (2009) showed that homogeneous and isotropic slip does not change the spectrum qualitatively no matter how large the slip length (i.e. operator perturbation) is.…”