Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid

Abstract

We re-examine fully developed isothermal unidirectional plane Couette-Poiseuille flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in [1]. We show that the conclusion made there that, in contrast to Newtonian and power-law fluids, piezo-viscous fluids allow multiple solutions is not justified, and that the inflection velocity profiles reported in [1] cannot exist. Subsequently, we undertake a systematic parametric study of these flows and identify three distinct families of solutions which can exist in the considered geometry. One of these families has no similar counterpart for fluids with pressure-independent viscosity. We also show that the critical wall speed exists beyond which Poiseuilletype flows are impossible regardless of the magnitude of the applied pressure gradient. For smaller wall speeds channel choking occurs for Poiseuille-type flows at large pressure gradients. These features distinguish drastically piezo-viscous fluids from their Newtonian and power-law counterparts.

Further remarks on simple flows of fluids with pressure-dependent viscosities

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AbstractSuslov and Tran [3] recently revisited the study carried out by Hron et al [1] and they on the basis of their analysis claim that some of conclusions concerning one specific example, amongst the many considered by Hron et al We have reexamined both papers, and we find that whether or not velocity profiles with inflection points exist depends on the class of functions to which the pressure belongs. If the pressure field is allowed to be discontinuous, which is in keeping with the class of functions to which pressure belongs to in the study of Hron et al

[1], such inflectional profiles are possible.

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“…In this study we re-study the problem and show that the conclusion of Suslov and Tran [3] is correct if one requires the pressure to be continuous, or if one requires that the constructed solution is a weak solution. However, on the other hand, if one allows for the pressure to be discontinuous, then such profiles with inflections are possible, and the conclusion drawn by Suslov and Tran [3] is incorrect. In this context, it ought to be borne in mind that the pressure field in the analysis of Hron et al [1] belongs to the class of functions that could be discontinuous 4 .…”

“…However, if one requires the pressure field to be continuous then as Suslov and Tran [3] claim, such inflectional profiles are not possible. We provide a detailed explanation for this phenomenon that goes beyond the discussion presented in the paper by Suslov and Tran [3], and concerns subtle mathematical issues. Among other results we show that the solution with the inflectional profile is-interestingly-not a weak solution of the governing equations.…”

“…Recently Suslov and Tran [3] re-examined the several solutions established by Hron et al [1] for the flow of fluids with pressure dependent viscosities, and claimed that one of them, namely the flow between parallel plates that showed the possibility of profiles with inflection, to be incorrect. In this study we re-study the problem and show that the conclusion of Suslov and Tran [3] is correct if one requires the pressure to be continuous, or if one requires that the constructed solution is a weak solution.…”

Laminar pipe flow with time-dependent viscosity

“…However, this is significant only at extremes of pressure far beyond flows of most common engineering interest. Consequences of pressure dependence in pipe flows have been investigated by, for example, Vasudevaiah & Rajagopal (2005), Massoudi & Phuoc (2006) and Suslova & Tran (2008).…”