Further remarks on simple flows of fluids with pressure-dependent viscosities Jaroslav Hron a,2 , Josef M´ alek a,1 , V´ ıt Pr ˚ uˇ sa a,2 , K.R. Rajagopal ∗ ,b,3 a Mathematical Institute of Charles University, Sokolovsk´ a 83, Praha 8 - Karl´ ın, Czech Republic b Department of Mechanical Engineering, Texas A&M University, Texas, 77843, United States of America Abstract Suslov 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. [1], are not justified. They claim that the class of simple flows of fluids with pressure dependent viscosity considered by Hron et al. [1] do not allow multiple solutions, and that the inflection velocity profiles reported in Hron et al. [1] cannot exist. 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. 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. Concerning the non-uniqueness of the solution, we show that if we explicitly—instead of assuming that constants are fixed by an unknown procedure—specify a procedure for fixing all the integration constants in the solution, for example by fixing the pressure at two points or fixing the pressure gradient and the pressure at one point, we get a unique solution to the problem, provided all relevant quantities are continuous. On the other hand, if we relax the assumption on continuity, we can get multiple solutions. Key words: pressure dependent viscosity, inflection velocity profiles, non-uniqueness 2000 MSC: 76D99, 35Q35 1. Introduction 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. 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 ∗ Corresponding author. Email addresses: hron@karlin.mff.cuni.cz (Jaroslav Hron), malek@karlin.mff.cuni.cz (Josef M´ alek), prusv@karlin.mff.cuni.cz (V´ ıt Pr˚ uˇ sa), krajagopal@tamu.edu (K.R. Rajagopal) 1 Josef M´ alek’s contribution is a part of the research project MSM 0021620839 financed by the M ˇ SMT; support of GA ˇ CR 201/09/0917 is also acknowledged. 2 Jaroslav Hron and V´ ıt Pr˚ uˇ sa thank to the Neˇ cas Center for Mathematical Modeling (project LC06052 finaced by the M ˇ SMT of the Czech Republic) for its support. 3 K.R. Rajagopal thanks to the National Science Foundation for its support. Preprint submitted to Journal of Non-Newtonian Fluid Mechanics August 6, 2009