Nonlinear stability analysis of convection for fluids with exponentially temperature-dependent viscosity

Abstract

A nonlinear energy stability analysis of the onset of convection for fluids with viscosity depending exponentially on temperature is performed. It is shown that the condition assuring linear stability assures nonlinear asymptotic stability, too

Stability analysis of the Rayleigh–Bénard convection for a fluid with temperature and pressure dependent viscosity

“…Moreover, since the stability of the conduction solution m 0 makes sense only within the class of solutions (9) in which the zero solution u = v = w = θ = P 1 = 0 is unique, for free boundaries we exclude any other solution by requiring the usual 'average velocity conditions' (see Kloeden and Wells (1983))…”

Unconditional non‐linear stability for a fluid of third grade

“…Employing a natural energy method, we obtain a critical Rayleigh is the relevant clue of this behaviour. The proportionality =B suggests that for B very large the coe cient tends to zero and the non-linear results are the same as the linear ones, as long as B does not dominate on 2 . As B will decrease enough, it will end by dominate the 2 term.…”

“…Much attention has recently been devoted to the subject of convection studies with variable viscosity, since viscosity is one of the uid properties which may dramatically change with temperature; see e.g. References [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and see also several references given in Reference [19, pp. 6-158], regarding the dynamic viscosity of glycerin; while glycerin exhibits a dramatic viscosity change there are many other examples, oily uids, quoted in the tables in Reference [19].…”

Non‐linear stability for convection with quadratic temperature dependent viscosity

“…The linearized stability has been studied in much detail but the non-linear stability problem remains to be explored completely. The non-linear stability for such convection problems in di erent settings has been treated extensively by Capone [4], Capone and Gentile [5], Carr and Straughan [6], Diaz and Straughan [7], Straughan [8], Straughan [9], Rionero [10] and Rionero and Mulone [11]. The readers are referred to Diaz and Straughan [7] for a thorough introduction to the past literature on this subject.…”