Instability thresholds for thermal convection in a Kelvin–Voigt fluid of variable order

Abstract

We present numerical techniques for calculating instability thresholds in a model for thermal convection in a complex viscoelastic fluid of Kelvin–Voigt type. The theory presented is valid for various orders of an exponential fading memory term, and the strategy for obtaining the neutral curves and instability thresholds is discussed in the general case. Specific numerical results are presented for a fluid of order zero, also known as a Navier–Stokes–Voigt fluid, and fluids of order 1 and 2. For the latter cases it is shown that oscillatory convection may occur, and the nature of the stationary and oscillatory convection branches is investigated in detail, including where the transition from one to the other takes place.

Competitive Double Diffusive Convection in a Kelvin–Voigt Fluid of Order One

“…1, with positions depending on λ. When C = 0 oscillatory convection may occur as shown in Straughan [64], Figs. 2 and 4.…”

“…In Straughan [64] γ is taken as 0.2 or 0.5 and λ values of 10 −3 , 10 −2 , 0.1 and 1 are investigated. Given this information we here concentrate on Pr = 25, Pc = 541.75 or 1750, with λ in the range [10 −4 , 2].…”

“…The thermal convection problem for a Navier-Stokes-Voigt fluid without a salt field or for a Kelvin-Voigt fluid of order one without a salt field was suggested by Sukacheva and Matveeva [27], Sukacheva and Kondyukov [29]. An analysis of the thermal convection problem for a Kelvin-Voigt fluid of order one without a salt field is given by Straughan [64], whereas an equivalent analysis of the thermal convection problem for a Navier-Stokes-Voigt fluid is detailed by Straughan [47]. The isothermal models for a Kelvin-Voigt fluid of orders zero, one and higher are explained at length by Oskolkov [25,26], see also Oskolkov and Shadiev [30].…”

“…One is manifest in the 12 and 21 terms and reflects the presence of the viscoelastic term in (5) 1 involving . In the absence of the salt field this term yields an increasing stability threshold Ra as increases, and may lead to oscillatory convection, Hopf bifurcation, as shown in figures 2 and 4 of Straughan [64]. The second skew symmetric term in L A is manifest in the 14 and 41 terms and is due to the fact that the basic layer is heavier in salt at the base.…”

Continuous dependence and convergence for a Kelvin–Voigt fluid of order one

“…A particular class of viscoelastic fluids of interest here are those associated with the names of Kelvin and of Voigt, see e.g. [16][17][18][19][20][21]. Much of the interest in these fluids stems from work of Russian writers in this field and analytical studies of Kelvin-Voigt fluids are contained in [22,23], with generalizations of these models and analytical results to encompass the non -isothermal situation in [24][25][26].…”