The Rayleigh–Bénard problem for a fluid with pressure- and temperature-dependent material properties

Abstract

The Rayleigh–Bénard problem in viscous fluids is one of the most intensely studied problems in fluid mechanics. The literature is rich in studies concerning Navier–Stokes fluids with constant material properties and fluids with variable viscosity and/or thermal conduction, whereas it lacks in results concerning the more realistic case in which all the material properties of the fluid are variable. In this paper, we re-examine the classical problem of the onset of thermal-convection for fluids whose material properties depend on the pressure and temperature by studying the linear stability of the conduction solution.

Appendix A: Depth-dependent material parameters for the mineral oil Shell T9

Taken $(p_\textrm{ref},\theta _\textrm{ref})=(1$ atm, 25 $^\circ$C) as reference state, we non-dimensionalize (66)–(70) by using (7) and find that the depth-dependent coefficients in (39) for the mineral oil Shell T9 are as follows

$$\begin{aligned} \hat{\alpha }(z)= & {} \dfrac{1}{\alpha _\textrm{ref}}\Bigg \{\dfrac{a_V}{1+a_V\theta _\textrm{ref}\big (\bar{\theta }(z)-1\big )}\nonumber \\{} & {} \quad -\dfrac{\rho _\textrm{ref}gd\beta _K}{K_{00}}\exp \big (\beta _K\theta _\textrm{ref}\bar{\theta }(z)\big )p_h(z)\left[ 1+\dfrac{K'_0}{K_{00}}\rho _\textrm{ref}gd \exp \big (\beta _K\theta _\textrm{ref}\bar{\theta }(z)\big )p_h(z)\right] ^{-1}\Bigg \}, \end{aligned}$$

(73a)

$$\begin{aligned} \hat{\mu }(z)= & {} \dfrac{\mu _\infty }{\mu _\textrm{ref}}\exp \left[ \dfrac{B_F\varphi _\infty }{\bar{\theta }(z)\mathcal {F}^g(z)-\varphi _\infty }\right] , \end{aligned}$$

(73b)

$$\begin{aligned} \hat{k}(z)= & {} \dfrac{B+C\left\{ \mathcal {F}(z)\left[ 1+A\bar{\theta }(z)\mathcal {F}^3(z)\right] \right\} ^{-s}}{k_\textrm{ref}}, \end{aligned}$$

(73c)

$$\begin{aligned} \hat{c}_p(z)= & {} \dfrac{C_0\mathcal {F}^{4}(z)+m\bar{\theta }(z)}{{c_p}_\textrm{ref}\rho _R \mathcal {F}^3(z)}, \end{aligned}$$

(73d)

$$\begin{aligned} \hat{\kappa }(z)= & {} -\dfrac{Cs}{k_\textrm{ref}}\left\{ \mathcal {F}(z)\left[ 1+A{\bar{\theta }}(z)\mathcal {F}^3(z)\right] \right\} ^{-s-1}\nonumber \\{} & {} \quad \times \left\{ \alpha _\textrm{ref}\theta _\textrm{ref}\mathcal {F}(z)\hat{\alpha }(z)\left[ 1+4A{\bar{\theta }}(z)\mathcal {F}^3(z)\right] +A\mathcal {F}^4(z)\right\} \bar{\theta }'(z), \end{aligned}$$

(73e)

where

$$\begin{aligned} \mathcal {F}(z)=\dfrac{1+a_V\theta _\textrm{ref}\big (\bar{\theta }(z)-1\big )}{\left\{ 1+\dfrac{K'_0}{K_{00}}\rho _\textrm{ref}gd \exp \big [\beta _K\theta _\textrm{ref}\bar{\theta }(z)\big ]p_h(z)\right\} ^{1/K'_0}}, \end{aligned}$$

(73f)

$\rho _\textrm{ref}=872.059$ kg/m$^3$, $\alpha _\textrm{ref}=0.77\cdot 10^{-3}$ K$^{-1}$, $\mu _\textrm{ref}=0.017$ Pa s, $k_\textrm{ref}=0.1114$ W m$^{-1}$ K$^{-1}$ and ${c_p}_{\textrm{ref}}=1.789$ kJ kg$^{-1}$ K$^{-1}$.

For the mineral oil Shell T9, from (51c) and (68) the reduced response function for the viscosity (51i) is found to be

$$\begin{aligned} \tilde{\mu }=\dfrac{\mu _\infty }{\mu _\textrm{ref}}\exp \left[ \dfrac{B_F\varphi _\infty }{h(z)\mathcal {H}^g(z)-\varphi _\infty }\right] , \end{aligned}$$

(74a)

with

$$\begin{aligned} h(z)=1+(\delta -1)(1-z) \quad \text {and} \quad \mathcal {H}=\dfrac{1+a_V\theta _\textrm{ref}[h(z)-1]}{\left\{ 1+\dfrac{K'_0}{K_{00}}p_\textrm{ref} \exp \big [\beta _K\theta _\textrm{ref}h(z)\big ]\right\} ^{1/K'_0}}. \end{aligned}$$

(74b)