Darcy–Brinkman analysis of thermo-vibrational convection in gyrotactic swimmers: an overstability theory

Abstract

This paper investigates Darcy–Brinkman thermal convection in the stratified porous saturated suspension of active particles subjected to vertical oscillation. For a heated layer, in the context of no-slip boundaries, the derived critical numbers are found to be real-valued, which signifies that the mechanism of convection is through the stationary mode, although for a certain range of heat parameters, oscillatory convection is inevitable. The dispersion expressions are developed to characterize the stationary and overstability thresholds of the system using the Galerkin method. An attempt has been made to analyze and substantiate the significance of important parameters such as modified Darcy number $(D_\textrm{a})$, wave number $(\alpha ^{\jmath })$, and Rayleigh numbers [bioconvection $(R_\textrm{b})$, thermal $(R_\textrm{a})$, and their vibrational analogs $(R_\textrm{v}, R_\textrm{t})$ ] for the representative ranges of Péclet $(1\le \hbox {Pe} \le 2)$ and gyrotactic $(1\le G\le 5)$ numbers. While incremental gyrotactic propulsion encourages the decrease in bioconvection strength, higher Péclet values induce the suspension to stabilize. Porosity has a destabilizing effect that considerably lessens the ability of vertical vibration to stabilize. The layer becomes unstable due to the thermal-oscillational connection of the thermal vibration parameter, which slows the development of bioconvection blooms.

Appendix

The functions $\psi$, $\psi _{1}$, $\psi _{2}$, and $\psi _{3}$ defined in Eq. (28), $\delta _{\mathrm{i}}$, $\gamma _{\mathrm{i}}$($i=1...7$) in Eq. (29) and $\varphi _{\mathrm{i}} = \delta _{\mathrm{i}}/\gamma _{\mathrm{i}},(i=1,2....7)$ in Eq. (36) are given as follows:

$$\begin{aligned} \psi&= {} \big (1+\alpha _{0}\big )G\psi _{1}+\big [1+a^{\jmath ^2}\big (1-\alpha _{0}\big )G\big ] \psi _{2} \end{aligned}$$

(38)

$$\begin{aligned} \psi _{1}&= {} 4\Big (e^\textrm{Pe}-1\Big )/Pe\end{aligned}$$

(39)

$$\begin{aligned} \psi _{2}&= {} \Big (e^\textrm{Pe}+1\Big )\Big (8/\big (Pe\big )^{2}\Big )-\Big (e^\textrm{Pe}-1\Big )\Big (16/\big (Pe\big )^{3}+1/Pe\Big )\end{aligned}$$

(40)

$$\begin{aligned} \psi _{3}&= {} \Big (e^\textrm{Pe}-1\Big )\Big (24/\big (Pe\big )^{5}+2/\big (Pe\big )^{3}\Big )-\Big (12/\big (Pe\big )^{4}\Big )\nonumber \\ & \quad \times & {} \Big (e^\textrm{Pe}+1\Big )\end{aligned}$$

(41)

$$\begin{aligned} f_{1}&= {} f_{1}\Big (R_\textrm{v}, R_\textrm{t}\Big )= 900 \psi _{3} R_\textrm{v}-30 \mho R_\textrm{t}\end{aligned}$$

(42)

$$\begin{aligned} f_{2}&= {} f_{2}\Big ( R_\textrm{a}, R_\textrm{b}, R_\textrm{v}, R_\textrm{t}\Big )= \Big [ 30\psi _{3}R_\textrm{v}- \mho R_\textrm{t}\Big ]\Big ( R_\textrm{a}/R_\textrm{b}\Big )\end{aligned}$$

(43)

$$\begin{aligned} f_{3}&= {} f_{3}\Big (R_\textrm{a}, R_\textrm{v}, R_\textrm{t}\Big )=\Big ( 30\psi _{3}R_\textrm{v}- \mho R_\textrm{t} \Big )R_\textrm{a}\end{aligned}$$

(44)

$$\begin{aligned} \delta _{1}&= {} D_\textrm{a}\Big [20\alpha ^{\jmath ^2}+\alpha ^{\jmath ^4}\Big ]+\Big (10+\alpha ^{\jmath ^2}\Big ),\hspace{0.5em} \gamma _{1}=\Big (10+\alpha ^{\jmath ^2}\Big )c_\textrm{a}Da\end{aligned}$$

(45)

$$\begin{aligned} \delta _{2}&= {} \Big (10+a^{\jmath ^2}\Big )^{2},\hspace{0.5em}\gamma _{2}= \Big (10+a^{\jmath ^2}\Big )Pr\end{aligned}$$

(46)

$$\begin{aligned} \delta _{3}&= {} \Big [\alpha ^{\jmath ^2}\Big (120+\big ({\text{Pe}}\big )^{4}-10\big ({\text{Pe}}\big )^{2}\Big )+10\big ({\text{Pe}}\big )^{4}\Big ]\end{aligned}$$

(47)

$$\begin{aligned} \gamma _{3}&= {} \Big (120+\big ({\text{Pe}}\big )^{4}-10\big ({\text{Pe}}\big )^{2}+Big){\text{LePr}}\phi \end{aligned}$$

(48)

$$\begin{aligned} \delta _{4}&= {} 30 \alpha ^{\jmath ^2}\psi \Big (10-\big ({\text{Pe}}\big )^{2}\Big )\Big (10+\alpha ^{\jmath ^2}\Big )^{2}\end{aligned}$$

(49)

$$\begin{aligned} \gamma _{4}&= {} 30 \alpha ^{\jmath ^2}\psi \Big (10-\big ({\text{Pe}}\big )^{2}\Big )\Big (10+\alpha ^{\jmath ^2}\Big )Pr\end{aligned}$$

(50)

$$\begin{aligned} \delta _{5}&= {} \alpha ^{\jmath ^2}\Big (10+\alpha ^{\jmath ^2}\Big )\Big [\alpha ^{\jmath ^2}\Big (120+\big ({\text{Pe}}\big )^{4}-10\big ({\text{Pe}}\big )^{2}\Big )+10\big ({\text{Pe}}\big )^{4}\Big ]\end{aligned}$$

(51)

$$\begin{aligned} \gamma _{5}&= {} \alpha ^{\jmath ^2}\Big (10+\alpha ^{\jmath ^2}\Big )\Big (120+\big ({\text{Pe}}\big )^{4}-10\big ({\text{Pe}}\big )^{2}\Big ){\text{LePr}}\phi \end{aligned}$$

(52)

$$\begin{aligned} \delta _{6}&= {} \alpha ^{\jmath ^4}\psi \Big (10-\big ({\text{Pe}}\big )^{2}\Big )\Big (10+ \alpha ^{\jmath ^2}\Big )f_{1}, \gamma _{6}= \alpha ^{\jmath ^4}\psi \Big (10-\big ({\text{Pe}}\big )^{2}\Big )f_{1}Pr\end{aligned}$$

(53)

$$\begin{aligned} \delta _{7}&= {} \alpha ^{\jmath ^4}\Big [\alpha ^{\jmath ^2}\Big (120+\big ({\text{Pe}}\big )^{4}-10\big ({\text{Pe}}\big )^{2}\Big )+10\big ({\text{Pe}}\big )^{4}\Big ]f_{3}\end{aligned}$$

(54)

$$\begin{aligned} \gamma _{7}&= {} \alpha ^{\jmath ^4}\Big (120+\big ({\text{Pe}}\big )^{4}-10\big ({\text{Pe}}\big )^{2}\Big )f_{3}{\text{LePr}}\phi \end{aligned}$$

(55)

$$\begin{aligned} \eta _{1}&= {} 12\Big [ D_\textrm{a}\Big (\alpha ^{\jmath ^4}+20 \alpha ^{\jmath ^2}\Big )+\Big (10+\alpha ^{\jmath ^2}\Big )\Big ]/10\end{aligned}$$

(56)

$$\begin{aligned} \eta _{2}&= {} \Big [ D_\textrm{a}\Big (\alpha ^{\jmath ^4}+20 \alpha ^{\jmath ^2}\Big )+\Big (10+\alpha ^{\jmath ^2}\Big )\Big ]\Big (10+ \alpha ^{\jmath ^2}\Big )/\alpha ^{\jmath ^2}\end{aligned}$$

(57)

$$\begin{aligned} \varphi _{2}&= {} \varphi _{4} = \varphi _{6}=Pr/\Big (10+\alpha ^{\jmath ^2}\Big ),\hspace{0.5em} \varphi _{3} = \varphi _{5} = \varphi _{7} = {\text{LePr}}\phi /\alpha ^{\jmath ^2}\end{aligned}$$

(58)

$$\begin{aligned} \varphi _{1}&= {} \Big (10+\alpha ^{\jmath ^2}\Big )c_\textrm{a}Da/\Big [D_\textrm{a}\Big (\alpha ^{\jmath ^4}+20 \alpha ^{\jmath ^2}\Big )+\Big (10+\alpha ^{\jmath ^2}\Big )\Big ] \end{aligned}$$

(59)