Modal analysis of a fluid flowing over a porous substrate

Abstract

We study the modal stability analysis for a three-dimensional fluid flowing over a saturated porous substrate where the porous medium is assumed to be anisotropic and inhomogeneous. A coupled system of time-dependent evolution equations is formulated in terms of normal velocity, normal vorticity, and fluid surface deformation, respectively, and solved numerically by using the Chebyshev spectral collocation method. Two distinct instabilities, the so-called surface mode instability and the shear mode instability, are identified. Modal stability analysis predicts that the Darcy number has a destabilizing influence on the surface mode instability but has a stabilizing influence on the shear mode instability. Similarly, the surface mode instability intensifies but the shear mode instability weakens with the increase in the value of the coefficient of inhomogeneity. Although the anisotropy parameter shows a stabilizing effect, increasing porosity exhibits a destabilizing effect on the shear mode instability. However, the anisotropy parameter and porosity have no significant impact on the surface mode instability. Spanwise wavenumber is found to have a stabilizing influence on both the surface mode and shear mode instabilities.

Graphical abstract

Appendix A: Squire’s transformation for the gravity-driven fluid flowing over a porous substrate

Using the normal mode solution, the three-dimensional perturbation Eqs. (18)–(31) can be written as

$$\begin{aligned} ik_x{\hat{u}}+\partial _y{\hat{v}}+ik_z{\hat{w}}= & {} 0,~ 0 \le y \le 1,\end{aligned}$$

(A1)

$$\begin{aligned} ik_x(U-c){\hat{u}}+\partial _yU{\hat{v}}+ik_x{\hat{p}}-[\partial _{yy}-(k_x^2+k_z^2)]{\hat{u}}/{\text {Re}}= & {} 0,~ 0 \le y \le 1,\end{aligned}$$

(A2)

$$\begin{aligned} ik_x(U-c){\hat{v}}+\partial _y{\hat{p}}-[\partial _{yy}-(k_x^2+k_z^2)]{\hat{v}}/{\text {Re}}= & {} 0,~0 \le y \le 1, \end{aligned}$$

(A3)

$$\begin{aligned} ik_x(U-c){\hat{w}}+ik_z{\hat{p}}-[\partial _{yy}-(k_x^2+k_z^2)]{\hat{w}}/{\text {Re}}= & {} 0, ~0 \le y \le 1, \end{aligned}$$

(A4)

$$\begin{aligned} ik_{x}{\hat{u}}_p+\partial _y{\hat{v}}_p+ik_{z}{\hat{w}}_p= & {} 0,~~ -\delta \le y \le 0,\end{aligned}$$

(A5)

$$\begin{aligned} -(1/\varepsilon )ik_{x}c{\hat{u}}_p+ik_{x}{\hat{p}}_p+[1/({\text {Re~Da}}\,\eta _x)]{\hat{u}}_p= & {} 0,~~ -\delta \le y \le 0,\end{aligned}$$

(A6)

$$\begin{aligned} -(1/\varepsilon )ik_{x}c{\hat{v}}_p+\partial _y{\hat{p}}_p+[\xi /({\text {Re~Da}}\,\eta _y)]{\hat{v}}_p= & {} 0,~~ -\delta \le y \le 0,\end{aligned}$$

(A7)

$$\begin{aligned} -(1/\varepsilon )ik_{x}c{\hat{w}}_p+ik_{z}{\hat{p}}_p+[1/({\text {Re~Da}}\,\eta _z)]{\hat{w}}_p= & {} 0,~~ -\delta \le y \le 0, \end{aligned}$$

(A8)

$$\begin{aligned} {\hat{v}}_p=0,{} & {} ~~\textrm{at}~~y=-\delta ,\end{aligned}$$

(A9)

$$\begin{aligned} \partial _y{\hat{u}}=(\alpha _{BJ}/\sqrt{{\text {Da}}\,\eta _x})({\hat{u}}-{\hat{u}}_p),{} & {} ~~ \textrm{at}~~ y=0,\end{aligned}$$

(A10)

$$\begin{aligned} \partial _y{\hat{w}}=(\alpha _{BJ}/\sqrt{{\text {Da}}\,\eta _z})({\hat{w}}-{\hat{w}}_p),{} & {} ~~ \textrm{at}~~ y=0,\end{aligned}$$

(A11)

$$\begin{aligned} {\hat{v}}={\hat{v}}_p, ~~ {\hat{p}}={\hat{p}}_p,{} & {} ~~\textrm{at}~~ y=0,\end{aligned}$$

(A12)

$$\begin{aligned} \partial _y{\hat{u}}+ik_x{\hat{v}}+{\hat{h}}\partial _{yy}U=0,{} & {} ~~ \textrm{at}~~ y=1,\end{aligned}$$

(A13)

$$\begin{aligned} \partial _y{\hat{w}}+ik_z{\hat{v}}=0,{} & {} ~~ \textrm{at}~~ y=1,\end{aligned}$$

(A14)

$$\begin{aligned} -{\hat{p}}+(2/{\text {Re}}) \partial _y{\hat{v}}=-{\text {We}}(k_x^2+k_z^2){\hat{h}}-(\cos \theta /{\text {Fr}}^2){\hat{h}},{} & {} ~~ \textrm{at}~~ y=1,\end{aligned}$$

(A15)

$$\begin{aligned} ik_x(U-c){\hat{h}}={\hat{v}},{} & {} ~~ \textrm{at}~~ y=1. \end{aligned}$$

(A16)

Now, we use the following extended Squire’s transformations [15, 49]: $k_x{\hat{u}}+k_z{\hat{w}}={\tilde{k}}{\tilde{u}}$, ${\hat{v}}={\tilde{v}}$, ${\tilde{k}}{\hat{p}}=k_x{\tilde{p}}$, ${\tilde{k}}=\sqrt{k_x^2+k_z^2}$, $c={\tilde{c}}$, $k_x{\text {Re}}={\tilde{k}}{\tilde{{\text {Re}}}}$, $k_{x}{\hat{u}}_p+k_{z}{\hat{w}}_p={\tilde{k}}{\tilde{u}}_p$, ${\hat{v}}_p={\tilde{v}}_p$, ${\tilde{k}}{\hat{p}}_p=k_x{\tilde{p}}_p$, Da $={\tilde{{\text {Da}}}}$, $\varepsilon ={\tilde{\varepsilon }}$, $\eta _x={\tilde{\eta }}_x$, $\eta _y={\tilde{\eta }}_y$, $\xi ={\tilde{\xi }}$, $\alpha _{BJ}={\tilde{\alpha }}_{BJ}$, ${\tilde{k}}^2 {\text {We}} =k_x^2 {\tilde{{\text {We}}}}$, $k_x^2 {\text {Fr}}^2={\tilde{k}}^2{\tilde{{\text {Fr}}}}^2$, and $k_x{\hat{h}}={\tilde{k}}{\tilde{h}}$. Using the above transformations, one can derive a similar set of perturbation equations for the two-dimensional disturbance with a normal mode solution ($\propto \exp [i{\tilde{k}}(x-{\tilde{c}}t)]$)

$$\begin{aligned} i{\tilde{k}}{\tilde{u}}+\partial _y{\tilde{v}}= & {} 0,~~ 0 \le y \le 1,\end{aligned}$$

(A17)

$$\begin{aligned} i{\tilde{k}}(U-{\tilde{c}}){\tilde{u}}+\partial _yU{\tilde{v}}+i{\tilde{k}}{\tilde{p}} -(\partial _{yy}-{\tilde{k}}^2){\tilde{u}}/{\tilde{{\text {Re}}}}= & {} 0,~~ 0 \le y \le 1,\end{aligned}$$

(A18)

$$\begin{aligned} i{\tilde{k}}(U-{\tilde{c}}){\tilde{v}}+\partial _y{\tilde{p}}+(\partial _{yy} -{\tilde{k}}^2){\tilde{v}}/{\tilde{{\text {Re}}}}= & {} 0,~~0 \le y \le 1, \end{aligned}$$

(A19)

$$\begin{aligned} i{\tilde{k}}{\tilde{u}}_p+\partial _y{\tilde{v}}_p= & {} 0,~~ -\delta \le y \le 0,\end{aligned}$$

(A20)

$$\begin{aligned} -(1/{\tilde{\varepsilon }})i{\tilde{k}}{\tilde{c}}{\tilde{u}}_p+i{\tilde{k}}{\tilde{p}}_p +[1/({\tilde{{\text {Re}}}}\,{\tilde{{\text {Da}}}}\,{\tilde{\eta }}_x)]{\tilde{u}}_p= & {} 0,~~ -\delta \le y \le 0,\end{aligned}$$

(A21)

$$\begin{aligned} -(1/{\tilde{\varepsilon }})i{\tilde{k}}{\tilde{c}}{\tilde{v}}_p+\partial _y{\tilde{p}}_p +[{\tilde{\xi }}/({\tilde{{\text {Re}}}}\,{\tilde{{\text {Da}}}}\,{\tilde{\eta }}_y)]{\tilde{v}}_p= & {} 0,~~ -\delta \le y \le 0, \end{aligned}$$

(A22)

$$\begin{aligned} {\tilde{v}}_p= 0,{} & {} ~~\textrm{at}~~ y=-\delta ,\end{aligned}$$

(A23)

$$\begin{aligned} \partial _y{\tilde{u}}={\tilde{\alpha }}_{BJ}/\sqrt{{\tilde{{\text {Da}}}}\,{\tilde{\eta }}_x} ({\tilde{u}}-{\tilde{u}}_p),{} & {} ~~ \textrm{at}~~ y=0,\end{aligned}$$

(A24)

$$\begin{aligned} {\tilde{v}}={\tilde{v}}_p,~~{\tilde{p}}={\tilde{p}}_p,{} & {} ~~ \textrm{at}~~ y=0,\end{aligned}$$

(A25)

$$\begin{aligned} \partial _y{\tilde{u}}+i{\tilde{k}}{\tilde{v}}+{\tilde{h}}\partial _{yy}U = 0,{} & {} ~~ \textrm{at}~~ y=1,\end{aligned}$$

(A26)

$$\begin{aligned} -{\tilde{p}}+(2/{\tilde{{\text {Re}}}})\partial _y{\tilde{v}}=-{\tilde{{\text {We}}}}\, {\tilde{k}}^2{\tilde{h}}-(\cos \theta /{\tilde{{\text {Fr}}}}^2){\tilde{h}},{} & {} ~~ \textrm{at}~~ y=1,\end{aligned}$$

(A27)

$$\begin{aligned} i{\tilde{k}}(U-{\tilde{c}}){\tilde{h}}={\tilde{v}},{} & {} ~~ \textrm{at}~~ y=1, \end{aligned}$$

(A28)

where ${\tilde{k}}$ is the wavenumber and ${\tilde{c}}$ is the wave speed of the two-dimensional disturbance. Here, all ‘tilde’ quantities represent the flow variables when the infinitesimal disturbance is two-dimensional. Clearly, the Reynolds number ${\tilde{{\text {Re}}}}$ for the two-dimensional disturbance is less than the Reynolds number Re for the three-dimensional disturbance because ${\tilde{{\text {Re}}}}=(k_x/{\tilde{k}}){\text {Re}} < {\text {Re}}$, where $k_z \ne 0$. Therefore, the modal instability corresponding to the two-dimensional infinitesimal disturbance occurs at a lower Reynolds number than that of the three-dimensional infinitesimal disturbance. In other words, compared to three-dimensional disturbance, one can infer that two-dimensional disturbance is more unstable.