Linear stability of a contaminated fluid flow down a slippery inclined plane

The linear stability analysis of a fluid flow down a slippery inclined plane is carried out when the free surface of the fluid is contaminated by a monolayer of insoluble surfactant. The aim is to extend the earlier study [Samanta et al., J. Fluid Mech. 684, 353 (2011)] for low to high values of the Reynolds number in the presence of an insoluble surfactant. The Orr-Sommerfeld equation (OSE) is derived for infinitesimal disturbances of arbitrary wave numbers. At low Reynolds number, the OSE is solved analytically by using the long-wave analysis, which shows that the critical Reynolds number decreases in the presence of a slippery plane but increases in the presence of an insoluble surfactant. This fact ensures a destabilizing effect of wall slip and a stabilizing effect of insoluble surfactant on the long-wave surface mode. Further, the Chebyshev spectral collocation method is implemented to tackle the OSE equation numerically for an arbitrary value of the Reynolds number, or equivalently, for an arbitrary value of the wave number. At moderate Reynolds number, wall slip exhibits a stabilizing effect on the surface mode as opposed to the result in the long-wave regime, while the insoluble surfactant exhibits a stabilizing effect on the surface mode as in the result of the long-wave regime. On the other hand, at high Reynolds number, both wall slip and insoluble surfactant exhibit a stabilizing effect on the shear mode. Further, it is shown that both surface and shear modes compete with each other to dominate the primary instability once the inclination angle is sufficiently small. In addition, new phase boundaries are identified to differentiate the regimes of surface and shear modes.

Figure 1

Schematic diagram of a surfactant-laden fluid flow down a slippery inclined plane.

Figure 2

(a) The variation of the streamwise base velocity $U/U_s$ as a function of $y/d$. (b) The variation of the shear stress $(dU/dy)\times (d/U_s)$ as a function of $y/d$. Here solid, dashed, and dotted lines stand for $\beta =0,\beta =0.04$, and $\beta =0.08$, respectively.

Figure 3

The variation of the critical Reynolds number ${\mathrm{Re}}_{\mathrm{c}}$ for the surface mode with the slip length $\beta$ when $\mathrm{Ma}=1,\mathrm{Ca}=2$, and $\theta =\pi /4$. The stars connected by straight line segments represent the present result. The circle is the result of Blyth and Pozrikidis [25]. The solid points connected by the blue line segments are the results of Samanta et al. [21] without surfactant, and the green diamond is the result of Yih [8].

Figure 4

(a) The variation of the neutral curve for the surface mode in the ($\mathrm{Re},k$) plane for different values of the slip length when $\theta =\pi /4,\mathrm{Ca}=2,\mathrm{Pe}=2$, and $\mathrm{Ma}=1$. Here “U” and “S” indicate unstable and stable regions, respectively. (b) The variation of the temporal growth rate $k{c}_i$ for the surface mode with wave number $k$ when $\mathrm{Re}=2.5$ (top) and $\mathrm{Re}=15$ (bottom). Here solid, dashed, and dotted lines stand for $\beta =0,\beta =0.04$, and $\beta =0.08$, respectively. The circles show the results of Blyth and Pozrikidis [25].

Figure 5

(a) The variation of the neutral curve for the surface mode in the ($\mathrm{Re},k$) plane for different values of the Marangoni number when $\theta =\pi /4,\mathrm{Ca}=2,\mathrm{Pe}=2$, and $\beta =0.04$. Here “U” and “S” indicate unstable and stable regions, respectively. (b) The variation of the temporal growth rate $k{c}_i$ for the surface mode with wave number $k$ when $\mathrm{Re}=15$. Here solid, dashed, and dotted lines stand for $\mathrm{Ma}=1,\mathrm{Ma}=2$, and $\mathrm{Ma}=3$, respectively. (c) The variation of the spatial growth rate $-k_i$ for the surface mode with the wave number $k_r$ when $\theta =4^{\circ },\beta =0,\mathrm{Ka}=13.3$, and $\mathrm{Re}=20$. Here solid, dashed, and dotted lines stand for $\mathrm{Ma}=0,\mathrm{Ma}=0.05$, and $\mathrm{Ma}=0.1$, respectively. The circles are the results of Samanta et al. [21]. (d) The variation of the spatial growth rate $-k_i$ for the surface mode with the wave number $k_r$ when $\theta =4^{\circ },\beta =0,\mathrm{Ka}=13.3$, and $\mathrm{Re}=40$. Here solid, dashed, and dotted lines stand for $\mathrm{Ma}=0,\mathrm{Ma}=1$, and $\mathrm{Ma}=2$, respectively.

Figure 6

(a) The variation of the neutral curve for the surface mode in the ($\mathrm{Re},k$) plane for different values of the Péclet number when $\theta =\pi /4,\mathrm{Ca}=2,\mathrm{Ma}=1$, and $\beta =0.04$. Here solid, dashed, dotted, and dash-dotted lines stand for $\mathrm{Pe}=2,\mathrm{Pe}=10,\mathrm{Pe}=20$, and $\mathrm{Pe}=100$, respectively. Here “U” and “S” indicate unstable and stable regions, respectively. (b) The variation of the temporal growth rate $k{c}_i$ for the surface mode with wave number $k$ when $\mathrm{Re}=5$ (top). The variation of the temporal growth rate $k{c}_i$ for the surface mode with wave number $k$ when $\mathrm{Re}=60$ (bottom). Here solid, dashed, dotted, and dash-dotted lines stand for $\mathrm{Pe}=2,\mathrm{Pe}=10,\mathrm{Pe}=20$, and $\mathrm{Pe}=100$, respectively.

Figure 7

(a) The variation of the spatial growth rate $-k_i$ for the surface mode with the wave number $k_r$ when $\theta =\pi /4,\beta =0.04,\mathrm{Ca}=2,\mathrm{Ma}=1$, and $\mathrm{Re}=5$. Here solid, dashed, dotted, and dash-dotted lines stand for $\mathrm{Pe}=2,\mathrm{Pe}=10,\mathrm{Pe}=20$, and $\mathrm{Pe}=100$, respectively. (b) The variation of the spatial growth rate $-k_i$ for the surface mode with the wave number $k_r$ when $\theta =\pi /4,\beta =0.04,\mathrm{Ca}=2,\mathrm{Ma}=1$, and $\mathrm{Re}=60$. Here solid, dashed, dotted, and dash-dotted lines stand for $\mathrm{Pe}=2,\mathrm{Pe}=10,\mathrm{Pe}=20$, and $\mathrm{Pe}=100$, respectively.

Figure 8

The variation of the relative error $E_N$ with the number of Chebyshev polynomials $N$ when $\theta =4^{\circ },\mathrm{Ca}=2,\mathrm{Re}={10}^4$, and $k=1$. (Top) Solid, cross, and star points stand for $\beta =0,\beta =0.04$, and $\beta =0.08$, respectively, when $\mathrm{Ma}=0$ and $\mathrm{Pe}=2$. (Middle) Solid, cross, and star points stand for $\mathrm{Ma}=0,\mathrm{Ma}=1$, and $\mathrm{Ma}=3$, respectively, when $\beta =0$ and $\mathrm{Pe}=2$. (Bottom) Solid, cross, and star points stand for $\mathrm{Pe}=2,\mathrm{Pe}=10$, and $\mathrm{Pe}=20$, respectively, when $\beta =0$ and $\mathrm{Ma}=1$.

Figure 9

(a) The spectrum of the eigenvalue problem (75) when $\mathrm{Re}=21000,k=1.2,\theta =1^{\prime },\mathrm{Ka}=51269.45$, and $\mathrm{Ma}=0$. Here circles, stars and squares represent the results of $\beta =0,\beta =0.02$, and $\beta =0.03$, respectively. (b) The spectrum of the eigenvalue problem (75) when $\mathrm{Re}=7000,k=2.3,\theta =1^{\prime },\mathrm{Ka}=51269.45$, and $\beta =0$. Here circles, stars, and squares represent the results of $\mathrm{Ma}=0,\mathrm{Ma}=10$, and $\mathrm{Ma}=20$, respectively.

Figure 10

(a) The variation of the neutral curve for the shear mode in the ($\mathrm{Re},k$) plane for different values of the slip length when $\theta =1^{\circ },\mathrm{Ka}=13096.3$, and $\mathrm{Ma}=0$. Solid, dashed, and dotted lines stand for $\beta =0,\beta =0.01$, and $\beta =0.02$, respectively. Here “U” and “S” indicate unstable and stable regions, respectively. The circles are the results of Chin et al. [11]. (b) The variation of the temporal growth rate $k{c}_i$ for the shear mode with the wave number $k$ when $\mathrm{Re}=15000$. Solid, dashed, and dotted lines stand for $\beta =0,\beta =0.01$, and $\beta =0.02$, respectively.

Figure 11

(a) The variation of the neutral curve for the shear mode in the ($\mathrm{Re},k$) plane for different values of the Marangoni number when $\theta =1^{\prime },\mathrm{Ka}=51269.45$, and $\beta =0$. Solid, dashed, and dotted lines stand for $\mathrm{Ma}=0,\mathrm{Ma}=10$, and $\mathrm{Ma}=20$, respectively. Here “U” and “S” indicate unstable and stable regions, respectively. The circles are the results of Floryan et al. [12]. (b) The variation of the temporal growth rate $k{c}_i$ for the shear mode with the wave number $k$ when $\mathrm{Re}=7400$. Solid, dashed, and dotted lines stand for $\mathrm{Ma}=0,\mathrm{Ma}=10$, and $\mathrm{Ma}=20$, respectively.

Figure 12

(a) The variation of the neutral curve for the shear mode in the ($\mathrm{Re},k$) plane for different values of the Péclet number when $\theta =1^{\prime },\mathrm{Ma}=1,\mathrm{Ca}=2$, and $\beta =0$. Solid, dashed, and dotted lines stand for $\mathrm{Pe}=1,\mathrm{Pe}=2$, and $\mathrm{Pe}=3$, respectively. (b) The variation of the temporal growth rate $k{c}_i$ for the shear mode with the wave number $k$ when $\mathrm{Re}=5450$. Solid, dashed, and dotted lines stand for $\mathrm{Pe}=1,\mathrm{Pe}=2$, and $\mathrm{Pe}=3$, respectively. (c) The variation of the neutral curve for the shear mode in the ($\mathrm{Re},k$) plane for different values of the Péclet number when $\theta =1^{\prime },\mathrm{Ma}=1,\mathrm{Ca}=2$, and $\beta =0$. Solid, dashed, dotted, dash-dotted, and thin lines stand for $\mathrm{Pe}=10,\mathrm{Pe}=20,\mathrm{Pe}=50,\mathrm{Pe}=80$, and $\mathrm{Pe}=100$, respectively. (d) The variation of the temporal growth rate $k{c}_i$ for the shear mode with the wave number $k$ when $\mathrm{Re}=5450$. Solid, dashed, dotted, dash-dotted, and thin lines stand for $\mathrm{Pe}=10,\mathrm{Pe}=20,\mathrm{Pe}=50,\mathrm{Pe}=80$, and $\mathrm{Pe}=100$, respectively. Here “U” and “S” indicate unstable and stable regions, respectively.

Figure 13

(a) The variation of the neutral curves for the surface and shear modes in the ($\mathrm{Re},k$) plane for different values of the slip length when $\theta =3^{\prime },\mathrm{Ka}=35548.25$, and $\mathrm{Ma}=0$. Solid, dashed, dotted, and dash-dotted lines stand for $\beta =0,\beta =0.01,\beta =0.02$, and $\beta =0.03$, respectively. The circles are results of Floryan et al. [12]. (b) The variation of the neutral curves for the surface and shear modes in the ($\mathrm{Re},k$) plane for different values of the slip length when $\theta =0.{25}^{\prime },\mathrm{Ka}=81385.18$, and $\mathrm{Ma}=0$. Solid, dashed, dotted, and dash-dotted lines stand for $\beta =0,\beta =0.01,\beta =0.02$, and $\beta =0.03$, respectively. (c) The critical Reynolds numbers ${\mathrm{Re}}_{\mathrm{c}}$ corresponding to the surface and shear modes. (d) The phase boundary between the surface and shear modes.

Figure 14

(a) The variation of the neutral curves for the surface and shear modes in the ($\mathrm{Re},k$) plane for different values of the Marangoni number when $\theta =1^{\prime },\mathrm{Ka}=51269.45$, and $\beta =0$. Solid, dashed, dotted, and dash-dotted lines stand for $\mathrm{Ma}=0,\mathrm{Ma}=0.5,\mathrm{Ma}=1$, and $\mathrm{Ma}=10$, respectively. The circles are results of Floryan et al. [12]. (b) The variation of the neutral curves for the surface and shear modes in the ($\mathrm{Re},k$) plane for different values of the Marangoni number when $\theta =0.{25}^{\prime },\mathrm{Ka}=81385.18$, and $\beta =0$. Solid, dashed, and dotted lines stand for $\mathrm{Ma}=0,\mathrm{Ma}=0.5,\text{and}\mathrm{Ma}=1$, respectively. (c) The critical Reynolds numbers ${\mathrm{Re}}_{\mathrm{c}}$ corresponding to the surface and shear modes. (d) The phase boundary between the surface and shear modes.

Figure 15

(a) The spectrum of the eigenvalue problem (75) for different values of the slip length when $k=0.9,\mathrm{Re}=30000,\theta =3^{\prime },\mathrm{Ka}=35548.25$, and $\mathrm{Ma}=0$. Here circles, stars, and squares represent the results of $\beta =0,\beta =0.02$, and $\beta =0.03$, respectively. (b) The variation of the absolute value of the normalized eigenfunction $\left|\varphi \right|$ as a function of $y$ when $\beta =0.02$. The solid and dashed lines represent the eigenfunctions corresponding to the shear and surface modes, respectively.

Figure 16

(a) The spectrum of the eigenvalue problem (75) for different values of the slip length when $k=0.8,\mathrm{Re}=80000,\theta =1^{\prime },\mathrm{Ka}=51269.4$, and $\beta =0$. Here circles, stars, and squares represent the results of $\mathrm{Ma}=0,\mathrm{Ma}=0.5$, and $\mathrm{Ma}=1$, respectively. (b) The variation of the absolute value of the normalized eigenfunction $\left|\varphi \right|$ as a function of $y$ when $\mathrm{Ma}=1$. The solid and dashed lines represent the eigenfunctions corresponding to the shear and surface modes, respectively.