Thin film flow down a porous substrate in the presence of an insoluble surfactant: Stability analysis

Anjalaiah,; Usha, R.; Millet, Séverine

doi:10.1063/1.4789459

Cited by 29 publications

(17 citation statements)

References 46 publications

(89 reference statements)

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“…However, the similar modes are observed by Anjalaiah et al. [16] while analyzing the effect of insoluble surfactant on a thin film flow down a porous plane. Instead of shear mode, stabilized Marangoni mode occurs due to the presence of insoluble surfactant.…”

Section: Introductionsupporting

confidence: 78%

“…The governing Navier–Stokes equations for the liquid layer is given by (following Pascal [23] and Anjalaiah et al. [16]),

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

ρ [\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}] = - \frac{\partial p}{\partial x} + μ [\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}] + ρ g \sin θ,

ρ [\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}] = - \frac{\partial p}{\partial y} + μ [\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}] - ρ g \cos θ,

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Differently, the equations governing the momentum transport through the porous layer use the Darcy–Brinkman approximation (Anjalaiah et al. [16]) and are,

\frac{\partial u_{p}}{\partial x_{p}} + \frac{\partial v_{p}}{\partial y_{p}} = 0,

\frac{ρ}{χ} \frac{\partial u_{p}}{\partial t_{p}} + \frac{μ}{K} u_{p} = - \frac{\partial p_{p}}{\partial x_{p}} + μ_{e} [\frac{\partial^{2} u_{p}}{\partial x_{p}^{2}} + \frac{\partial^{2} u_{p}}{\partial y_{p}^{2}}] + ρ g \sin θ,

\frac{ρ}{χ} \frac{\partial v_{p}}{\partial t_{p}} + \frac{μ}{K} v_{p} = - \frac{\partial p_{p}}{\partial y_{p}} + μ_{e} [\frac{\partial^{2} v_{p}}{\partial x_{p}^{2}} + \frac{\partial^{2} v_{p}}{\partial y_{p}^{2}}] - ρ g \cos θ,

where the subscript “ p ” prevails the porous medium. Note that, the flow in the porous region is assumed to be slow enough to neglect inertia (Anjalaiah et al.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Note that, the flow in the porous region is assumed to be slow enough to neglect inertia (Anjalaiah et al. [16] and Nield and Bejan [32]). The components of filtration velocity in the porous layer are indicated by

u_{p}

and

v_{p}

, respectively and

p_{p}

is the pressure.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Later on, several mechanisms have been employed to control the surface wave instability of fluid flows by employing either surface active or inactive agents depending on the applications (Li and Pozrikidis [14], Blyth and Pozrikidis [15], and Anjalaiah et al. [16]).…”

Section: Introductionmentioning

confidence: 99%

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Effect of a floating elastic membrane for stabilizing the film flow down a porous inclined plane

2020

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The effect of a floating elastic membrane on a gravity‐driven flow down an inclined porous plane is examined. The present work extends the earlier work of Anjalaiah et al. [Phys. Fluid, Vol. 25, 022101 (2013)] by including the thin membrane and excluding the insoluble surfactant at the top surface of the flow. The membrane tension emergence from the elasticity, is used to monitoring the surface tension on the top surface of the fluid layer. The fluid flow below the elastic membrane is governed by the Navier–Stokes equations and flow through the porous medium is governed by the Darcy–Brickmann equations. Using the normal mode analysis, two Orr–Sommerfeld systems are derived for each layers and solved by using the spectral collocation method to capture the linear instability. Further, the temporal growth rate and marginal stability curve are analyzed for different set of structural parameters. In general, the instability of free surface flow is more due to the presence of the unstable Yih mode. Both, the membrane tension and the uniform mass distribution rate along the length of the membrane assure to stabilize the surface waves by lowering the temporal growth rate of the disturbances. Consequently, the floating elastic membrane and the porosity of the inclined substrate pretend to suppress the unstable Yih mode and stabilize the flow anatomy. Hence, the floating elastic membrane can be disposed for the lethargic control of the free‐surface flows' instability.

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Section: Introductionsupporting

confidence: 78%

“…The governing Navier–Stokes equations for the liquid layer is given by (following Pascal [23] and Anjalaiah et al. [16]),

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

ρ [\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}] = - \frac{\partial p}{\partial x} + μ [\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}] + ρ g \sin θ,

ρ [\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}] = - \frac{\partial p}{\partial y} + μ [\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}] - ρ g \cos θ,

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Differently, the equations governing the momentum transport through the porous layer use the Darcy–Brinkman approximation (Anjalaiah et al. [16]) and are,

\frac{\partial u_{p}}{\partial x_{p}} + \frac{\partial v_{p}}{\partial y_{p}} = 0,

\frac{ρ}{χ} \frac{\partial u_{p}}{\partial t_{p}} + \frac{μ}{K} u_{p} = - \frac{\partial p_{p}}{\partial x_{p}} + μ_{e} [\frac{\partial^{2} u_{p}}{\partial x_{p}^{2}} + \frac{\partial^{2} u_{p}}{\partial y_{p}^{2}}] + ρ g \sin θ,

\frac{ρ}{χ} \frac{\partial v_{p}}{\partial t_{p}} + \frac{μ}{K} v_{p} = - \frac{\partial p_{p}}{\partial y_{p}} + μ_{e} [\frac{\partial^{2} v_{p}}{\partial x_{p}^{2}} + \frac{\partial^{2} v_{p}}{\partial y_{p}^{2}}] - ρ g \cos θ,

where the subscript “ p ” prevails the porous medium. Note that, the flow in the porous region is assumed to be slow enough to neglect inertia (Anjalaiah et al.…”