Dynamics of an Electrified Multi-layer Film Down a Porous Incline
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ORIGINAL ARTICLE
Dynamics of an Electrified Multi-layer Film Down a Porous Incline Magdy A. Sirwah1 · Ahmed Assaf2 Received: 28 February 2020 / Accepted: 6 October 2020 / Published online: 5 November 2020 © Springer Nature B.V. 2020
Abstract The nonlinear dynamics of gravity-driven two-layers flow through an oblique microchannel of porous walls subjected to an external electric potential is examined. The evolution equation governing the surface wave deflection is derived in the frame of long wave theory. The stability criteria of the linearized system are investigated. As permeability, inclination or dielectric constant increase, the disturbances become stronger. However, the viscosity ratio plays an irregular role on the stability. Resonant waves propagating on the fluid interface are introduced. The instability of the base flow is simulated. It is observed that the instability onset can be controlled by many physical properties related with the model. The effect of permeability as well as dielectric constant corresponds to linear processing expectations. Viscosity ratio improves stability in certain situations. However, the electric role is generally dominant in the current model. Such results may be useful in practical applications by designing a device in order to control the instability. Solitary waves propagating on the interface are studied. The presence of stable stationary solitons is shown in certain statuses of the model. Keywords Long-wave technique · Multilayer flow · Porous walls · Resonance · Solitons
Introduction In practical processes, the surface of a thin layer of liquid is required to be flat, such as in coating and painting applications. However, the instability is required in another applications like liquids cooling (Anjalaiah, Usha and Millet 2013). Moreover, another potential application is in slow flow of crude oil over long distances through pipes, where the flow can be considered inertialess, the power needed to pump the viscous liquid may be reduced by injecting water to represent a permanent lubrication film which isolates the oil from the channel wall (Papanastasiou et al. 2000). Thin film flows approximation is suitable to describe processing of several materials, like composites, polymers
Ahmed Assaf
[email protected] Magdy A. Sirwah [email protected] 1
Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt
2
Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt
and metals which assume liquid form (Papanastasiou et al. 2000). longwave flows are applicable for thin films or subject to microgravity conditions. Sometimes it is called small wavenumber or lubrication approximation (DavalosOrozco 2018; Nepomnyashchy et al. 2019). The last term characterizes flow in which normal velocity gradients are ∂u dominant; i.e. ∂x ∂u ∂y , whereas in extensional or stretching flows the parallel velocity gradients dominate; i.e. ∂u ∂u ∂x ∂y (Papanastasiou et al. 2000). For such reasons, the subject of thin layer flow
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