6 results
Role of odd viscosity in falling viscous fluid
- Arghya Samanta
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- Journal:
- Journal of Fluid Mechanics / Volume 938 / 10 May 2022
- Published online by Cambridge University Press:
- 09 March 2022, A9
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The aim of the present study is to investigate the linear and nonlinear wave dynamics of a falling incompressible viscous fluid when the fluid undergoes an effect of odd viscosity. In fact, such an effect arises in classical fluids when the time-reversal symmetry is broken. The motivation to study this dynamics was raised by recent studies (Ganeshan & Abanov, Phys. Rev. Fluids, vol. 2, 2017, p. 094101; Kirkinis & Andreev, J. Fluid Mech., vol. 878, 2019, pp. 169–189) where the odd viscosity coefficient suppresses thermocapillary instability. Here, we explore the linear surface wave and shear wave dynamics for the isothermal case by solving the Orr–Sommerfeld eigenvalue problem numerically with the aid of the Chebyshev spectral collocation method. It is found that surface and shear instabilities can be weakened by the odd viscosity coefficient. Furthermore, the growth rate of the wavepacket corresponding to the linear spatio-temporal response is reduced as long as the odd viscosity coefficient increases. In addition, a coupled system of a two-equation model is derived in terms of the fluid layer thickness $h(x,t)$ and the flow rate $q(x,t)$. The nonlinear travelling wave solution of the two-equation model reveals the attenuation of maximum amplitude and speed in the presence of an odd viscosity coefficient, which ensures the delay of transition from the primary parallel flow with a flat surface to secondary flow generated through the nonlinear wave interactions. This physical phenomenon is further corroborated by performing a nonlinear spatio-temporal simulation when a harmonic forcing is applied at the inlet.
Effect of surfactant on the linear stability of a shear-imposed fluid flowing down a compliant substrate
- Arghya Samanta
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- Journal:
- Journal of Fluid Mechanics / Volume 920 / 10 August 2021
- Published online by Cambridge University Press:
- 10 June 2021, A23
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We study the linear stability of a surfactant-laden shear-imposed fluid flowing down a compliant substrate. The aim is to extend the earlier and recent studies (Carpenter & Garrad, J. Fluid Mech., vol. 155, 1985, pp. 465–510; Alexander et al., J. Fluid Mech., vol. 900, 2020, A40) in the presence of insoluble surfactant when an external streamwise imposed shear stress acts at the fluid surface. In other words, the current study expands the earlier study (Wei, Phys. Fluids, vol. 17, 2005, 012103) in the presence of a flexible substrate. The Orr–Sommerfeld-type boundary value problem is derived and solved by using the long-wave series expansion as well as the Chebyshev spectral collocation method for disturbances of arbitrary wavenumbers. The long-wave result reveals the existence of two dominant temporal modes, the so-called surface mode and surfactant mode, where the surface mode propagates faster than the surfactant mode. It is found that the surface mode can be stabilized by introducing an insoluble surfactant at the fluid surface even though the spring stiffness $C_K$ keeps a lower value than its critical value $C_K^*$. But the imposed shear stress exhibits a dual role in the surface mode in two different regimes of spring stiffness $C_K$, i.e. a stabilizing effect when $C_K< C_K^*$ and a destabilizing effect when $C_K>C_K^*$. Further, the surfactant mode becomes more unstable with the increasing values of spring stiffness $C_K$ and damping coefficient $C_D$. On the other hand, the numerical result in the arbitrary wavenumber regime reveals the existence of subcritical instability induced by the surface mode. Furthermore, a different temporal mode, the so-called wall mode, appears in the finite wavenumber regime for special values of $C_K$ and $C_D$, which becomes weaker with increasing values of the wall parameters $C_K$, $C_D$, $C_B$ and $C_T$, but becomes stronger with increasing values of the inclination angle $\theta$ and wall parameter $C_I$. Moreover, the temporal growth rate associated with the wall mode enhances with the increasing value of the Marangoni number but attenuates with the increasing value of imposed shear stress. In addition, another temporal mode, the so-called shear mode, emerges in the finite wavenumber regime when the Reynolds number is high and the inclination angle is small. The unstable region generated by the shear mode magnifies with the increasing value of the imposed shear stress but decays with the increasing value of Marangoni number. Further, the shear mode becomes more unstable as soon as the spring stiffness $C_K$ and damping coefficient $C_D$ increase.
Instability of a shear-imposed flow down a vibrating inclined plane
- Arghya Samanta
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- Journal:
- Journal of Fluid Mechanics / Volume 915 / 25 May 2021
- Published online by Cambridge University Press:
- 24 March 2021, A93
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A linear stability of a shear-imposed viscous liquid flowing down a vibrating inclined plane is deciphered for disturbances of arbitrary wavenumbers. The main purpose of this study is to expand the model of Woods & Lin (J. Fluid Mech., vol. 294, 1995, pp. 391–407) for a shear-imposed flow (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485) when the inclined plane oscillates in streamwise and cross-stream directions, respectively. The time-dependent Orr–Sommerfeld-type boundary value problem is derived and solved numerically based on the Chebyshev spectral collocation method along with Floquet theory. Numerical results corresponding to the cross-stream oscillation disclose that there exist three different types of instabilities, the so-called gravitational, subharmonic and harmonic instabilities, which can be resonated in separate unstable ranges of wavenumber by varying the amplitude of cross-stream oscillation. In fact, the subharmonic and harmonic resonances occur once the forcing amplitude exceeds the respective critical amplitudes for the subharmonic and harmonic instabilities. At low Reynolds number, the subharmonic resonance excited at low forcing amplitude intensifies but attenuates in the presence of imposed shear stress when the forcing amplitude is high. However, the harmonic resonance excited solely at high forcing amplitude intensifies in the presence of imposed shear stress. In contrast, at moderate Reynolds number, the subharmonic resonance excited at low forcing amplitude can be weakened by incorporating an imposed shear stress at the liquid surface. Furthermore, at high Reynolds number, a new instability, the so-called shear instability, arises along with the aforementioned three instabilities and becomes stronger in the presence of imposed shear stress. However, the gravitational and shear instabilities become weaker as soon as the forcing amplitude of cross-stream oscillation increases. On the other hand, numerical results for a streamwise oscillatory flow reveal that there exist three distinct unstable zones separated by stable ranges of Reynolds number. The resonated unstable zone induced by the streamwise oscillation attenuates, but the unstable zone responsible for the gravitational instability enhances in the presence of imposed shear stress. As soon as the Reynolds number is large and the inclination angle is sufficiently small, a new instability, the so-called shear instability, occurs in the finite wavenumber regime along with the resonated and gravitational instabilities. Further, the shear instability also intensifies in the presence of imposed shear stress for a streamwise oscillatory flow.
Linear stability of a viscoelastic liquid flow on an oscillating plane
- Arghya Samanta
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- Journal:
- Journal of Fluid Mechanics / Volume 822 / 10 July 2017
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- 31 May 2017, pp. 170-185
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Linear stability of a viscoelastic liquid on an oscillating plane is studied for disturbances of arbitrary wavenumbers. The main aim is to extend the earlier study of Dandapat & Gupta (J. Fluid Mech., vol. 72, 1975, pp. 425–432) to the finite wavenumber regime, which has not been attempted so far in the literature. The Orr–Sommerfeld boundary value problem is formulated for an unsteady base flow, and it is resolved numerically based on the Chebyshev spectral collocation method along with the Floquet theory. The analytical solution predicts that U-shaped unstable regions appear in the separated bandwidths of the imposed frequency, and the dominant mode of the long-wave instability intensifies in the presence of the viscoelastic parameter. The numerical solution shows that oblique neutral curves come out from the branch points of the U-shaped neutral curves at finite wavenumber and continue with the imposed frequency until the curves cross the next U-shaped neutral curve. As a consequence, in the finite wavenumber regime, no stable bandwidth of the imposed frequency is predicted by the long-wavelength analysis. Further, in some frequency ranges, the finite wavenumber instability is more dangerous than the long-wave instability.
Shear-imposed falling film
- Arghya Samanta
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- Journal:
- Journal of Fluid Mechanics / Volume 753 / 25 August 2014
- Published online by Cambridge University Press:
- 21 July 2014, pp. 131-149
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The study of a film falling down an inclined plane is revisited in the presence of imposed shear stress. Earlier studies regarding this topic (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485; Wei, Phys. Fluids, vol. 17, 2005a, 012103), developed on the basis of a low Reynolds number, are extended up to moderate values of the Reynolds number. The mechanism of the primary instability is provided under the framework of a two-wave structure, which is normally a combination of kinematic and dynamic waves. In general, the primary instability appears when the kinematic wave speed exceeds the speed of dynamic waves. An equality criterion between their speeds yields the neutral stability condition. Similarly, it is revealed that the nonlinear travelling wave solutions also depend on the kinematic and dynamic wave speeds, and an equality criterion between the speeds leads to an analytical expression for the speed of a family of travelling waves as a function of the Froude number. This new analytical result is compared with numerical prediction, and an excellent agreement is achieved. Direct numerical simulations of the low-dimensional model have been performed in order to analyse the spatiotemporal behaviour of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. It is noticed that the presence of imposed shear stress in the upstream (downstream) direction makes the evolution of spatially growing waves weaker (stronger).
Effect of surfactant on two-layer channel flow
- Arghya Samanta
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- Journal:
- Journal of Fluid Mechanics / Volume 735 / 25 November 2013
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- 25 October 2013, pp. 519-552
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The effect of insoluble surfactant on the interfacial waves in connection with a two-layer channel flow is investigated for low to moderate values of the Reynolds number. Previous studies focusing on Stokes flow (Frenkel & Halpern, Phys. Fluids, vol. 14, 2002, p. L45; Halpern & Frenkel, J. Fluid Mech., vol. 485, 2003, pp. 191–220) are extended by including the inertial effect and the study of low-Reynolds-number flow (Blyth & Pozrikidis, J. Fluid Mech., vol. 521, 2004b, pp. 241–250) is enlarged up to moderate Reynolds number. Linear stability analysis based on the Orr–Sommerfeld boundary value problem identifies a surfactant mode together with an interface mode. The presence of surfactant on the interfacial mode is stabilizing at high viscosity ratio and destabilizing at low viscosity ratio. The threshold of instability is determined as a function of the Marangoni number. A long-wave model is developed to predict the families of nonlinear waves in the neighbourhood of the threshold of instability. Far from the threshold, wave dynamics is explored under the framework of a three-equation model in terms of lower layer flow rate ${q}_{2} (x, t)$, lower liquid-layer thickness $h(x, t)$ and surfactant concentration $\Gamma (x, t)$. Primary instability analysis of a three-equation model captures the result of the Orr–Sommerfeld boundary value problem very well for quite large values of wavenumber. In the nonlinear regime, travelling wave solutions demonstrate deceleration of maximum amplitude and acceleration of speed with the Marangoni number at high viscosity ratio $m\gt 1$ and show completely the opposite behaviour at low viscosity ratio $m\lt 1$. However, both maximum amplitude and speed attain a fixed value with increasing Reynolds number and this leads to saturation of instability.