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IIT Hyderabad

The present study investigates the linear spatio-Temporal and weakly nonlinear stability of a pressure-driven two-layer channel flow subjected to a wall-normal temperature gradient commonly encountered in industrial applications. The liquid-liquid interface tension is assumed to be a linearly decreasing function of temperature. The study employs both numerical (pseudo-spectral method) and long-wave approaches. The general linear stability analysis (GLSA) predicts shear-flow and thermocapillary modes that arise due to the imposed pressure and temperature gradients, respectively. The previous stability analyses of the same problem predicted a negligible effect of the pressure-driven flow on the linear stability of the system. However, the GLSA reveals stabilising and destabilising effects of the pressure-driven flow depending on the viscosity ratio , thermal conductivity ratio , interface position and the sign of the imposed temperature gradient . The analysis predicts a range of for given and, which can not be stabilised by the thermocapillarity. The numerically predicted long-wave instability is then captured using the long-wave asymptotic approach. The arguments based on the physical mechanism further successfully explain the role of, the sign of and the interaction between the velocity and temperature perturbations in stabilising/destabilising the flow. The spatio-Temporal analysis reveals the dominance of the spanwise mode in causing the absolutely unstable flow. The weakly nonlinear analysis reveals a subcritical pitchfork bifurcation without shear flow. However, with the shear flow, the streamwise mode undergoes a supercritical Hopf bifurcation. © The Author(s), 2023. Published by Cambridge University Press.

Department of Chemical Engineering

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Journal of Fluid Mechanics

Spatio-Temporal dynamics of a two-layer pressure-driven flow subjected to a wall-normal temperature gradient

We investigate the temporal and spatio-temporal buoyancy instabilities in a horizontal liquid layer supported by a poorly conducting substrate and subjected to an oblique temperature gradient (OTG) with horizontal and vertical components, denoted as HTG and VTG, respectively. General linear stability analysis (GLSA) reveals a strong stabilizing effect of the HTG on the instabilities introduced by the VTG for Prandtl numbers 1$]]> via inducing an extra vertical temperature gradient opposing the VTG through energy convection. For <![CDATA[$Pr, a new mode of instability arises as a result of a velocity jump in the liquid layer caused by cellular circulation. A long-wave weakly nonlinear evolution equation governing the spatio-temporal dynamics of the temperature perturbations is derived. Spatio-temporal stability analysis reveals the existence of a convectively unstable long-wave regime due to the HTG. Weakly nonlinear stability analysis reveals the supercritical type of bifurcation changing from pitchfork in the presence of a pure VTG to Hopf in the presence of the OTG. Numerical investigation of the spatio-temporal dynamics of the temperature disturbances in the layer in the weakly nonlinear regime reveals the emergence of travelling wave regimes propagating against the direction of the HTG and whose phase speed depends on. In the case of a small but non-zero Biot number, the wavelength of these travelling waves is larger than that of the fastest-growing mode obtained from GLSA. © 2022 Cambridge University Press. All rights reserved.

Buoyancy instabilities in a liquid layer subjected to an oblique temperature gradient

The temporal stability of a two-layer plane Couette flow with an interfacial heat source/sink is studied, in the presence of a viscosity stratification, predicting unstable streamwise and spanwise perturbations. Based on the wavenumber at onset, the perturbations are further classified as long-wave and finite-wave, where the long-wave mode is investigated using asymptotic and numerical analyses. The finite-wave instability is expected to be relevant for highly viscous fluids. Generally, a heat source (sink) at the interface is found to have a stabilising (destabilising) effect on the flow. However, if the rate of heat release exceeds a certain threshold, a new instability emerges, referred to here as an 'explosive' instability, characterised by an unbounded growth rate of the perturbations. The interaction between the viscosity stratification, thermocapillarity and inertia leads to the emergence of another new mode of streamwise instability, termed the 'interaction' instability. The spanwise perturbations possess higher growth rates than the streamwise perturbations for the heat sink along the interface when inertia has a stabilising influence, and vice versa. The heat sink leads to the emergence of thermocapillary instability by generating a negative temperature gradient across the fluids. In contrast, a heat source increases the temperature of the interface, resulting in a positive temperature gradient across the fluids, leading to the suppression of the instabilities by the thermocapillary effect. If the heat source provides energy at a faster rate than the energy lost due to Marangoni convection and heat diffusion, then it leads to an accumulation of energy, which presumably leads to the explosive instability. © 2022 The Author(s). Published by Cambridge University Press.

Fulltext

Dynamics of a two-layer flow with an interfacial heat source/sink: Viscosity stratification

The present study considers a shear-thinning viscoelastic liquid layer sheared by the air and flowing past a deformable-solid layer in the presence of a surfactant at the air-liquid interface to model the airflow in the oral area and airways. The stability analysis reveals the existence of purely elastic and unconditionally unstable 'liquid elastic' and 'solid elastic' modes. The mechanism responsible for the destabilisation of the solid elastic mode is the shear stresses exerted by the air on the liquid and by the liquid on the deformable solid while for the liquid elastic mode, the mechanism is the first normal stress difference across the air-liquid interface. The liquid and solid elastic modes undergo resonance, resulting in the 'resonance mode' of instability. The resonance mode exhibits a much higher growth rate than the liquid and solid elastic modes. The shear-thinning characteristic of the liquid and presence of the surfactant leads to enhancement in the growth rate of the resonance mode. An estimate shows a good correlation between the exhaled fluid particle (i.e. droplets and aerosols) diameters and the wavelength of the perturbations with maximum growth rate. In essence, the present analysis predicts that the airflow in the airways and oral area could lead to an elastic instability arising due to the elastic nature of the saliva, mucus and underlying muscle layers. © The Author(s), 2021.

Purely elastic instabilities in the airways and oral area

The linear stability analysis of a viscoelastic (Oldroyd-B) liquid layer subjected to an oblique temperature gradient (OTG) is investigated numerically. For the case of low liquid elasticity, the analysis shows a strong stabilizing effect of the horizontal component (HTG) of the OTG on the two elastic modes emerging due to the presence of the vertical component (VTG) of the OTG. However, if the liquid elasticity is sufficiently large, the HTG fails to stabilize the upstream elastic mode. The liquid elasticity stabilizes the Newtonian interaction mode arising out of the interaction between the HTG and the VTG. The thermocapillary flow introduced by the HTG leads to the development of a new elastic mode absent in the case of a Newtonian liquid layer. The present paper thus shows that the elasticity of the liquid plays a major role in the competition between various instability modes to determine the dominant mode of instability. © 2021 Author(s).

Physics of Fluids

Thermocapillary instability in a viscoelastic liquid layer under an imposed oblique temperature gradient

The linear stability of "sliding Couette flow"of a Newtonian fluid through the annular gap formed by two concentric cylinders having a ratio of inner to outer cylinder radii, β, and driven by the axial motion of the inner cylinder is studied in the low Reynolds number (<1) regime. The inner wall of the outer cylinder is lined by a deformable neo-Hookean solid layer of dimensionless thickness H. This flow configuration is encountered in medical procedures such as thread-injection and angioplasty, where the inserted needle is surrounded by the deformable wall of blood vessels. In stark contrast to the configuration with rigid cylinders, we predict the existence of finite- A nd short-wave linear instabilities even in the creeping-flow limit, driven by the deformable nature of the outer cylinder. Interestingly, these instabilities exist for arbitrary β, and even for non-axisymmetric perturbations, in parameter regimes where the flow is stable for the configuration with a rigid outer cylinder. For the finite-wave instability, the axisymmetric mode is the most critical mode of the instability, while the non-axisymmetric mode with azimuthal wavenumber n = 4 is the critical mode for the short-wave instability. By replacing the outer rigid boundary surrounding the deformable wall by an "unrestrained"stress-free boundary, we demonstrate that the flow becomes significantly more unstable. Thus, the present study shows that sliding Couette flow with a deformable wall can be linearly unstable at an arbitrarily low Reynolds number, in direct contrast to the stability of the same configuration with a rigid cylinder. © 2020 Author(s).

Instability induced by wall deformability in sliding Couette flow

Stability analysis of a liquid layer subjected to an oblique temperature gradient (OTG) is carried out. The general linear stability analysis reveals a stabilization effect of the imposed horizontal component (horizontal temperature gradient, HTG) of the OTG on the long-wave instabilities introduced by the vertical component (vertical temperature gradient, VTG) of the OTG. This stabilization is due to the VTG induced by the prescribed HTG, which counteracts the imposed VTG. The induced VTG arises due to the presence of advection of the energy. As a result of the stabilization, the long-wave mode forms an island of instability in the-plane, where and are the ratio of the strength of the imposed HTG to imposed VTG components of the OTG, and the critical Marangoni number, respectively. However, for sufficiently high, a new class of modes emerge with the critical Marangoni number scaling as. These modes originate as a result of the interaction between the thermocapillary flow caused by the imposed HTG on the one hand, and the VTG on the other, and remain the dominant modes of instability at higher. The long-wave analysis is carried out and, in its framework, the nonlinear evolution equation is derived, and, based on it, linear and weakly nonlinear analyses are performed. An increase in changes the type of bifurcation from subcritical to supercritical. The numerical solution of the evolution equation around the critical parameter values validates the predictions of the weakly nonlinear analysis. The present study illustrates a possible use of imposing the HTG to prevent dry-spot formation and rupture of the film caused by the imposed VTG. © 2020 American Medical Association. All rights reserved.

Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient

Viscoelastic polymer films are present in numerous industrial applications such as in paint coatings, lubricants, and optoelectronics which involve a large variation in temperatures. The present study deals with the general linear stability analysis (GLSA) of a Jeffreys viscoelastic fluid layer having a deformable free surface deposited on a perfectly conducting slippery plate. Variation in temperature results in variation in surface tension thereby inducing Marangoni stresses leading, under certain conditions, to the emergence of instability of the quiescent, heat-conducting base state which may be either stationary or oscillatory. Accounting for the deformability of the free surface of the layer, the GLSA reveals the existence of the long-wavelength Marangoni instability absent in the previous studies that assumed a nondeformable free surface. The analysis reveals a strong effect of variation in the capillary number, Ca, on the stationary mode, whereas the oscillatory mode is moderately affected. The critical parameters for the stationary mode remain unaffected by variation in the Prandtl number, Pr, and the dimensionless relaxation L1 and retardation L2 parameters; however, their impact on the oscillatory modes is significant. Variation in the Biot number, Bi, has a stabilizing effect on the system. Variation in the Bond number, Bo, is found to strongly influence the long-wave regime of the stationary mode, but it does not influence the short-wave regime of the stationary mode and oscillatory modes. Variation in the slip coefficient leads to a moderate increase in Mac in the case of the stationary mode, whereas in the case of the oscillatory mode variation of both Mac and kc is nonmonotonic. The energy balance analysis suggests that viscoelasticity of the fluid is responsible for the emergence of oscillatory instability. Our analysis reveals the existence of a wide parameter range where oscillatory modes determine the stability properties of the system. This result emphasizes that neglecting viscoelasticity of the fluid may lead to significant errors when the dynamics of a viscoelastic fluid is investigated. The present study further demonstrates that even for a highly elastic fluid, the interface deformability prevails and promotes the long-wave stationary instability at the expense of the oscillatory instability. © 2020 American Physical Society.

Physical Review Fluids

Marangoni instability in the linear Jeffreys fluid with a deformable surface

Fluid flows coupled by a deformable solid layer (DS) may be found in both natural and industrial settings. To analyse the stability of such systems, we consider plane Couette flows coupled by a DS. The shear stress exerted by the flowing fluids on the DS leads to its deformation, coupling the motion in each fluid. We demonstrate that, as a consequence of the coupling and irrespective of the thickness of the DS, a viscous instability exists, which can lead to absolute instability at sufficiently high dimensionless speed of the lower plate. The predicted instability apparently arises due to the energy exchange between the fluids and DS via the tangential velocities at the interface. A linear elasticity model for the DS is employed, removing the non-physical growth rate predicted by previous models. A spatio-temporal analysis further reveals the existence of an absolute instability. Insight on this instability could be potentially utilised in manipulating mixing in microfluidic systems and membrane separation as well as for understanding morphologies evolving during membrane formation via interfacial polymerisation. © 2020 Cambridge University Press. All rights reserved.

Stability of fluid flows coupled by a deformable solid layer

We consider thermocapillary instability in a three-dimensional liquid layer with a deformable interface with an ambient gas phase and subjected to an oblique temperature gradient when the temperature gradient at the substrate is prescribed. We demonstrate that this configuration leads to a drastic change in the instability features with respect to those emerging when either a purely vertical temperature gradient (VTG) or a purely horizontal temperature gradient (HTG) is present. In the case of the return flow as the base state, the spanwise long-wave instability mode dominates except for the range of small Bond numbers Bo. Slippage at the substrate has a stabilizing (destabilizing) effect on streamwise (spanwise) long-wave modes in the presence of a HTG. In the case of linear flow as the base state, both streamwise and spanwise long-wave modes play a major role in the instability onset depending on the ratio between the HTG and the VTG η for higher values of the capillary number Ca, e.g., Ca > 0.001. However, for lower values of Ca, e.g., Ca < 0.001, streamwise and spanwise instability modes become finite-waves at large η. In contrast to the return flow, for the linear flow, slippage at the substrate destabilizes both long-wave modes. © 2020 American Institute of Physics Inc.. All rights reserved.

Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient: Effect of a prescribed normal temperature gradient at the substrate

The present study is aimed at assessing the existing results concerning the stability of canonical shear flows in channels and tubes with deformable walls, in light of consistent formulations of the nonlinear solid constitutive model and linearised interface conditions at the fluid-solid interface. We show that a class of unstable shear-wave modes at low Reynolds number, predicted by previous studies for pressure-driven flows through neo-Hookean tubes and channels, is absent upon use of consistent interfacial conditions. Furthermore, we analyse the consequences of the change in solid model on the stability of the canonical shear flows by using both neo-Hookean and Mooney-Rivlin models. We show that the salient features of the stability of the system are adequately captured by a consistent formulation of the neo-Hookean solid model, thus precluding the need to employ more detailed solid models. The stability analysis of planar flows past a neo-Hookean solid subjected to three-dimensional disturbances showed that two-dimensional disturbances are more unstable than the corresponding three-dimensional disturbances within the consistent formulations. We show that prior inconsistent formulations of the solid constitutive equation predict a physically spurious spanwise instability in disagreement with experiments thereby demonstrating their inapplicability to predict instabilities in flow past deformable solid surfaces. Using the consistent formulation, the present work provides an accurate picture, over a range of Reynolds numbers, of the stability of canonical shear flows through deformable channels and tubes. Importantly, it is shown how inconsistencies in either the bulk constitutive relation or in the linearisation of the interface conditions can separately lead to physically spurious instabilities. The predictions of this work are relevant to experimental studies in flow through deformable tubes and channels in the low and moderate Reynolds number regime. © 2018 Cambridge University Press.

Stability of flow through deformable channels and tubes: Implications of consistent formulation

We study the linear stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel. The analysis reveals the existence of a finite-wave instability in the creeping-flow limit, which is absent for Newtonian fluid flow in the same geometry. Indeed, the instability in plane Poiseuille flow of a Bingham fluid closely resembles the finite-wave instability exhibited by plane Couette flow of a Newtonian fluid past a deformable solid. The motion of the unyielded region near the channel centerline effectively results in a Couette-like drag flow of the fluid in the yielded region, albeit with a parabolic base flow velocity profile. More importantly, the boundary conditions at the interface between yielded and unyielded zones is found to be crucial for the existence of the finite-wave instability. This is in stark contrast to Newtonian channel flow where the conditions at the channel centerline are dictated by the symmetry or antisymmetry of the velocity perturbations. The continuation of the predicted instability in the creeping-flow limit is also shown to determine the stability of the flow of a Bingham fluid at higher Reynolds number. In addition to treating the plug as a rigid solid, we also consider the plug to be an elastic solid and show that the elastic nature of the plug does not have any effect on the predicted finite-wave instability. We further argue, using the elastic plug scenario, that some of the regularized models can introduce model-dependent instabilities which are specific to the parameters used in the models. © 2019 American Physical Society.

Stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel

The linear stability of the plane Couette flow of both power-law and Carreau fluids past a deformable, neo-Hookean solid is analyzed at arbitrary Reynolds numbers. An algebraic error in the mathematical formulation of the earlier studies (for the power-law fluid) is corrected and is shown to result in quantitative differences in the predictions for the stability of the flow. Due to the lack of a proper (zero-shear) viscosity scale and a time scale for the onset of shear thinning in the power-law model, we show that the stability analysis of the flow yields vastly different scalings for the unstable mode depending on the way the problem is scaled to render it dimensionless. When the deformable solid properties are used to non-dimensionalize, we show that for the unstable modes (the so-called “wall modes” at high Re) Γc∝Re−1(2n+1), while when flow properties are used to non-dimensionalize, Γc∝Re−13 much akin to a Newtonian fluid, where Γ=Vm*η*/G*R* is the dimensionless shear rate in the flow, and Γc denotes the minimum value required for instability. Here, Vm* is the velocity of the top plate, G* is the shear modulus of the solid, R* is the fluid thickness, and η* is the (arbitrary) viscosity scale in the power-law model. Within the framework of the power-law model, it is not possible to discriminate between the two predicted scalings. To resolve this in an unambiguous manner, we used the Carreau model to account for shear thinning and to study its role on the stability of flow past deformable solid surfaces. The Carreau model has a well-defined zero-shear viscosity η0* as well as a time scale λ* that characterizes the onset of shear thinning. For fixed λ*η0*/(ρ*R*2), we show that the unstable wall modes scale as Γc∼Re(1−2n)3 at high Re, thus providing a resolution to the ambiguity in the results obtained using the power-law model. The present work thus shows that, at moderate to high Re, shear thinning has a strongly stabilizing effect on the wall mode instability in flow past deformable solid surfaces. © 2018 Author(s).

Stability of plane Couette flow of Carreau fluids past a deformable solid at arbitrary Reynolds numbers

In the formulation of stability of fluid flow through channels and tubes with deformable walls, while the fluid is naturally treated in an Eulerian framework, the solid can be treated either in a Lagrangian or Eulerian framework. A consistent formulation, then, should yield results that are independent of the chosen framework. Previous studies have demonstrated this consistency for the stability of plane Couette flow past a deformable solid layer modelled as a neo-Hookean solid, in the creeping-flow limit. However, a similar exercise carried out in the creeping-flow limit for the stability of pressure-driven flow in a neo-Hookean tube shows that while the flow is stable in the Lagrangian formulation, it is unstable in the existing Eulerian formulation. The present work resolves this discrepancy by presenting consistent Lagrangian and Eulerian frameworks for performing stability analyses in flow through deformable tubes and channels. The resolution is achieved by making important modifications to the Lagrangian formulation to make it fundamentally consistent, as well as by proposing a proper formulation for the neo-Hookean constitutive relation in the Eulerian framework. In the neo-Hookean model, the Cauchy stress tensor in the solid is proportional to the Finger tensor. We demonstrate that the neo-Hookean constitutive model within the Eulerian formulation used in the previous studies is a special case of the Mooney-Rivlin solid, with the Cauchy stress tensor being proportional to the inverse of the Finger tensor unlike in a true neo-Hookean solid. Remarkably, for plane Couette flow subjected to two-dimensional perturbations, there is perfect agreement between the results obtained using earlier Eulerian and Lagrangian formulations despite the crucial difference in the constitutive relation owing to the rather simple kinematics of the base state. However, the consequences are drastic for pressure-driven flow in a tube even for axisymmetric disturbances. We propose a consistent neo-Hookean constitutive relation in the Eulerian framework, which yields results that are in perfect agreement with the results from the Lagrangian formulation for both plane Couette and tube flows at arbitrary Reynolds number. The present study thus provides an unambiguous formulation for carrying out stability analyses in flow through deformable channels and tubes. We further show that unlike plane Couette flow and Hagen-Poiseuille flow in rigid-walled conduits where there is a remarkable similarity in the linear stability characteristics between these two flows, the stability behaviour for these two flows is very different when the walls are deformable. The instability of plane Couette flow past a deformable wall is very robust and is not sensitive to the constitutive nature of the solid, but the stability of pressure-driven flow in a deformable tube is rather sensitive to the constitutive nature of the deformable solid, especially at low Reynolds number. © 2017 Cambridge University Press.

Consistent formulations for stability of fluid flow through deformable channels and tubes

Temporal and spatio-temporal stability analyses are carried out to characterize the occurrence of convective and absolute instabilities in combined Couette-Poiseuille flow of a Newtonian fluid past a deformable, neo-Hookean solid layer in the creeping-flow limit. Plane Couette flow of a Newtonian fluid past a neo-Hookean solid becomes temporally unstable in the inertia-less limit when the parameter Γ = V η/(GR) exceeds a critical value. Here, V is the velocity of the top plate, η is the fluid viscosity, G is the shear modulus of the solid layer, and R is the fluid layer thickness. The Kupfer-Bers method is employed to demarcate regions of absolute and convective instabilities in the Γ-H parameter space, where H is the ratio of solid to fluid thickness in the system. For certain ranges of the thickness ratio H, we find that the flow could be absolutely unstable, and the critical Γ required for absolute instability is very close to that for temporal instability, thus making the flow absolutely unstable at the onset of temporal instability. In some cases, there is a gap in the parameter Γ between the temporal and absolute instability boundaries. The present study thus shows that absolute instabilities are possible, even at very low Reynolds numbers in flow past deformable solid surfaces. The presence of absolute instabilities could potentially be exploited in the enhancement of mixing at low Reynolds numbers in flow through channels with deformable solid walls. © 2017 Author(s).