Base-state Solution Research Articles

Transverse electric-thermal-fluid instabilities in an electromagnetic heat exchanger

Electromagnetic (EM) heat exchangers (HX) are critical components in power beaming applications where EM waves are radiated towards an EM HX, which then converts incident energy into heat or mechanical work. An EM HX consists of a lossy ceramic and a fluid flow that maintains thermal contact with and transfers heat from the ceramic. These materials have loss factors which increase with temperature, so that beyond a critical temperature thermal runaway can take place. Stable characteristic temperatures, which depend on the rate of energy removal from the system, during high-power EM heating of ceramic materials suggest that coolants would be in the gaseous phase. As a first step, we consider a model EM HX system consisting of a horizontal channel containing a viscous, dielectric fluid with a constant coefficient of thermal expansion, bounded from below by a grounded ceramic receiver of finite thickness. The system is subject to plane EM waves, propagating normally to the channel from above, and polarized in the same direction as a plane Poiseuille flow of the coolant. With the Boussinesq approximation, we calculate the base state solution of the system and then investigate the linear stability of this base-state. We find three modes of instability: thermal runaway, Rayleigh-Bénard convection, and a novel instability which we call the fringe-field instability, that takes place in the plane normal to base-state flow direction.

The Role of Buoyancy Induced Instability in Transpirational Cooling Applications

Transpirational cooling is an effective thermal protection method in hypersonic vehicles. In order to properly manage the high heat load, an understanding of the convective flow regimes as well as the thermophysical properties of the working fluid are required. Often, the vehicle’s fuel is re-purposed as the coolant or working fluid that is passed through the porous media. If the geometry is such that the coolant is heated from below, buoyancy-induced instability can ensue resulting in a mixed convection phenomena. Transpirational cooling applications require a unique analysis which combines a Darcy–Forchheimer relationship for the momentum relation, a flowing base state which introduces non-negligible convective terms for the energy equation, and a novel consideration of a cubic density dependence on temperature. This latter feature is justified by fitting thermodynamic data for typical transpirational cooling conditions. A base state solution is provided and the onset of instability is investigated using linear stability analysis. The governing equations are solved utilizing multiple methods, comparing results from a combination of analytical solutions, finite difference, power series, and Chebyshev methods. Results demonstrate excellent consistency in predictions across these methods and indicate that including non-linear density effects promote a stabilizing effect. Finally, the effect of varying the net through-flow in the porous media is investigated.

Linear stability analysis of evaporating falling liquid films

We consider the linear stability of evaporating thin films falling down an inclined plate. The one sided-model presented first by “Burelbach, J.P., Bankoff, S.G., Davis, S.H., 1988, Nonlinear stability of evaporating/condensing liquid films, Journal of Fluid Mechanics 195, 463–494. ” was implemented to decouple the dynamics of the liquid than those of the vapor at the interface, at which the evaporation is modeled based on a thermal equilibrium approach. We consider the base state solution derived by “Joo, S., Davis, S.H., Bankoff, S., 1991, Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers, Journal of Fluid Mechanics 230, 117–146. ” which is based on the slow evaporation assumption. In previous works, only low dimensional models. i.e. the long wave theory, have been analysed for evaporating liquid films. Conversely in this paper, we extend the Orr-Sommerfeld eigenvalue problem for a film falling down a heated wall to include evaporation effects namely, vapor recoil and mass loss. As expected, we observe that the long wave theory fails in predicting the correct behavior when the inertia is strong or the wavenumber k is large. We confirm that the instability induced by vapor recoil (E-mode) behaves in a similar fashion to the instability due to the thermocapillary effect (S-mode). Both the S-mode and the E-mode can enhance each other, specially, at low Reynolds numbers Re. Moreover, we examine the perturbation energy budget to have an insight into the instabilities mechanism. We show that the presence of evaporation adds a new term corresponding to the work done by vapor recoil at the interface (VRE). We also find that the main contributor to the perturbation kinetic energy in the unstable E-mode is the work done by shear stress while VRE is negligible, unless Re << 1. Simpler analytical expressions of the energy balance terms near the instability threshold are obtained by using a long wave approximation.

Three-dimensional electroconvective vortices in cross flow.

This study focuses on the three-dimensional (3D) electrohydrodynamic flow instability between two parallel electrodes driven by unipolar charge injection with and without cross flow. Lattice Boltzmann method with a two-relaxation time model is used to compute flow patterns. In the absence of cross flow, the base-state solution is hydrostatic, and the electric field is one-dimensional. With strong charge injection and high electrical Rayleigh number, the system exhibits electroconvective vortices. Disturbed by perturbation patterns, such as rolling pattern, square pattern, and hexagon pattern, the flow develops corresponding to the most unstable mode. The growth rate and pattern transitions are studied using dynamic mode decomposition of the transient numerical solutions. The interactions between cross flow and electroconvective vortices lead to suppression and disappearance of structures with velocity components in the direction of cross flow, while the other components are not affected. Surprisingly, the transition from a 3D to a 2D flow pattern enhances the convective charge transport, marked by an increase in the electric Nusselt number. Hysteresis in the 3D to 2D transition is characterized by the nondimensional parameter Y, a ratio of the electrical force term to the viscous term in the momentum equation.

Stability of concentrated suspensions under Couette and Poiseuille flow

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. A linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated, and the connections to the stability problem for the related classical Bingham flow problem are discussed.

Surface nonuniformities in latex paints due to evaporative mechanisms

A model is developed for predicting long‐wavelength nonuniformities in the thickness of drying latex paint films. The model includes the effects of temperature, latex particle volume fraction, surface surfactant density, bulk surfactant density, and several material and environmental factors. After the model is simplified by applying the lubrication approximation, equations for spatially independent base state solutions are derived. The base state solutions describe a drying latex paint film of uniform thickness. The equations for the base states are solved numerically and a linear stability analysis is conducted. This analysis indicates that evaporation, slow surfactant kinetics, low initial surface tension, substrate permeability, and high initial latex particle volume fractions destabilize the uniform film, while fast surfactant kinetics, high initial surface tension, and high viscosity are stabilizing. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1841–1858, 2018

Stability analysis of rimming flow inside a horizontally rotating cylinder in the presence of an insoluble surfactant

Two-dimensional base state solutions for rimming flows and their stability analysis to small axial perturbations are analyzed numerically. A thin liquid film which is uniformly covered with an insoluble surfactant flows inside a counterclockwise rotating horizontal cylinder. In the present work, a mathematical model is obtained which consists of coupled thin film thickness and surfactant concentration evolution equations. The governing equations are obtained by simplifying the momentum and species transport equations using the thin-film approximation. The model equations include the effect of gravity, viscosity, capillarity, inertia, and Marangoni stress. The concentration gradients generated due to flow result in the surface tension gradient that generates the Marangoni stress near the interface region. The oscillations in the flow due to inertia are damped out by the Marangoni stress. It is observed that the Marangoni stress has stabilizing effect, whereas inertia and surface tension enhance the instability growth rate. In the presence of low diffusion of the surfactant or large value of the Péclet number, the Marangoni stress becomes more effective. The analytically obtained eigenvalues match well with the numerically computed eigenvalues in the absence of gravity.

Inertia flows of Bingham fluids through a planar channel: Hydroelastic instability analysis

In this paper, the effect of inertial terms on hydroelastic stability of a pressure-driven flow of a viscoplastic fluid flowing through a channel lined with a highly compliant polymeric gel is investigated. It is assumed that the fluid obeys the Bingham constuitive equation and the polymeric gel follows a two-constant Mooney–Rivlin material, which is used for modeling a nonviscous hyperelastic polymeric coating. A base-state solution is obtained for the fluid motion and solid deformation, simultaneously. Next, some infinitesimally small two-dimensional disturbances are imposed on the base-state solution. Dropping out all nonlinear perturbation terms, the modal linear stability analysis of the channel flow is conducted. The effects of the Bingham number and material constants are then examined on the critical Reynolds number. It is found that the yield stress has a stabilizing effect while the Mooney–Rivlin parameters have destabilizing effects on the pressure-driven flow of Bingham fluids.

Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer

We consider a model for the solidification of an ideal ternary alloy in a mushy layer that incorporates the effects of thermal and solutal diffusion, convection and solidification. Our results reveal that although the temperature and solute fields are constrained to the liquidus surface of the phase diagram, the system still admits double-diffusive modes of instability. Additionally, modes of instability exist even in situations in which the thermal and solute fields are each individually stable from a static point of view. We identify these instabilities for a general model in which the base-state solution and its linear stability are computed numerically. We then highlight these instabilities in a much simpler model that admits an analytical solution.

Effect of normal and parallel magnetic fields on the stability of interfacial flows of magnetic fluids in channels

The effect of an imposed magnetic field on the linear stability of immiscible two-fluid Poiseuille flow in a channel is examined for low Reynolds numbers. Surface tension acts on the interface, the fluids have different densities and viscosities, and one fluid is magnetic (ferrofluid). A Langevin function is used to model the fluid magnetization, resulting in a nonlinear permeability; the stability properties depend on this permeability relation both directly and indirectly, through the base state solution. Uniform magnetic fields applied normal or parallel to the interface both lead to an interfacial instability. Normal fields excite longer wavelength modes, generally having higher growth rates, but parallel fields can excite faster growing modes in high permeability fluids at large applied field strength. Whether or not the field stabilizes or destabilizes the flow depends on the viscosity and layer thickness ratios in a simple way, while the placement of the magnetic fluid layer does not play a major role. Growth rates predicted for realistic microchannel conditions are shown to be large enough to make ferrofluid manipulation a practical method of control.

On the instability of a falling film due to localized heating

We analyse the stability of a thin film falling under the influence of gravity down a locally heated plate. Marangoni flow, due to local temperature changes influencing the surface tension, opposes the gravitationally driven Poiseuille flow and forms a horizontal band at the upper edge of the heater. The thickness of the band increases with the surface tension gradient, until an instability forms a rivulet structure periodic in the transverse direction. We study the dependence of the critical Marangoni number, a non-dimensional measure of the surface tension gradient at the onset of instability, on the associated Bond and Biot numbers, non-dimensional measures of the curvature pressure and heat-conductive properties of the film respectively. We develop a model based on long-wave theory to calculate base-state solutions and their linear stability. We obtain dispersion relations, which give us the wavelength and growth rate of the fastest growing mode. The calculated film profile and wavelength of the most unstable mode at the instability threshold are in quantitative agreement with the experimental results. We show via an energy analysis of the most unstable linear eigenmode that the instability is driven by gravity and an interaction between base-state curvature and the perturbation thickness. In the case of non-zero Biot number transverse variations of the temperature profile also contribute to destabilization.

Local linear stability characteristics of viscoelastic periodic channel flow

Linear stability of periodic channel (PC) flow of an upper convected Maxwell (UCM) liquid is investigated in inertial (Reynolds number Re⪢0, Weissenberg number We∼O(1)) and in purely elastic ( Re≡0, We∼O(1)) flow regimes. Base state solution is evaluated using O( ε 2) domain perturbation analysis where ε denotes the channel wall amplitude. Significant destabilization, i.e. reduction in critical Reynolds number Re c, with increasing ε is predicted for the Newtonian flow, especially in the diverging section of the channel. Introduction of elasticity E≡ We/ Re, representing the ratio of fluid relaxation time to a time scale of viscous diffusion based on channel half height, leads to further destabilization. However, the minimum in the Re c– E curve reported for plane channel flow is not observed. Analysis of the budget of perturbation kinetic energy shows that this minimum in the plane channel flow results from two competing contributions to kinetic energy: a normal stress contribution that increases with increasing E and a shear stress contribution that decreases monotonically with increasing E with the two curves intersecting for E≈0.002. This value is approximately equal to the value of E for which the plane shear layer is maximally destabilized. When this happens, the critical Deborah number, defined as the ratio of the fluid relaxation time to time scale of the perturbation, is O(1). Comparison of results obtained for the UCM and second order fluid (SOF) models shows that the latter model does not predict a minimum Re c for E≤0.003. Moreover, eigenspectrum for the SOF contains a set of eigenvalues with positive real parts equal to 1/ We. Results obtained for the eigenspectrum in the purely elastic limit indicate that the PC flow is linearly stable for O(1) axial wavenumbers for We≤10, ε≤0.1 and n≤0.1, although the decay rates of the perturbation are smaller than that of the plane channel flow. The local eigenspectrum could contain spurious eigenvalues with positive real parts that could lead to erroneous predictions of flow instability. Using a contour mapping technique, it is shown that deformation of the flow domain can lead to spurious eigenvalues.

Long-scale evolution of thin liquid films

Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Stability of bounded rapid shear flows of a granular material

This paper presents a linear stability analysis of a rapidly sheared layer of granular material confined between two parallel solid plates. The form of the steady base-state solution depends on the nature of the interaction between the material and the bounding plates and three cases are considered, in which the boundaries act as sources or sinks of pseudo-thermal energy, or merely confine the material while leaving the velocity profile linear, as in unbounded shear. The stability analysis is conventional, though complicated, and the results are similar in all cases. For given physical properties of the particles and the bounding plates it is found that the condition of marginal stability depends only on the separation between the plates and the mean bulk density of the particulate material contained between them. The system is stable when the thickness of the layer is sufficiently small, but if the thickness is increased it becomes unstable, and initially the fastest growing mode is analogous to modes of the corresponding unbounded problem. However, with a further increase in thickness a new mode becomes dominant and this is of an unusual type, with no analogue in the case of unbounded shear. The growth rate of this mode passes through a maximum at a certain value of the thickness of the sheared layer, at which point it grows much faster than any mode that could be shared with the unbounded problem. The growth rate of the dominant mode also depends on the bulk density of the material, and is greatest when this is neither very large nor very small.

The mean evershed flow

An explanation for the mean Evershed flow is given in this paper. Because the inertial force of the Evershed flow is small, we can divide the relevant dynamic process into a base state and a perturbation. The base state solution gives the equilibrium relations between the pressure gradient, Lorentz force and gravity and the law of flow. The flow in the theoretical model agrees with the observed Evershed flow. The perturbation describes the force driving the Evershed flow and a specific ease is discussed. Because in the base state, the pressure gradient is already in equilibrium with the Lorentz force and the gravity, the driving force of the mean Evershed flow is small.