Crispation Number Research Articles

Onset of surface driven convection in selfrewetting fluid layer overlying a porous medium

The onset of thermocapillary convective motion in a self-rewetting fluid layer overlying a porous medium with thermally dependent surface tension is studied analytically. Surface tension is assumed to be a quadratic function of temperature. The top surface of a fluid layer is deformably free and the bottom is rigid. We considered boundaries to be insulating to temperature perturbations. The governing equation that satisfies the composite system is analyzed by the normal mode approach and solved by the regular perturbation technique for linear stability. A mathematical expression is derived for the critical Marangoni number by solving coupled equations. The influence of crispation number, thermal diffusivity ratio, and other physical parameters involved therein are analyzed for the convective stability of the bilayer system. It has been found that the start of convection is delayed when the crispation number goes down and the thermal diffusivity ratio goes up. Also, the impact of the ratio of the thickness of the fluid to the thickness of the porous matrix and the other physical parameters on controlling the convective motion of the configuration is examined in detail.

Understanding the dynamic pore wetting by 1H LF NMR characterization. Part 2: Effect of liquid surface tension

The first paper (Part 1) of this study fully described the effect of the dynamic viscosity of liquids on dynamic pore wetting of porous materials. The surface tension, as an important physical properties of liquids may also play a vital role in the pore wetting process. However, the quantitative characterization of dynamic pore wetting is difficult. Thus, this paper (Part 2) investigated the effect of liquid surface tension (mixture of 1-propanol and pure water in different proportions) on dynamic wetting of pores of different sizes in hydrophilic and hydrophobic active carbon by a 1H low-field nuclear magnetic resonance (1H LF NMR), and the quantitative model between surface tension and pore wetting percentage was built. The Crispation number of liquids in the pores of different sizes was calculated. For the hydrophilic samples, the wetting processes of pores of different sizes (except for mesopores) were not changed with the surface tension. For the hydrophobic sample, the liquid with lower surface tension achieved a higher pore wetting percentage and the order of pore wetting rate was: micropores > transition pores > mesoproes. The 60-min wetting percentages in pores of different sizes showed a linear decrease relationship with the surface tension. As the Crispation number increased, the 60-min wetting percentages in different pore sizes had a first-order linear increase trend. This paper not only provides a novel method for characterizing the dynamic pore wetting of porous materials, but also have a significance for regulating pore wetting process.

Understanding the dynamic pore wetting by 1H LF-NMR characterization. Part 1: Effect of dynamic viscosity of liquid

Dynamic pore wetting is significant for understanding the wetting behavior of fluid in porous materials. This study of dynamic pore wetting includes two parts (Part 1 and Part 2). The first paper (Part 1) of this study mainly investigated the effect of liquid viscosity on dynamic wetting of pores of different sizes in hydrophilic and hydrophobic active carbon by a 1H low-field nuclear magnetic resonance (1H LF-NMR). In order to fully understand the role of physical properties of liquids in dynamic pore wetting, the mechanism of surface tension affecting the pore wetting process was revealed in the second paper (Part 2). In Part 1, the feasible of 1H LF-NMR characterizing the dynamic pore wetting process was firstly discussed by comparing the measured results of 1H LF-NMR and weighting method. The quantitative model between dynamic viscosity and pore wetting percentage was built and the Crispation number of liquids in the pores of different sizes were calculated to reveal the degree of interface deformation inside the pore. As the viscosity increased, the pore wetting percentages in the hydrophilic and hydrophobic samples were decreased significantly. The decreasing degree of pore wetting percentage was in the order: transition pores > micropores > mesopores. The correlation between the viscosity and 60 min pore wetting percentage was the first-order linear decreasing except for mesopores. Compared with the hydrophobic sample, the 60 min pore wetting percentages (except for mesopores) in the hydrophilic sample had a better linear decreasing relationship with the Crispation number.

A local and parallel Uzawa finite element method for the generalized Navier–Stokes equations

In this paper, we propose and develop a local and parallel Uzawa finite element method for the generalized Navier–Stokes equations. The Uzawa finite element method is no need to deal with the saddle point problem, and only solves one vector-valued elliptic equation and one simple scalar-valued equation. It has the geometric convergence with a crispation number γ what has nothing to do with the mesh size h. As for the local and parallel Uzawa finite element method, each subproblem is a global problem, but most of degrees of freedom originate from the subdomain. Moreover, the presented method is easy to be applied with less communication requirements and has good parallelism. Finally, numerical results verify the performance of the proposed method.

Combined Effect of Magnetic field and Internal Heat Generation on the Onset of Marangoni Convection

Marangoni convection in a horizontal layer with a uniform internal heat source and vertical magnetic field is analyzed. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The upper surface of a fluid layer is deformably free. The eigen value equations of the perturbed state obtained from the normal mode analysis are solved by using regular perturbation method with as wave number. The results show that the critical Marangoni number <i>M_c</i> become larger as the Chandrasekhar number <i>Q</i> increases, internal heat source and the Crispation number <i>C_r</i> decreases.

Incipient mixing by Marangoni effects in slow viscous flow of two immiscible fluid layers

Ignoring inertia, a deformable interface separating two fluid films is considered, subject to non-uniform tension driven by the solutal Marangoni effect in the presence of a scalar concentration field. Detailed description of adsorption kinetics is abrogated by a simple ansatz directly relating interfacial tension and bulk solute concentration. Consequently, the formal mathematical treatment and some of the results share features in common with the Rayleigh–Benard–Marangoni thermocapillary problem. Normal mode perturbation analysis in the limit of small interface deformations establishes the existence of an unstable response for low wavenumber excitation. In the classification of Cross & Hohenberg (1993, Pattern formation outside of equilibrium. Rev. Mod. Phys., 65, 851–1112), both type I and type II behaviour are observed. By considering the zero wavenumber situation exactly, it is proved that all eigenvalues are purely imaginary with non-positive imaginary part; hence, a type III instability is not possible. For characteristic timescales of mass diffusion much shorter than the relaxation time of interfacial fluctuations (infinite crispation number): the response growth rate is obtained explicitly; only a single excitation mode is available, and a complete stability diagram is constructed in terms of the relevant control parameters. Otherwise, from a quiescent base state, an infinite discrete spectrum of modes is observed that exhibit avoided crossing and switching phenomena, as well as exceptional points where stationary state pairs coalesce into a single oscillatory standing wave pattern. A base state plane Poiseuille flow, driven by an external pressure gradient, generally exaggerates the response: growth rates of instabilities are enhanced, and stable decay is further suppressed with increasing base flow speed, but the inherent symmetry breaking destroys stationary and standing wave response. Results are obtained in this most general situation by implementing a numerical Chebyshev collocation scheme. The model was motivated by hydrodynamic processes supposed to be involved in gastric digestion of humans.

An experimental study on dynamic pore wettability

Dynamic pore wettability is important for understanding fluid behavior and adsorption in porous media for enhanced oil recovery, groundwater movements, nanofluidics and nanolubrication. Although dynamic wetting of different liquids has been extensively studied on various plane substrates, dynamic wettability in small pores has not been systematically investigated. In this paper, we measured dynamic contact angles in single glass capillaries with a size range from 100 to 250μm, in order to investigate the relationship between dynamic contact angle and spontaneous imbibition of various liquids, the effects of surface tension and viscosity of liquids on dynamic contact angle, and the dependence of contact angle on contact-line velocity in a pore. The liquids used are silicone oils with various viscosities, deionized water, 1-propanol, n-decane, crude oil, propanol-water mixtures with various surface tensions. The results indicate that the dynamic contact angle of liquids in a pore increases monotonically with the contact-line velocity at low capillary numbers, and this increase becomes more significant when more viscous liquids or liquids with lower surface tensions are used. A new empirical correlation based on the obtained experimental data has been proposed to describe dynamic pore wettability in a low capillary number range (1.0×10−7<Ca<1.8×10−5). Finally, we show that all our results can be summarized by a master curve relating the contact angle variation at a specific velocity with the Crispation number (Cr) signifying the importance of the interfacial deformation of liquid in a pore.

Role of Marangoni Instability in Fabrication of Axially and Internally Grooved Hollow Fiber Membranes

Hollow fiber membranes (HFMs) are extensively used in different industrial applications. Under some controlled fabrication conditions, axially aligned grooves can be formed on the HFM inner surface during typical immersion precipitation-based phase inversion fabrication processes. Such grooved HFMs are found to be promising for nerve repair and regeneration. The axially aligned grooves appearing on the inner surface of the membrane are considered as hydrodynamic instability patterns. During the immersion precipitation process, a transfer of solvent takes place across the interface between a polymer solution and a nonsolvent. This solvent transfer induces gradients of interfacial tension that are considered to be the driving mechanism for Marangoni instability. The onset of the stationary instability is studied by means of a linear instability theory, and the critical and maximum wavenumbers are determined and discussed in terms of the dimensionless groups characterizing the system: viscosity ratio, diffusivity ratio, Schmidt number, crispation number, adsorption number, Marangoni number, and the polymer bulk concentration. A good agreement is found between the predicted wavelength of the most dangerous wave and the experimental groove width. Consequently, solutal Marangoni instability can explain the groove formation mechanism in HFM fabrication.

Convective Instabilities in Two Liquid Layers.

We perform linear stability calculations for horizontal fluid bilayers, taking into account both buoyancy effects and thermocapillary effects in the presence of a vertical temperature gradient. To help understand the mechanisms driving the instability, we have performed both long-wavelength and short-wavelength analyses. The mechanism for the large wavelength instability is complicated, and the detailed form of the expansion is found to depend on the Crispation and Bond numbers. The system also allows a conventional Rayleigh-Taylor instability if heavier fluid overlies lighter fluid, and the long-wavelength analysis describes this case as well. In addition to the asymptotic analyses for large and small wavelengths, we have performed numerical calculations using materials parameters for a benzene-water system.

Introductory analysis of Bénard–Marangoni convection

We describe experiments on Bénard–Marangoni convection which permit a useful understanding of the main concepts involved in this phenomenon such as, for example, Bénard cells, aspect ratio, Rayleigh and Marangoni numbers, Crispation number and critical conditions. In spite of the complexity of convection theory, we carry out a simple and introductory analysis which has the additional advantage of providing very suggestive experiments. As a consequence, we recommend our device for use as a laboratory experiment for undergraduate students of the thermodynamics of nonlinear and fluid physics.

Feedback Control of the Marangoni–Bénard Instability in a Fluid Layer with a Free-Slip Bottom

Feedback control was applied to the steady Marangoni–Benard convection in a horizontal layer of fluid with a free-slip bottom heated from below and cooled from above. The critical values of the Marangoni numbers for the onset of steady convection are calculated and the latter is found to be critically dependent on the Crispation and Bond numbers. It is shown that the onset of instability can be delayed and the critical Marangoni number can be increased through the use of feedback control.

Marangoni Convection in a Composite Porous Layer and a Fluid Layer with a Deformable Free Surface

The linear stability analysis of Marangoni convection in a composite system comprised of an incompressible fluid-saturated porous layer underlying a layer of the same fluid is considered. The upper fluid surface, free to the atmosphere, is considered to be deformable and subjected to temperature- dependent surface tension. The fluid flow in the porous layer is governed by the Forchheimer-extended-Darcy equation, and both Beavers-Joseph and Jones conditions are employed at the interface between the two layers. The resulting eigenvalue problem is solved exactly and also by the regular perturbation technique when the boundaries of the system are insulated to temperature perturbations. The effect of the crispation number, Bond number, and the other physical parameters involved therein are analyzed for the stability of the system. It is found that a decrease in the crispation number and an increase in the Bond number delay the onset of convection. Also, the effect of the ratio of the fluid to the porous layer thickness along with the other physical parameters on the control (suppress or augment) of convection is analyzed in detail. © 2007 Begell House, Inc

Effect of a non-uniform basic temperature gradient on the onset of Bénard-Marangoni convection: Stationary and oscillatory analyses

The qualitative effect of a non-uniform basic temperature gradient on the stationary and oscillatory stability analyses of the onset of Bénard-Marangoni convective in a horizontal fluid layer is investigated numerically using the fourth order Runger-Kutta-Gill's method coupled with the iterative Broyden's method. The effects of crispation at a deformable upper free surface and the conductive effect of non-steady conditions within the fluid layer on onset are discussed. Results show that the effect of crispation (i.e., Crispation number C) is a clearly destabilizing factor, but the conductive effect (i.e., a i * of non-steady conditions within the fluid layer does play a stabilizing state. The system becomes more stable at larger values of the Prandtl number Pr, the Biot number B i and the Bond number B o . We obtain a set of characteristic curves in the ( M C, R)-plane that satisfy the linear relation on the onset of the stationary and oscillatory Bénard-Marangoni convective instability.

Oscillatory Marangoni convection in a conducting fluid layer with a deformable free surface in the presence of a vertical magnetic field

Linear stability theory is applied to the problem of the onset of oscillatory Marangoni convection in a horizontal layer of electrically conducting fluid heated from below in the presence of a vertical magnetic field. The fluid layer is bounded from below by a rigid boundary and from above by a deformable free surface. The critical Marangoni number M c, the critical wavenumber a c and the critical frequency ωc are obtained for wide ranges of the Prandtl number #E5/E5#1, the magnetic Prandtl number #E5/E5#2, the crispation number C r and the Chandrasekhar number Q. We present numerically a necessary and sufficient condition for oscillatory Marangoni convection to occur.

Effect of a non-uniform temperature gradient on the onset of oscillatory Bénard–Marangoni convection of an electrically conducting liquid in a magnetic field

This study deals theoretically with the effect of a non-uniform basic temperature gradient on the linear stability of the oscillatory Bénard–Marangoni convection in a horizontal layer of a viscous quiescent, electrically conducting fluid in the presence of a uniform vertical magnetic field. The upper surface of a fluid layer is deformably free and the lower surface is rigid. The eigenvalue equations of the perturbed state obtained from the normal mode analysis are solved by using the fourth order Runge–Kutta–Gill’s method with the shooting technique. The results show that the critical Rayleigh number Ra c and the critical Marangoni number − Ma c become larger as the Chandrasekhar number Q or the Biot number Bi of the upper free surface increases, and the Crispation number Cr decreases. When compared with the linear temperature profile, the inverted parabolic temperature profile indicates a reinforcement of stability, while the parabolic temperature profile indicates a diminution of stability in both Rayleigh–Bénard and Bénard–Marangoni convections. In addition, for the piecewise linear temperature profiles, the influences of thermal depth on the critical conditions are also obtained. Comparisons are also made of the critical conditions [− Ma c, a c, q ic] between the present results and published data sets for the linear basic temperature profile case, and the agreement is found out to be generally good.

Effects of nonuniform temperature gradients on the onset of oscillatory Marangoni convection in a magnetic field

This article theoretically studies the onset of oscillatory Marangoni convection in a horizontal layer of an electrically conducting fluid, to which a nonuniform thermal gradient and a uniform magnetic field are applied. The top surface of a fluid layer is deformably free and the bottom is rigid. By means of the linear stability theory and a normal mode analysis, the eigenvalue equations of the perturbed state are solved by using the fourth-order Runge-Kutta-Gill's method with the shooting technique. The computational results are compared with those known from the literature, and the agreement is found out to be generally good. The results indicate that the critical Marangoni number −Ma c increases with increasing the Chandrasekhar number Q, the Prandtl number Pr, or the Biot number Bi of the upper free surface, but decreases with increasing the Crispation number Cr. As compared with the linear temperature profile, the inverted parabolic temperature profile shows higher −Ma c values, while the parabolic temperature profile shows lower −Ma c values. In addition, for the piecewise linear temperature profiles, the influences of thermal depth on the critical Marangoni number are also obtained.

Effects of a.c. Electric Field and Rotation on Bénard–Marangoni Convection

The coupled buoyancy and thermocapillary instability, the Benard–Marangoniproblem, in an electrically conducting fluid layer whose upper surface is deformed and subject to a temperature gradient is studied. Both influences of an a.c. electric field and rotation are investigated. Special attention is directed at the occurrence of convection both in the form of stationary motion and oscillatory convection. The linear stability problem is solved for different values of the relevant dimensionless numbers, namely the a.c. electric Rayleigh number, the Taylor, Rayleigh, Biot, Crispation and Prandtl numbers. For steady convection, it is found that by increasing the angular velocity, one reinforces the stability of the fluid layer whatever the values of the surface deformation and the applied a.c. electric field. We have also determined the regions of oscillatory instability and discussed the competition between both stationary and oscillatory convections.

Linear and nonlinear Rayleigh–Bénard–Marangoni instability with surface deformations

Thermoconvective instabilities in a bilayer liquid–gas system with a deformed interface are investigated. In the first part of the work which is devoted to a linear approach, emphasis is put on the role of the upper gas layer on the instability phenomenon. The condition to be satisfied by the gas to remain purely conductive is established. The so-called Oberbeck–Boussinesq approximation is discussed and its range of validity is carefully defined. Instead of the classical Rayleigh, Marangoni, crispation, and Galileo numbers, new dimensionless groups are introduced. A critical comparison with several previous works is made. The nonlinear analysis consists in studying the different convective patterns which can appear above the threshold. Particular attention is devoted to the shape of the interface and the so-called “hybrid” relief. The amplitude of the deformation is also determined and comparison with experimental data is discussed.

Oscillatory instability analysis of Bénard-Marangoni convection in a rotating fluid under a uniform magnetic field

The effects of rotation and magnetic field on the onset of oscillatory modes of Bénard-Marangoni instability in a horizontal fluid layer with a deformably free surface are investigated numerically. The derived eigenvalue equations are solved using the fourth order Runge-Kutta-Gill's method coupled with the Broyden's method. The results show that the Crispation number C, associated with the deformation of the upper surface, is significant on the Bénard-Marangoni instability. The system is stabilizing, as the Biot number Bi, the Bond number Bo, the Taylor number F and the Chandrasekhar number Q increase. The absolute value of critical Marangoni number | M c| in the | M c| R)-plane decreases linearly with increasing Rayleigh number R.

Effects of Coriolis force and surface tension on Bénard-Marangoni convective instability

Abstract The effect of the Coriolis force on the onset of the Benard-Marangoni convective instability in a horizontal fluid layer, subject to a rotation, is investigated. The upper surface is deformably free and the lower surface is bounded with a solid plate, heated from below. The principle of exchange of stability is applied. The characteristic problem is solved numerically, using the Runge-Kutta-Gill's shooting method. The numerical results obtained compared very well with previously published ones. The thickness ratio d r , conductivity ratio k r , Biot number Bi , Crispation number C and Taylor number Ta are significant on determining the critical Rayleigh number R c or Marangoni number M c with the corresponding critical wavenumber a c . The results show that the system is more stabilizing at larger values of the thickness ratio d r , conductivity ratio k r , Biot number Bi and Taylor number Ta .