Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Free convection from one thermal and solute source in a confined porous medium

This paper reports a numerical study of double diffusive natural convection in a vertical porous enclosure with localized heating and salting from one side. The physical model for the momentum conservation equation makes use of the Darcy equation, and the set of coupled equations is solved using the finite-volume methodology together with the deferred central difference scheme. An extensive series of numerical simulations is conducted in the range of −10 ⩽ N ⩽ + 10, 0 ⩽ R _t ⩽ 200, 10⁻² ⩽ Le ⩽ 200, and 0.125 ⩽ L ⩽ 0.875, where N, R _t , Le, and L are the buoyancy ratio, Darcy-modified thermal Rayleigh number, Lewis number, and the segment location. Streamlines, heatlines, masslines, isotherms, and iso-concentrations are produced for several segment locations to illustrate the flow structure transition from solutal-dominated opposing to thermal dominated and solutal-dominated aiding flows, respectively. The segment location combining with thermal Rayleigh number and Lewis number is found to influence the buoyancy ratio at which flow transition and flow reversal occurs. The computed average Nusselt and Sherwood numbers provide guidance for locating the heating and salting segment.     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

A Numerical Study Promoting Algebraic Models for the Lewis Number Effect in Atmospheric Turbulent Premixed Bunsen Flames

This numerical investigation carried out on turbulent lean premixed flames accounts for two algebraic – the Lindstedt–Vaos (LV) and the classic Bray–Moss–Libby (BML) – reaction rate models. Computed data from these two models is compared with the experimental data of Kobayashi et al. on 40 different methane, ethylene and propane Bunsen flames at 1 bar, where the mean flame cone angle is used for comparison. Both models gave reasonable qualitative trend for the whole set of data, in overall. In order to characterize quantitatively, firstly, corrections are made by tuning the model parameters fitting to the experimental methane–air (of Le = 1.0) flame data. In case of the LV model, results obtained by adjusting the pre-constant, i.e., reaction rate parameter, C_R, from its original value 2.6 to 4.0, has proven to be in good agreement with the experiments. Similarly, for the BML model, with the tuning of the exponent n, in the wrinkling length scale, L_y = C_l⋅ l_x(s_L/u′)ⁿ from value unity to 1.2, the outcome is in accordance with the measured data. The deviation between the measured and calculated data sharply rises from methane to propane, i.e., with increasing Lewis number. It is deduced from the trends that the effect of Lewis number (for ethylene–air mixtures of Le = 1.2 and propane–air mixtures of Le = 1.62) is missing in both the models. The Lewis number of the fuel–air mixture is related to the laminar flame instabilities. Second, in order to quantify for its influence, the Lewis number effect is induced into both the models. It is found that by setting global reaction rate inversely proportional to the Lewis number in both the cases leads to a much better numerical prediction to this set of experimental flame data. Thus, by imparting an important phenomenon (the Lewis number effect) into the reaction rates, the generality of the two models is enhanced. However, functionality of the two models differs in predicting flame brush thickness, giving scope for further analysis.     

Numerical study on heat/mass transfer by shear/thermal/solutal-convection in a cavity under counteracting buoyancies

This study provides physical insights into a lid-driven square cavity filled with a mixture of a solvent vapor and non-condensable gas, subjected to the vertically parallel thermal and solutal gradients. The top lid is maintained at constant speed while bottom lid and the other two walls are kept fixed. Zero heat and mass fluxes are imposed on the vertical side walls. The transport equations are solved numerically through a pressure-correction-based iterative algorithm (SIMPLE) with the QUICK scheme for convective terms. The diffusivities of heat and salt are assumed to be equal throughout this investigation. The essential details of flow, temperature and concentration fields are presented for the opposing buoyancy forces ratio (B < 0) with special attention being given for the values of parameters for which either the flow inside the cavity is operated by the mechanically induced convection; or the flow structure inside the cavity is akin to a single-diffusive thermal or solutal convection. The variations of average rates of heat and mass transfer are uniform with the Reynolds number, while the variations of these quantities against the solutal Richardson number (Ri _C ) and thermal Richardson number (Ri _T ) point to the existence of the local minimum/maximum. Finally, two linear relations between Ri _C and Ri _T at a constant sliding speed are proposed to identify the above points of high and low transport phenomena and justified with the exhibition of the flow structures inside the cavity.     

The optimal dimensions of convective-radiating circular fins

The optimal dimensions of convective-radiating circular fins with variable profile, heat-transfer coefficient and thermal conductivity, as well as internal heat generation are obtained. A profile of the form y=(w/2) [1+(r _o/r) ⁿ ] is studied, while variation of thermal conductivity is of the form k=k _o[1+ɛ((T−T _∞)/ (T _b−T _∞)) ^m ]. The heat-transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h=K[(r−r _o)/(r _e−r _o)]^λ. The results for λ=0 to λ=1.9, and −0.4≤ɛ≤0.4, have been expressed by suitable dimensionless parameters. A correlation for the optimal dimensions of a constant and variable profile fins is presented in terms of reduced heat-transfer rate. It is found that a (quadratic) hyperbolic circular fin with n=2 gives an optimum performance. The effect of radiation on the fin performance is found to be considerable for fins operating at higher base temperatures, whereas the effect of variable thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed, in general, that the optimal fin length and the optimal fin base thickness are greater when compared to constant fin thickness. Received on 22 February 1999     

Thermophoresis particle deposition in a non-Darcy porous medium under the influence of Soret, Dufour effects

Thermophoresis particle deposition in free convection on a vertical plate embedded in a fluid saturated non-Darcy porous medium is studied using similarity solution technique. The effect of Soret and Dufour parameters on concentration distribution, wall thermophoretic deposition velocity, heat transfer and mass transfer is discussed in detail for different values of dispersion parameters (Ra _γ, Ra _ξ) inertial parameter F and Lewis number Le. The result indicates that the Soret effect is more influential in increasing the concentration distribution in both aiding as well as opposing buoyancies. Also, the non-dimensional heat transfer coefficient and non-dimensional mass transfer coefficient changes according to different values of thermophoretic coefficient k.     

Slip-flow heat transfer in a microchannel with viscous dissipation

Forced convection flow in a microchannel with constant wall temperature is studied, including viscous dissipation effect. The slip-flow regime is considered by incorporating both the velocity-slip and the temperature-jump conditions at the surface. The energy equation is solved for the developing temperature field using finite integral transform. To increase β_v Kn is to increase the slip velocity at the wall surface, and hence to decrease the friction factor. Effects of the parameters β_v Kn, β, and Br on the heat transfer results are illustrated and discussed in detail. For a fixed Br, the Nusselt number may be either higher or lower than those of the continuum regime, depending on the competition between the effects of β_v Kn and β. At a given β_v Kn the variation of local Nusselt number becomes more even when β becomes larger, accompanied by a shorter thermal entrance length. The fully developed Nusselt number decreases with increasing β irrelevant to β_v Kn. The increase in Nusselt number due to viscous heating is found to be more pronounced at small β_v Kn.     

Soret effect and slow mass diffusion as a catalyst for overstability in Marangoni-Bénard flows

We study the onset of time dependent Marangoni-Bénard convection in binary mixtures subject to Soret effect by numerical computation of linear instability thresholds in infinite fluid layers and two-dimensional boxes. The calculations are done for positive Marangoni numbers (Ma > 0) and negative Marangoni Soret parameters S _M = –(D _S γ _c )/(Dγ _T ) where D _S and D are the Soret and mass diffusion coefficients, respectively, and γ _T , γ _c are the first derivatives of the surface tension with respect to temperature and concentration. Our purpose is to understand why for particular choices of Prandtl and Schmidt numbers, the increase of the stabilizing solutal contribution leads to a decrease of the critical temperature difference, a phenomenon already reported by Chen & Chen [5] and Skarda et al. [12] For various choices of Prandtl and Schmidt numbers we analyze the evolution of the critical Marangoni number Ma _c , critical wavenumber k _c and angular frequency ω _c with S _M and compute the corresponding eigenvectors. We next propose a physical mechanism which explains how the stabilizing solutal contribution acts as a catalyst for overstability. Finally, we extend our results to two dimensional boxes of small aspect ratio.     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

A simple model for the refractive index of sodium iodide aqueous solutions

A model for predicting the refractive index of sodium iodide (NaI) aqueous solution n _NaI as a function of temperature T, NaI concentration c and wavelength λ was determined for moderate parameter variations. The equation accurately predicted the salt concentration required to match n _NaI to the refractive index of Pyrex n _P. Received: 30 July 1998/Accepted: 14 December 1998     

Thermal convection in a porous layer: Effects of anisotropy and surface boundary conditions

The theory describing the onset of convection in a homogeneous porous layer bounded above and below by isothermal surfaces is extended to consider an upper boundary which is partly permeable. The general boundary condition p + λ ∂p/∂n = constant is applied at the top surface and the flow is investigated for various λ in the range 0 ⩽ λ < ∞. Estimates of the magnitude and horizontal distribution of the vertical mass and heat fluxes at the surface, the horizontally-averaged heat flux (Nusselt number) and the fraction of the fluid which recirculates within the layer are found for slightly supercritical conditions. Comparisons are made with the two limiting cases λ → ∞, where the surface is completely impermeable, and λ = 0, where the surface is at constant pressure. Also studied are the effects of anisotropy in permeability, ξ = K _H /K _V , and anisotropy is thermal conductivity, η = k _H /k _V , both parameters being ratios of horizontal to vertical quantities. Quantitative results are given for a wide variety of the parameters λ, ξ and η. In the limit ξ/η → 0 there is no recirculation, all fluid being converted out of the top surface, while in the limit ξ/η → ∞ there is full recirculation.     

Reynolds Number Dependence of Velocity Structure Functions in a Turbulent Pipe Flow

Low-order moments of the increments δu andδv where u and v are the axial and radial velocity fluctuations respectively, have been obtained using single and X-hot wires mainly on the axis of a fully developed pipe flow for different values of the Taylor microscale Reynolds numberR _λ. The mean energy dissipation rate〉ε〈 was inferred from the uspectrum after the latter was corrected for the spatial resolution of the hot-wire probes. The corrected Kolmogorov-normalized second-order structure functions show a continuous evolution withR _λ. In particular, the scaling exponentζ _v , corresponding to the v structure function, continues to increase with R _λ in contrast to the nearly unchanged value of ζ _u . The Kolmogorov constant for δu shows a smaller rate of increase with R _λ than that forδv. The level of agreement with local isotropy is examined in the context of the competing influences ofR _λ and the mean shear. There is close but not perfect agreement between the present results on the pipe axis and those on the centreline of a fully developed channel flow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Variable permeability effect on convection in binary mixtures saturating a porous layer

The Darcy Model with the Boussinesq approximation is used to study natural convection in a shallow porous layer, with variable permeability, filled with a binary fluid. The permeability of the medium is assumed to vary exponentially with the depth of the layer. The two horizontal walls of the cavity are subject to constant fluxes of heat and solute while the two vertical ones are impermeable and adiabatic. The governing parameters for the problem are the thermal Rayleigh number, R _T, the Lewis number, Le, the buoyancy ratio, φ, the aspect ratio of the cavity, A, the normalized porosity, ε, the variable permeability constant, c, and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained on the basis of the parallel flow approximation. The onset of supercritical convection, or subcritical, convection are predicted by the present theory. A linear stability analysis of the parallel flow model is conducted and the critical Rayleigh number for the onset of Hopf’s bifurcation is predicted numerically. Numerical solutions of the full governing equations are found to be in excellent agreement with the analytical predictions.     

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients,and Horizontal Throughflow

Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R _v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R _vc, we determine the value R _vt of R _v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R _vt, is only slightly greater than the critical value, R _vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R _vt is significantly greater than the critical value, R _vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.     

Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations

We consider reaction diffusion equations of the prototype form u _t = u _xx + λ u + |u| ^p-1 u on the interval 0 < x < π, with p > 1 and λ > m ². We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u ₁, . . . ,u _m . Since u _k = ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x ₁, . . . ,x _m of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation). Dedicated to Palo Brunovsky on the occasion of his birthday.     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.     

Analytic homotopy solution of generalized three-dimensional channel flow due to uniform stretching of the plate

In this communication a generalized threedimensional steady flow of a viscous fluid between two infinite parallel plates is considered. The flow is generated due to uniform stretching of the lower plate in x- and y-directions. It is assumed that the upper plate is uniformly porous and is subjected to constant injection. The governing system is fully coupled and nonlinear in nature. A complete analytic solution which is uniformly valid for all values of the dimensionless parameters β, Re and λ is obtained by using a purely analytic technique, namely the homotopy analysis method. Also the effects of the parameters β, Re and λ on the velocity field are discussed through graphs.     

A Study on Elliptical Gas Flow in Tri-porosity Gas Reservoirs

Elliptical flow is common in the near vertical fracture area and in anisotropic reservoirs. However, the classical radial flow models cannot provide a complete analysis for elliptical flow. This article presents a new mathematic model for gas elliptical flow in tri-porosity gas reservoirs. The differential equation of the new model is written in Mathieu equation, so that the solution can also be expressed by Mathieu functions. The numerical solution of the corresponding Mathieu functions ce _2n (ξ, −q), Ke _2n (ξ, −q) and their derivatives are obtained to derive the dimensionless pseudo pressure drop in Laplace space. The sensitivities of tri-porosity systems, including the parameters related to anisotropies C _De^2S and ξ _w, the storativity ratios ω _f and ω _m, and the interporosity flow coefficients λ_vf and λ_mf, are studied using Laplace numerical inversion. The new solution includes not only the factors considered in classic solutions in previous articles, but also incorporates the effect of reservoir anisotropy.