Theoretical Analysis of Anion Exclusion and Diffusive Transport Through Platy-Clay Soils

Diffusive transport through geosynthetic clay liners and engineered compacted clay landfill liners is the primary mechanism for mass transport from well-engineered modern landfills. For this reason, accurate estimates of diffusion coefficients for clay soils are essential for the design of engineered liner systems. A long-standing theoretical problem is the role of anion exclusion on the estimation of diffusion coefficients for ionic solutes migrating through charged porous media. This paper describes the steady-state solution of a fully coupled set of transport equations modeling ion movement through a permanently charged platy-clay soil. The microscale analysis takes into account the actual diffusion coefficient for each ion species, ion-pairing (as required by electroneutrality of the solution), as well as anion exclusion and cation inclusion ,arising from the permanent charge on clay particles. To render the problem tractable, the theoretical analysis focuses on an extremely small two-dimensional unit cell in an ideal, saturated, two-phase porous medium. The analysis presented here is limited to a particular geometrical example, but this example is sufficiently general for characteristic behaviours of systems of this kind to be identified. Most importantly, new insight is gained into the mechanism of ion migration through a charged platy-clay soil. The numerical results obtained from this study show that the identification of macroscopic transport quantities such as effective diffusion coefficients and membrane potentials from diffusion cell tests using standard diffusion theory only hold for a specific system. While ion exclusion behaviours are often referred to in the literature, as far as the authors are aware there has been no previous detailed microscale analysis of their role in steady-state diffusion through a charged platy-clay soil.     

Monitoring of Pollutant Diffusion into Clay Liners by Electrical Methods

A non-destructive test was carried out on a liner material—sand bentonite mixture (SB) with a continuous concentration diffusion of NaCl electrolyte. The work reported studied the spacio-temporal variation of the electrical conductivity $\sigma ^{*}_{\mathrm{s}}$ (z, t) in a diffusion soil column with different heights. A relationship between the interstitial pore fluid concentration of SB and the electrical conductivity of the solution has been established by mixing and compacting samples of sand bentonite with NaCl electrolytes at different concentrations. Electrical conductivity of compacted specimens was measured with a two-electrode cell. The conductivity measurements were used to quantify the pore fluid concentration and effective diffusion coefficient of SB liners. It is concluded here that the electrical conductivity of compacted specimens depends mainly on the salt concentration in the pore fluid and it could be used to measure ionic movement through liners during diffusion. The experimental diffusion coefficient reached theoretical diffusion coefficient when sample height is equal to 40 cm.     

Antiplane internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane

This paper deals with the antiplane magnetoelectroelastic problem of an internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane. The properties of the material such as elastic modulus, piezoelectric constant, dielectric constant, piezomagnetic coefficient, magnetoelectric coefficient and magnetic permeability are assumed in exponential forms and vary along the crack direction. Fourier transforms are used to reduce the impermeable and permeable crack problems to a system of singular integral equations, which is solved numerically by using the Gauss-Chebyshev integration technique. The stress, electric displacement and magnetic induction intensity factors at the crack tips are determined numerically. The energy density theory is applied to study the effects of nonhomogeneous material parameter β, edge conditions, location of the crack and load ratios on the fracture behavior of the internal crack.     

Heat,Water, and Solute Transfer in Unsaturated Porous Mdeia: II -- Compacted Soil Beneath Plastic Cover

Soil surface dynamics involve coupled transferof heat, water, and solute. An experimental andtheoretical study of heat, water, and solute transferin closed compacted soil columns under surfacetemperature wave amplitudes is presented. Thetemperature wave amplitudes ranged from 17.9 to 21.0°C. Potassium chloride solution was used tomoisten Clarinda clay and Fayette silty clay loamsoils. Initial water contents of 0.403 and 0.279 andinitial solute concentrations of 0.062 and 0.052 mol kg^-1were used in Clarinda and Fayette soils,respectively. The moistened soils were packed andcompacted in PVC columns (0.075 m diameter and 0.30 mhigh). Bulk densities of the compacted Clarinda andFayette soils were 1403 and 1585 kg m^-3,respectively. The columns were buried in soil suchthat column surfaces were exposed to natural as wellas artificial radiation and thermal conditions. Thecoupled nonsteady-state balance equations of mass andenergy were solved numerically to predict soiltemperature, water content, and solute concentrationdistributions. The theoretical model described soiltemperature, water content, and solute concentrationwell as compared with the measured values. TheFickian diffusive solute flux was one or two orders ofmagnitude greater than salt-sieving and thermal-diffusion solute fluxes.     

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.     

The Falkner–Skan Wedge Flows of Power-Law Fluids Embedded in a Porous Medium

The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.     

Numerical simulation of transpiration cooling through porous material

Transpiration cooling using ceramic matrix composite materials is an innovative concept for cooling rocket thrust chambers. The coolant (air) is driven through the porous material by a pressure difference between the coolant reservoir and the turbulent hot gas flow. The effectiveness of such cooling strategies relies on a proper choice of the involved process parameters such as injection pressure, blowing ratios, and material structure parameters, to name only a few. In view of the limited experimental access to the subtle processes occurring at the interface between hot gas flow and porous medium, reliable and accurate simulations become an increasingly important design tool. In order to facilitate such numerical simulations for a carbon/carbon material mounted in the side wall of a hot gas channel that are able to capture a spatially varying interplay between the hot gas flow and the coolant at the interface, we formulate a model for the porous medium flow of Darcy–Forchheimer type. A finite‐element solver for the corresponding porous medium flow is presented and coupled with a finite‐volume solver for the compressible Reynolds‐averaged Navier–Stokes equations. The two‐dimensional and three‐dimensional results at Mach number Ma = 0.5 and hot gas temperature T_HG=540 K for different blowing ratios are compared with experimental data. Copyright © 2014 John Wiley & Sons, Ltd.     

A modified finite element method for solving the time-dependent,incompressible Navier-Stokes equations. Part 1: Theory

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.     

Effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux

The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field. It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field. The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF). The parabolic equations obtained from FVF are numerically integrated with the help of a straightforward finite difference method. Moreover, the nonsimilar system of equations obtained from SFF is solved by using a local nonsimilarity method, for the whole range of the local transpiration parameter ζ. Consideration is also given to the regions where the local transpiration parameter ζ is small or large enough. However, in these particular regions, solutions are acquired with the aid of a regular perturbation method. The effects of the magnetic field M and the Hall parameter m on the local skin friction coefficient and the local Nusselt number coefficient are graphically shown for smaller values of the Prandtl number Pr (= 0.005, 0.01, 0.05). Furthermore, the velocity and temperature profiles are also drawn from various values of the local transpiration parameter ζ.     

On the Stress Tensor Jump at the Interface between a Polyatomic Gas and a Condensed Phase

For the components of the hydrodynamic stress tensor the boundary condition at the interface between a polyatomic gas and a condensed phase is obtained. The boundary-value problem is solved within the framework of the previously proposed kinetic model by the method of semi-spatial moments, taking into account the rotational degrees of freedom of the gas molecules. The gas-kinetic coefficient C _p entering into the boundary condition for the stress tensor components depends on the accommodation coefficient of the tangential momentum, q, the accommodation coefficients of the translational, _t , and rotational, _r , energy components, and the Prandtl number. This coefficient is calculated for several polyatomic gases.     

Efficient retrieval of the thermo‐acoustic flame transfer function from a linearized CFD simulation of a turbulent flame

The stability of thermo‐acoustic pressure oscillations in a lean premixed methane‐fired generic gas turbine combustor is investigated. A key element in predicting the acoustically unstable operating conditions of the combustor is the flame transfer function. This function represents the dynamic relationship between a fluctuation in the combustor inlet conditions and the flame's acoustic response. A transient numerical experiment involving spectral analysis in computational fluid dynamics (CFD) is usually conducted to predict the flame transfer function. An important drawback of this spectral method application to numerical simulations is the required computational effort. A much faster and more accurate method to calculate the transfer function is derived in this paper by using a most important basic assumption: the fluctuations must be small enough for the system to behave linear. This alternative method, which is called the linear coefficient method, uses a linear representation of the unsteady equations describing the CFD problem. This linearization is applied around a steady‐state solution of the problem, where it can consequently describe the dynamics of the system. Finally, the flame transfer function can be calculated from this linear representation. The advantage of this approach is that one only needs a steady‐state solution and linearization of the unsteady equations for calculating a dynamic transfer function, i.e. no time‐consuming transient simulations are necessary anymore. Nevertheless, as a consequence of the large number of degrees of freedom in a CFD problem, an extra order reduction step needs to be performed prior to calculating the transfer function from the linear representation. Still, the linear coefficient method shows a significant gain in both speed and accuracy when calculating the transfer function from the linear representation as compared to a spectral analysis‐based calculation. Hence, this method gives a major improvement to the application of the flame transfer function as a thermo‐acoustic design tool. Copyright © 2007 John Wiley & Sons, Ltd.     

Heat transfer in a thin layer of a viscous heavy noncompressible liquid under wavy flow conditions

The article describes a method for calculating the flow of heat through a wavy boundary separating a layer of liquid from a layer of gas, under the assumption that the viscosity and heat-transfer coefficients are constant, and that a constant temperature of the fixed wall and a constant temperature of the gas flow are given. A study is made of the equations of motion and thermal conductivity (without taking the dissipation energy into account) in the approximations of the theory of the boundary layer; the left-hand sides of these equations are replaced by their averaged values over the layer. These equations, after linearization, are used to determine the velocity and temperature distributions. The qualitative aspect of heat transfer in a thin layer of viscous liquid, under regular-wavy flow conditions, is examined. Particular attention is paid to the effect of the surface tension coefficient on the flow of heat through the interface.Notation x, y coordinates of a liquid particle - t time - v and u coordinates of the velocity vector of the liquid - p pressure in the liquid - c_v, , T,, andv heat capacity, thermal conductivity coefficient, temperature, density, and viscosity of the liquid, respectively - g acceleration due to gravity - surface-tension coefficient - c phase velocity of the waves at the interface - T_w wall temperature - h₀ thickness of the liquid layer - u₀ velocity of the liquid over the layer Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 147–151, July–August, 1970.     

Micromechanical computational modeling of expansive porous media

A modified Terzaghi principle is proposed to describe the influence of locally coupled electro-chemo-mechanical processes in highly compacted swelling clays upon the form of the macroscopic modified effective stress principle. The two-scale model is derived using the homogenization procedure to upscale the microscopic behavior of a two-phase system composed of clay particles saturated by a completely dissociated electrolyte aqueous solution. Numerical experiments are performed to illustrate the results in a particular cell geometry. To cite this article: M.A. Murad, C. Moyne, C. R. Mecanique 330 (2002) 865–870.     

Kinetic and phenomenological theories for the linearized Burnett equations of a molecular gas

The linearized Burnett equations for a molecular gas are obtained from a kinetic theory based on the Boltzmann equation, and from a phenomenological theory based on extended thermodynamics. The constitutive equation for the pressure tensor of a molecular gas has three terms that do not have appeared in the corresponding equation for a monatomic ideal gas. One is the well-known term proportional to divergence of velocity whose coefficient is the volume viscosity. The two others are proportional to Laplacians of the temperature and of the density, and are associated with athermal (or temperature) pressure and with adensity pressure, respectively.     

On the modeling and simulation of a laser‐induced cavitation bubble

Single cavitation bubbles exhibit severe modeling and simulation difficulties. This is due to the small scales of time and space as well as due to the involvement of different phenomena in the dynamics of the bubble. For example, the compressibility, phase transition, and the existence of a noncondensable gas inside the bubble have strong effects on the dynamics of the bubble. Moreover, the collapse of the bubble involves the occurrence of critical conditions for the pressure and temperature. This adds extra difficulties to the choice of equations of state. Even though several models and simulations have been used to study the dynamics of the cavitation bubbles, many details are still not clearly accounted for. Here, we present a numerical investigation for the collapse and rebound of a laser‐induced cavitation bubble in liquid water. The compressibility of the liquid and vapor are involved. In addition, great focus is devoted to study the effects of phase transition and the existence of a noncondensable gas on the dynamics of the collapsing bubble. If the bubble contains vapor only, we use the six‐equation model for two‐phase flows that was modified in our previous work [A. Zein, M. Hantke, and G. Warnecke, J. Comput. Phys., 229(8):2964‐2998, 2010]. This model is an extension to the six‐equation model with a single velocity of Kapila et al. (Phys. Fluid, 13:3002‐3024, 2001) taking into account the heat and mass transfer. To study the effect of a noncondensable gas inside the bubble, we add a third phase to the original model. In this case, the phase transition is considered only at interfaces that separate the liquid and its vapor. The stiffened gas equations of state are used as closure relations. We use our own method to determine the parameters to obtain reasonable equations of state for a wide range of temperatures and make them suitable for the phase transition effects. We compare our results with experimental ones. Also our results confirm some expected physical phenomena. Copyright © 2013 John Wiley & Sons, Ltd.     

Singularly perturbed feedback linearization with linear attitude deviation dynamics realization

A new approach for feedback linearization of attitude dynamics for rigid gas jet-actuated spacecraft control is introduced. The approach is aimed at providing global feedback linearization of the spacecraft dynamics while realizing a prescribed linear attitude deviation dynamics. The methodology is based on nonuniqueness representation of underdetermined linear algebraic equations solution via nullspace parametrization using generalized inversion. The procedure is to prespecify a stable second-order linear time-invariant differential equation in a norm measure of the spacecraft attitude variables deviations from their desired values. The evaluation of this equation along the trajectories defined by the spacecraft equations of motion yields a linear relation in the control variables. These control variables can be solved by utilizing the Moore–Penrose generalized inverse of the involved controls coefficient row vector. The resulting control law consists of auxiliary and particular parts, residing in the nullspace of the controls coefficient and the range space of its generalized inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a nullprojection matrix, and is designed to yield exponentially stable spacecraft internal dynamics, and singularly perturbed feedback linearization of the spacecraft attitude dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore–Penrose generalized inverse, in addition to the concepts of singularly perturbed controls coefficient nullprojection and damped controls coefficient nullprojection to disencumber the nullprojection matrix from its rank deficiency, and to enhance the closed loop control system performance. The methodology yields desired linear attitude deviation dynamics realization with globally uniformly ultimately bounded trajectory tracking errors, and reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability. The paper bridges a gap between the nonlinear control problem applied to spacecraft dynamics and some of the basic generalized inversion-related analytical dynamics principles.     

Heat-Mass-Transport and Thermal Stresses in Porous Charring Materials

The aim of this paper is to develop a method of asymptotic averaging for processes occurring in porous charring materials under high temperatures. The advantage of the method is the ability to calculate not only averaged macrocharacteristics of the processes, namely internal gas generation, filtration and deforming processes, but also microcharacteristics, such as microstresses in phases of charring material, gas velocity in a pore, etc. To determine microcharacteristics, the method allows us to formulate special mathematical problems on a periodic cell. To calculate macrocharacteristics, such as pore pressure of filtrating gas, rate of charring and macrostresses, with the help of asymptotic averaging method, averaged global equations are formulated. Here effective characteristics of porous medium (gas permeability coefficient, rate of charring, elasticity modulus, thermal expansion coefficient) are determined not empirically, as in most works on porous materials, but on the basis of solving the local problems. Solution of these problems over the periodic cell allows us to derive analytically the law of the Darcy type for a gas phase flow in porous media, to obtain an expression for intensive mass transfer between solid and gas phases, to set the form of constitutive relations for charring porous media, and also to calculate microstresses in a vicinity of a growing pore. As an example of solving a global averaged problem, the problem on one-sided high-temperature heating of a plate made of epoxy binder has been solved numerically.     

Fractal model for virgin compression of pure clays

The well-known linear relation between the void ratio and the logarithm of pressure for the one-dimensional consolidation has widely incorporated into elastoplastic constitute equation of soils, however, it is not at all straight line for more compressible clays and has several physically unaccepted properties. The compression of clays is explained by fractal theory, and is seen as a balance between the vertical pressure and repulsive force between clay plates. The distance between clay plates depends on the structure formation of the clay flocculation and clay surface. The structure formation of the clay flocculation and clay surface can be modeled by fractal approach. A new formulation of one-dimensional virgin compression of pure clays, expressed by the linear function of ln ν–ln p or ln e–ln p, is proposed based on the fractal model for clay structure. The linear function of ln ν–ln p is suitable to express the virgin compression of the extremely high plasticity clays up to the high pressure of 5000 kPa. For low plasticity clays, the virgin compression satisfies with the linear relationship of ln e–ln p at the pressure higher than 100 kPa. These too are show to be in satisfactory agreement with experimental data. The gradient of the plots of ln ν–ln p and ln e–ln p is related to the fractal dimensions of clay structure, the fundamental material parameters for the first time.     

Thermal slip of a nonuniformly heated gas along a solid planar surface

Two methods are used to study the solution of a linearized model of the Boltzmann equation in the problem of thermal slip of a nonuniformly heated gas along a solid flat wall.The first method involves analytic solution of the integral equation for the average gas velocity. In the case of purely diffuse or purely specular reflection of the molecules from the wall surface the first method makes it possible to obtain analytically two important results; namely, the average gas velocity at the surface and at a large distance from the wall. The average gas velocity profile cannot be constructed analytically with this method. The second approximate method involves expanding the distribution function into a series in Sonine polynomials in velocity space and formulation of half-space moment equations from which the correction to the distribution function is determined. This method is used to obtain a simple analytic expression for the distribution function, from which we can find the average velocity profile for the gas for any arbitrary tangential momentum accommodation coefficient. In particular cases in which analytic solution of the problem by the first method is possible, good agreement is obtained between the two computational methods.It is known that a gas in a temperature gradient field tangent to the wall must begin to move in the direction of the temperature gradient (thermal slip). The first attempt to solve the thermal slip problem was made by Maxwell [1]. In his analysis Maxwell assumed that the distribution function of the molecules incident on the wall near the surface does not differ from the bulk distribution at a large distance from the wall. As a result Maxwell obtained the following expression for the thermal slip velocity for any tangential momentum accommodation coefficient u ^*=3/4 grad lnT.Here is the kinematic viscosity.However, in the case of molecular reflection from the wall which is not purely specular, the distribution of the incident molecules in the Knudsen layer differs from the bulk distribution because of collisions with the molecules reflected from the wall. Thus, Maxwell's assumption is not valid in the general case.For the exact solution of the problem it is necessary to find the distribution function in the Knudsen layer by solving the Boltzmann equation. Several investigators have used the Grad method [2] to find the distribution function in the Knudsen layer. However, the use of Grad's method in the thermal-slip problem leads to Maxwell's result [3].The solution of the thermal-slip problem obtained by Sone [4] is more exact than the analyses noted above. A comparison of the results obtained by Sone with those of this investigation is given at the end of our paper.