The crumbling of a viscous prism with an inclined free surface

Summary We study the two-dimensional instantaneous Stokes flow driven by gravity in a viscous triangular prism supported by a horizontal rigid substrate and a vertical wall. The oblique side of the prism, inclined at an angle α with respect to the substrate, is a fluid-air interface, where the stresses are zero and surface tension is neglected. We develop the stream function ψ in polar coordinates (r,θ) centered at the vertex of α and split it into an inhomogeneous part, which accounts for gravity effects, and a homogeneous part, which is expressed as a series expansion. The inhomogeneous part and the first term of the expansion may be envisioned, respectively, as self-similar solutions of the first kind and of the second kind for r→0, each one holding in complementary α domains with a crossover at α _c =21.47^∘, which we study in some detail. The coefficients of the series are calculated by truncating the expansion and using the method of direct collocation with a suitable set of points at the wall. The solution strictly holds for t=0, because later the free surface ceases to be a plane; nevertheless, it provides a good approximation for the early time evolution of the fluid profile, as shown by the comparison with numerical simulations. For 0<α<45^∘, the vertex angle remains constant and the edge remains strictly at rest; the transition at α _c manifests itself through a change in the rate of growth of the curvature. The time at which the edge starts to move (waiting time) cannot be calculated since the instantaneous solution ceases to be valid. For α>45^∘, the instantaneous local solution indicates that the surface near the vertex is launched against the substrate so that the edge starts to move immediately with a power law dependence on time t. However, due to the high value of the exponent, the vertex may seem to be at rest for some finite time even in this case. Received 29 August 1997; accepted for publication 21 January 1998     

Mobile and immobile fluid transport: Coupling framework

We introduce a solver method for mobile and immobile transport regions. The motivation is driven by transport processes in porous media (e.g. waste disposal, chemical deposition processes). We analyze the coupled transport‐reaction equation with mobile and immobile areas. We apply analytical methods, such as Laplace‐transformation, and for the numerical methods we apply Godunov's scheme, see (Mat. Sb. 1959; 47 :271–306; Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, 2002). The method is based numerically on flux‐based characteristic methods and is an attractive alternative to the classical higher‐order TVD methods, see (J. Comput. Phys. 1993; 49 :357–393). In this paper, we will focus on the derivation of analytical solutions for general and special solutions of the characteristic methods that are embedded in a finite‐volume method. At the end of the paper, we illustrate the higher‐order method for different benchmark problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Mixed‐order interpolation for the Galerkin coarse‐grid approximations in algebraic multigrid solvers

An empirical investigation is made of AMG solver performance for the fully coupled set of Navier–Stokes equations. The investigation focuses on two different FV discretizations for the standard driven cavity test problem. One is a collocated vertex‐based discretization; the other is a cell‐centred staggered‐grid discretization. Both employ otherwise identical orthogonal Cartesian meshes. It is found that if mixed‐order interpolation is used in the construction of the Galerkin coarse‐grid approximation (CGA), a close‐to‐optimum mesh‐independent scaling of the AMG convergence is observed with similar convergence rates for both discretizations. If, on the other hand, an equal‐order interpolation is used, convergence rates are mesh‐dependent but the scaling differs in each case. For the collocated‐grid case, it depends both on the mesh size, h (or bandwidth Q～h^?1) and on the total number of grids, G, whereas for the staggered‐grid case it depends only on Q. Comparing the two characteristics reveals that the Q‐dependent parts are very similar; it is only in the G‐dependent convergence for the collocated‐grid case that they differ. This takes the form of stepped reductions in the AMG convergence rate (implying step reductions in the quality of the Galerkin CGA that correlate exactly with step increases in G). These findings reinforce previous evidence that, for optimum mesh‐independent performance, mixed‐order interpolations should be used in forming Galerkin CGAs for coupled Navier–Stokes problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

Series solutions for steady unsaturated flow in irregular porous domains

For the most part, analytical solutions for steady unsaturated infiltration have been restricted to infinite and semi-infinite seepage geometries, using the quasi-linear approximation for the hydraulic conductivity. We provide analytical series methods to solve the steady quasi-linear flow equations, in finite irregular seepage geometries. Unlike the classical approach, the series method has been modified, to cater for arbitrary boundary geometry and surface recharge distributions. The matrix flux potential and the stream function both satisfy the same governing partial differential equation, and the stream function formulation is used to estimate the series coefficients. For a finite vadose zone, the stream function solution does not uniquely determine the matrix flux potential, when flux boundary conditions are used. Consequently, the stream function solution applies to a range of moisture distributions, for given infiltration and evapotranspiration rates through the surface.     

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The non-linear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0<t^*<1 and reach the steady-state values for t^*≥4.     

Anomalous diffraction model of linear optical dichroism in shear-thickening polymer solutions

The anomalous diffraction approximation (ADA) has bees recently applied to interpret measurements of the linear optical dichroism induced by shear in shear-thickening polymer solutions. A conceptual problem in this application is discussed, and a minor modification to the interpretation is proposed which is concordant with earlier magneto-optic results, but retains the correct use of the ADA.     

Electro-osmotic pore pressures in soil due to an alternating electrical field

Pore pressure development in a soil specimen due to electro-osmosis under alternating current conditions is examined theoretically. Solutions to the governing equation are derived for one-dimensional flow with boundary conditions corresponding to an impervious (conventional no-flow boundary), a partially drained boundary, and a partially drained boundary with an intervening permeable zone between the boundary and the soil. These latter two boundary conditions can arise from details of pore pressure measuring systems at the specimen boundaries during laboratory experiments. An analysis of the solutions indicates that for a perfect no-flow boundary, excess pore pressures measured at an electrode consist of a steady state and rapidly-decaying transient response. The pore pressures exhibit a 45 degree phase shift relative to the applied electric current. The effect of the partially drained boundary is to reduce the peak to peak amplitude of the pore pressure and to increase the phase shift to as much as 90 degrees depending on the compressibility of the pore pressure measuring system. The effect of the impeded and partially drained boundary is to further reduce the amplitude of the pore pressures and to increase the phase shift to as much as 180 degrees depending on the relative permeability of the impeded boundary.     

Investigations on vibration characteristics of two-layered piezoceramic disks

This paper investigates the transverse and planar vibration characteristics of two-layered piezoceramic disks for traction-free boundary conditions by theoretical analysis, finite element numerical calculation, and experimental measurements. Amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), laser Doppler vibrometer (LDV), and impedance analysis were used to perform measurements and verify the theoretical solutions for extensional, tangential, and transverse vibrations. The poling direction of piezoelectric elements determines whether they are denoted as either of series- or parallel-type. This study observed that the resonant frequencies and mode shapes of the series- and parallel-type piezoceramic disks present different dynamic characteristics in resonance. Planar and transverse vibrations are coupled in series-type piezoceramic disks and uncoupled in those of parallel-type. Good agreements of dynamic characteristics determined by theoretical analysis, experimental measurements, and numerical calculation are presented for series- and parallel-type piezoceramic disks.     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.