On the steady motion of two immiscible fluids in an undulating porous reservoir

We describe steady two-dimensional flows of two immiscible fluids through an undulating porous medium of constant thickness, with impermeable or slightly permeable boundaries. Flows in the same or opposite directions are called, respectively, direct or counter flows. Three special classes of flow are determined:

1. The pressure dominated case occurs for high direct flows and has the interface approximately a constant vertical distance from the impermeable boundaries.
2. The gravity dominated case occurs for low direct flows and has the interface very close to the lower (upper) boundary for downward (upward) sloping boundaries except at crossovers.
3. Counter flows require the interface to decrease in the direction of flow of the lower fluid.

Numerical examples illustrate the three classifications above. For incompressible flows the interface and pressure equations uncouple. A stability analysis shows that the direction of integration of the differential equation for the interface must be opposite to the flow direction for direct flows; for counter flows the direction of integration depends on whether the interface is above or below a critical height. Direct flows through cyclic geometries are asymptotically cyclic upstream. If the reservoir is ‘leaky’, asymptotically self-similar flows result when the (small) permeability ratio is scaled to the dynamical flow parameters.     

On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).     

The initial-value problem for elastodynamics of incompressible bodies

We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials. Our method consists in:

1. Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.
2. Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:

1. Using energy estimates to bound the growth of various Sobolev norms of solutions.
2. Finding the solution as the limit of a sequence of solutions of linearized problems.

Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.     

Flow of a dipolar fluid between parallel plates

The flow of a dipolar fluid between two parallel plates with and without heat transfer is studied. The following cases are discussed:

1. Isothermal flow due to the relative motion of the plates,
2. Isothermal flow due to a constant pressure gradient with the plates at rest,
3. Nonisothermal flow with linearly varying plate temperatures.

Case (ii) is of particular interest to the experimentalists as it shows the effect of the material constants even when there are no externally applied dipolar tractions on the plates.     

Guidance properties of nonlinear planar waveguides

We discuss the propagation of electromagnetic waves through a stratified dielectric. The ability of such a device to support guided waves depends upon the way in which the refractive index varies across the layers. In the present discussion, we show how nonlinear effects and appropriate stratification can be used to obtain any one of the following behaviours:

1. guidance occurs only at low power.
2. guidance occurs only at high power.
3. guidance occurs at all powers.
4. there is no guidance.

The situation (i) is obtained by using materials with a defocusing dielectric response, whereas the situation (ii) is obtained for suitable configurations of self-focusing materials. The situations (iii) and (iv) can be obtained by using either defocusing or self-focusing materials. By seeking solutions of a particular form, we reduce the problem to the study of solutions in the Sobolev space H ¹(?) of a second-order differential equation. The discussion of defocusing nonlinearities is based in the study of the global behaviour of the branch of solutions bifurcating from a simple eigenvalue. For self-focusing nonlinearities we use a variational approach.     

An implicit Lagrangian lattice Boltzmann method for the compressible flows

In this paper, we propose a new Lagrangian lattice Boltzmann method (LBM) for simulating the compressible flows. The new scheme simulates fluid flows based on the displacement distribution functions. The compressible flows, such as shock waves and contact discontinuities are modelled by using Lagrangian LBM. In this model, we select the element in the Lagrangian coordinate to satisfy the basic fluid laws. This model is a simpler version than the corresponding Eulerian coordinates, because the convection term of the Euler equations disappears. The numerical simulations conform to classical results. Copyright © 2006 John Wiley & Sons, Ltd.     

Die Berechnung des Blasenwachstums beim Verdampfen von Flüssigkeiten an Heizflächen als numerische Lösung der Erhaltungsgleichungen

A numerical procedure on the basis of the Marker and Cell-method [1] was developed in order to solve the conservation equations for mass, momentum and energy for the case of bubble growth on a heating surface. This procedure was used to calculate steam bubble growth on a horizontal stainless steel heating surface under saturated pool boiling conditions at a system pressure of 1 bar and different superheatings. The essential results obtained are:

-Good agreement was found between calculations and experiments concerning bubble growth rates, bubble shape and temperature field in the liquid surrounding the bubble.

-During its growth the bubble penetrates the temperature boundary layer formed in the liquid on the heating surface, simultaneously liquid is displaced aside.

-The microlayer evaporation fraction of the total bubble growth increases with growth time from 20 % to 50%.

     

Flow past a suddenly cooled horizontal plate

The problem of unsteady laminar, incompressible free convection above a horizontal semi-infinite flat plate is studied theoretically. It is assumed that for timet<0 the plate is hotter than its surroundings and at timet=0 the plate is suddenly cooled to the same temperature of its surroundings. Three solutions of the momentum and energy equations are obtained, namely

1. an analytical solution which is valid for small time,
2. an asymptotic analytical solution which is valid for large time, and
3. a numerical solution which matches these two limiting analytical solutions.

It is found that the numerical solution matches the small and large time solutions accurately. Finally, the variation of the velocity, temperature, skin friction and heat transfer on the plate with time are discussed.     

Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.     

Variable isochromatic/isopachic-fringe visibility in photoholoelasticity

This paper presents a new “hybrid” method whereby the ratio of the isochromatic-fringe visibility/isopachic-fringe visibility may be easily and continuously varied. This simple procedure merely combines a conventional polariscope with a holographic system. A variable beam splitter permits an incoherent superposition of the reconstruction of a doubly exposed hologram with real-time isochromatics, either dark or light field. By varying the ratios of the above two, in the image plane, numerous interesting results may be obtained including:

1. Isochromatics only, without errors in position
2. Isochromatics-isopachic fringes identical to those obtained through classical interferometry
3. Isochromatic-isopachic fringes whereby the amplitude modulation between the two may be minimized
4. Continuously variable isopachic/isochromatic-fringe visibility.

     

Some pending problems in polymer melt rheology,as seen from the point of view of Doi's slip-link model

An exposition is given of results, as obtained with the aid of Doi's sliplink model, being considered as the most simple version of the famous “reptation model”. It turns out that this model which exhibits three distinct phases of relaxation (an extremely fast phase, an equilibration phase and a slow disengagement phase) is capable of explaining several peculiar features of polymer melt rheology:

1. The molecular mass dependence of the breadth of the rubber plateau in the storage modulus, of the zero-shear viscosity and of the normal-stress coefficients.
2. The molecular mass independence of the equilibrium (shear and tensile) compliances for monodisperse polymers (semi-quantitative prediction).
3. The seemingly contradictory sensitivity of these compliances for the breadth of the molecular mass distribution.
4. The critical value of the shear stress at which melt fracture occurs in capillary flow.
5. An equilibration phase in tensile experiments on unvolcanized rubber.

In this evaluation optical (flow birefringence) measurements are preferentially used.     

A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range

Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.     

On the Singular Incompressible Limit of Inviscid Compressible Fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In this paper we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.     

Determination of stress-intensity factors from photoelastic data with applications to surface-flaw problems

ATaylor-series correction to the maximum inplane shear stress was studied as a means of extending the data zone in photoelastic determination of stress-intensity factors beyond the singular region of a two-degree-of-freedom analysis. Convergence properties were obtained by comparing with several complete two-dimensional solutions. Experiments were performed on two kinds of three-dimensional problems, plates containing surface flows in both bending and extension. Results were analyzed by both a two-degree-of-freedom and aTaylor-series correction method (TSCM). Results were compared to theories of F. W. Smith and A. S. Kobayashi and R. C. Shah. It was concluded that:

1. The TSCM program converges rapidly to accurateK _I values and will accommodate the scatter inherent in experimental data if the series is properly truncated.
2. The TSCM program is essentially equivalent to the two-parameter representation when only the crack-surface effects dominate.
3. When effects other than crack surfaces are important, TSCM requires more terms but still predictsK _I with reasonable accuracy.

     

Glass-bonding techniques for semiconductor strain gages

Conventional organic-epoxy adhesives outgas when exposed to ultra-high vacuum and, as operating temperatures are increased, they begin to exhibit plastic behavior causing hysteresis and zero instability in the transducer. The use of an inorganic glass as the bonding material has resulted in a significant advance in transducer-fabrication technology for the following reasons:

1. The outgassing of transducers in high-vacuum applications is minimized.
2. Mechanical properties of the transducer such as hysteresis and repeatability are improved.
3. The electrical isolation of the strain gages from the metallic elements of the transducer is increased at high temperatures over that provided by epoxy. Also, the glass bond can survive and operate in severe radiation environments, wherein the epoxy adhesive will suffer either temporary or permanent loss of its dielectric strength.
4. Glass-bonding techniques are particularly useful for the extension of the temperature range of operation of silicon-strain-gage transducers.

Nispan C and 440-C stainless-steel substrates were successfully used with glass-bonded silicon strain gages to fabricate transducers for evaluation.     

Oscillation modes of laval nozzle flow

Experimental investigations of Laval nozzle flow show for relatively low supply to exit pressure ratios, which correspond to shock wave positions close to the nozzle throat, three different, oscillatory instabilities.

1. Shock pattern oscillations where the root of a λ-like shock front remains nearly in constant position, but where the proportion between the normal part and the oblique part of the shock changes periodically.
2. Shock wave and separation bubble oscillations where the motion of the shock wave is accompanied by displacements of the separation bubble.
3. Flow rate oscillations where the shock waves leave periodically through the nozzle throat in upstream direction.

     

A fast and accurate graphical method for designing an air conditioning cooling coil

The aim of this paper is to develop a fast and simple accurate graphical method for designing the required cooling coil for an air conditioning system in which both the sensible and latent heat are transferred. The method has the following advantages:

1. Direct solution utilizing only Psychrometer charts.
2. Solution of combined heat and mass transfer problems occuring at pressures other than that of atmosphere.
3. Direct determination of boundary temperature at which dehumidification begins for that type of problem where a portion of the surface is in a dry condition.
4. Less effort required than trial and error method in determination of air condition leaving a counterflow coil of a given area.

The calculated values of the method showed a good agreement with the experimental results. The average deviation for the total heat is about + 15 % and for sensible heat is + 9 %, which are of positive nature and on the safe side for practical design purposes. The method is also useful for similar practical application.     

Marangoni convection at alcohol aqueous solutions-air interfaces

For aqueousn-heptanol solutions and in a nearly two-dimensional flow, two strikingfeatures have been detected:

1. a shift of the minimum of the surface tension
2. a discrepancy between the observed Marangoni flow velocities and the expected ones from static surface tension values.

A qualitative explanation is given.     

Interferogram processing by spatial filtering

This paper describes a technique of interferogram processing by the use of spatial-frequency filtering. Proper part of Fourier spectrum of the interferogram is blocked out by a matched mask located in the focal plane of the imaging system. Depending on the mask shape and dimensions one can obtain

1. interferogram images with delineating fringes or
2. interferogram images of one component of strain.

     

Rapid interpretation of moiré photographs

The paper describes the possibilities of rapidly and conveniently obtaining results by means of the reflection moiré method. The moiré photographs are produced with the aid of a screen provided with a finely ruled grid. The contrast of the photographs is increased by optical means. By displacement of two equal moiré photographs, moiré fringes of second order for the curvature or the twist are obtained. It is shown how charts can be produced which indicate:

1. The distribution of the bending moments
2. The distribution of the bending moments for which the reinforcement should be designed
3. The distribution of the shear forces and the magnitude of the bearing reactions.