Application of liquid-crystal thermometry to drop temperature measurements

A technique has been developed that enables remote sensing of the temperatures of liquid drops in a medium of an immiscible, transparent liquid with the aid of dispersing microcapsules of thermochromic liquid crystal in each drop under illumination by either a planar floodlight or a light sheet which cuts the drop at a meridian. Based on appropriate hue-temperature calibrations made with an isothermal, stationary drop/medium system, one can analyze spatial and time variations of temperature within drops in motion under transient convective heating (or cooling) from the medium.List of Symbols B tristimulus blue component - D drop diameter - G tristimulus green component - H hue angle - h ₀ constant term in the expression for H - H _m mean value of H over drop surface - h vertical coordinate fixed onto test column - R tristimulus red component - Re drop Reynolds number, UD/v _c - r, g, b chromaticity coordinates - r drop radius - s penetration depth of light into drop - T temperature - T _d instantaneous drop temperature - T _d0 initial drop temperature - T _c continuous-phase temperature - U velocity of rise of drop - z vertical coordinate laid on drop - v _c kinematic viscosity of continuous phase - angle of lighting measured from camera axis - _s local view angle (azimuth) - polar angle     

A Numerical Study of Micro-Heterogeneity Effects on Upscaled Properties of Two-Phase Flow in Porous Media

Commonly, capillary pressure–saturation–relative permeability (P ^c–S–K _r) relationships are obtained by means of laboratory experiments carried out on soil samples that are up to 10–12 cm long. In obtaining these relationships, it is implicitly assumed that the soil sample is homogeneous. However, it is well known that even at such scales, some micro-heterogeneities may exist. These heterogeneous regions will have distinct multiphase flow properties and will affect saturation and distribution of wetting and non-wetting phases within the soil sample. This, in turn, may affect the measured two-phase flow relationships. In the present work, numerical simulations have been carried out to investigate how the variations in nature, amount, and distribution of sub-sample scale heterogeneities affect P ^c–S–K _r relationships for dense non-aqueous phase liquid (DNAPL) and water flow. Fourteen combinations of sand types and heterogeneous patterns have been defined. These include binary combinations of coarse sand imbedded in fine sand and vice versa. The domains size is chosen so that it represents typical laboratory samples used in the measurements of P ^c–S–K _r curves. Upscaled drainage and imbibition P ^c–S–K _r relationships for various heterogeneity patterns have been obtained and compared in order to determine the relative significance of the heterogeneity patterns. Our results show that for micro-heterogeneities of the type shown here, the upscaled P ^c–S curve mainly follows the corresponding curve for the background sand. Only irreducible water saturation (in drainage) and residual DNAPL saturation (in imbibition) are affected by the presence and intensity of heterogeneities.     

Capillary Pressure Curve for Liquid Menisci in a Cubic Assembly of Spherical Particles Below Irreducible Saturation

The capillary pressure–saturation relationship, P _c(S _w), is an essential element in modeling two-phase flow in porous media (PM). In most practical cases of interest, this relationship, for a given PM, is obtained experimentally, due to the irregular shape of the void space. We present the P _c(S _w) curve obtained by basic considerations, albeit for a particular class of regular PM. We analyze the characteristics of the various segments of the capillary pressure curve. The main features are the behavior of the P _c(S _w) curve as the wetting-fluid saturation approaches zero, and as this saturation is increased beyond a certain critical value. We show that under certain conditions (contact angle, distance between spheres, and saturation), the value of the capillary pressure may change sign.     

On the First Critical Field in Ginzburg–Landau Theory for Thin Shells and Manifolds

In this article, we investigate the response of a thin superconducting shell to an arbitrary external magnetic field. We identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H _c1 in Ginzburg–Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a simply connected surface of revolution and vertical and constant field, we further determine the exact number of vortices in the sample as the intensity of the applied field is raised just above H _c1. Finally, we derive via Γ-convergence similar statements for three-dimensional domains of small thickness, where in this setting point vortices are replaced by vortex lines.     

Flow of a yield stress fluid over a rotating surface

We study the flow of yield stress fluids over a rotating surface when both the viscoelastic solid behavior below a critical deformation (γ _c) and liquid properties beyond γ _c can play a significant role. We review the detailed characteristics of the flow in the solid regime in the specific case of a pure elongational strain (large height to radius ratio). We, in particular, show that there exists a critical rotation velocity (ω _c) associated with the transition from the solid to the liquid regime. We then consider the specific case of lubricational regime (small height to radius ratio) in the liquid regime. In that case we describe the different possible evolutions of the equilibrium shape of the material as a function of the rotation velocity (ω), from which we extrapolate the transient shape evolutions as ω increases. We show that for a sufficiently large rotation velocity the sample separates into two parts, one remaining at rest around the rotation axis, the other going on moving radially. These predictions are then compared with systematic spin-coating tests under increasing rotation velocity ramps followed by a plateau at ω _f with typical yield stress fluids. It appears that there exists a critical velocity below which the material undergoes a limited elongation and beyond which it starts to spread significantly over the solid surface. For a larger ω _f value the sample forms a thick peripheral roll, leaving behind it a thin layer of fluid at rest relatively to the disc. These characteristics are in qualitative agreement with the theoretical predictions. Beyond a sufficiently large ω _f value this roll eventually spreads radially in the form of thin fingers. Moreover, in agreement with the theory in the lubricational regime, the different curves of deformation vs ω fall along a master curve when the rotation velocity is scaled by ω _c for different accelerations, different sample radii, or different material yield stress. The final thickness of the deposit seems to be mainly governed by the displacement of the roll, the characteristics of which take their origin in the initial stage of the spreading, including the solid–liquid transition.     

Modeling Thermal-Hydrologic Processes for a Heated Fractured Rock System: Impact of a Capillary-Pressure Maximum

Various thermal-hydrologic models have been developed to simulate thermal-hydrologic conditions in emplacement drifts and surrounding host rock for the proposed high-level nuclear waste repository at Yucca Mountain, Nevada. The modeling involves two-phase (liquid and gas) and two-component (water and air) transport in a fractured-rock system, which is conceptualized as a dual-permeability medium. Simulated hydrologic processes depend upon calibrated system parameters, such as the van Genuchten α and m, which quantify the capillary properties of the fractures and rock matrix. Typically, these parameters are not calibrated for strongly heat-driven conditions, i.e., conditions for which boiling and rock dryout occur. The objective of this study is to modify the relationship between capillary pressure and saturation, P _c(S), for strongly heated conditions that drive saturation below the residual saturation (S → 0). We offer various extensions to the van Genuchten capillary-pressure function and compare results from a thermal-hydrologic model with data collected during the Drift-Scale Test, an in situ thermal test at Yucca Mountain, to investigate the suitability of these various P _c extension methods. The study suggests that the use of extension methods and the imposition of a capillary-pressure cap (or maximum) improve the agreement between Drift-Scale Test data and model results for strongly heat-driven conditions. However, for thermal-hydrologic models of the Yucca Mountain nuclear waste repository, temperature and relative humidity are insensitive to the choice of extension method for the capillary-pressure function. Therefore, the choice of extension method applied to models of drift-scale thermal-hydrologic behavior at Yucca Mountain can be made on the basis of numerical performance.     

Shear viscosity of hydrogen and helium fluids at extended regions of temperatures and pressures

Shear viscosities of hydrogen (H₂) and helium (He) fluids are investigated at extended regions of temperature and pressure using an improved equation of state, in which the the long-range attractive forces are incorporated into the formalism via the double Yukawa (DY) potential fitted to the empirical H₂ and He potentials. Quantum effects, which become paramount at low temperatures, are taken into account through a first-order quantum correction based on the Wigner–kirkwood expansion. The dimerization of H₂ is treated as a hard convex body for which an equation of state can be derived based on the scaled particle theory.     

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.     

Percolation Approach to the Anisotropy Factor in an Unsaturated Porous Medium

The anisotropy factor is defined as the ratio of the effective (macroscopic) conductivities parallel to the bedding plane and perpendicular to it. The anisotropy factor A(p,a) is a function of both the saturation degree, p, in the void space of the disordered medium and the anisotropy parameter, a, that characterizes the ratio of the local conductivities parallel and normal to the bedding plane. There are two opposite behaviors of the anisotropy factor as a function of the saturation degree described in literature. One presents the anisotropy factor as a curve with a maximum and the other as a curve with a minimum. The main result of calculating the conductivities of a uniaxial percolation anisotropic model is that at the saturation threshold value, p_c, A(p_c, a) = 1, wherefrom it increases at a >1 (or decreases at a < 1) with saturation. An extension of the computed results below the threshold is also proposed.     

Observations on Mode I and III fracture of steel and PMMA

Considered in this work are the Mode I and III fractures of W18Cr4V steel, 60Si2Mn steel and PMMA specimens. The mixed mode critical stress intensity factor, denoted by K_1f, is shown to be greater than K_1c for Mode I. The ratio K_1f/K_1c depends on the ways in which K₁ and K₃ interact and is affected by the position of the load with reference to the crack position. Analytical and experimental results are presented and discussed in connection with the microfracture surface observed experimentally.     

Multiscale Creep Compliance of Epoxy Networks at Elevated Temperatures

Epoxy networks are thermoset polymers for which an important structural length scale, molecular weight between crosslinks (M _c), influences physical and mechanical properties. In the present work, creep compliance was measured for three aliphatic epoxy networks of differing M _c using both macroscale torsion and microscale depth-sensing indentation at temperatures of 25 and 55°C. Analytical relations were used to compute creep compliance (J(t)) for each approach; similar results were observed for the two techniques at 25°C, but not at 55°C. Although creep compliance measurement differed at elevated temperatures, there were clear correlations between M _c, glass transition temperature, T _g, and the observed time-dependent mechanical behavior via both techniques at 55°C, but these correlations could not be seen at 25°C. This work demonstrates the capacity of depth-sensing indentation to differentiate among epoxy networks of differing structural configurations via J(t) for small material volumes at elevated temperatures.     

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Re _c, the critical wave number α _c and the critical wave speed c _c are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.     

Near-field of coaxial jets with large density differences

 The paper describes an experimental investigation of coaxial jets with large density differences. Measurements by various techniques show that density effects on the flow dynamics are taken into account to first order by considering the specific outer to inner jet momentum flux ratio M and not separately the density and velocity ratios. A regime of recirculation occurs for M higher than a critical value (M _c ≈50). For a given value of M, however, the position of the recirculation bubble is slightly shifted in the upstream direction for density ratios much smaller than one. An unexpected result is obtained for an extremely low density ratio: the onset of recirculation occurs for a significantly higher value of M (100<M _c <140) for helium/SF₆ jets (density ratio : 0.028). Received: 1 October 1997 / Accepted: 8 March 1998     

Experimental estimation of wall slip for filled rubber compounds

The phenomenon of wall slip during flow of rubber compounds through capillaries is investigated for a typical styrene-butadiene elastomer with carbon black. It was found that at low temperature (110°C) the dependencies of slip velocity V _c on shear stress are described by the power law but, additionally, V _c depends on radius of a channel. At high temperatures there is a critical shear stress below which sliding is absent. Sliding appears only at higher shear stresses where, again, V _c depends on shear stress and the radius of a channel.     

Geometric properties of two phase flow in geothermal reservoirs

The two phase flow equations frequently used in geothermal engineering ignore capillary pressure, which results in a singular system of equations. Analysis of these equations reveals three mechanisms for altering saturation: local boiling, the spatial dependence of flowing enthalpy due to the convective transport of fluid, and counterflow. A scalar function is associated with each of these three mechanims. At each point in space, flows are essentially two dimensional, with gravity establishing a vertical hierarchy, in that volumetric, energy and mass fluxes can never point below a lower member in this triple. With increasing liquid saturation, the characteristics associated with the saturation equation move up from below this grouping of directions, and eventually may even point above volumetric fluxes. Finally, weak shocks and the associated entropy condition are considered. The characteristics of the saturation equation coincide with the velocity of extremely weak shocks, and saturation increases with the passage of a weak shock, provided the magnitude of the characteristic speed increases with saturation.Notation C_l liquid heat capacity - C_m rock heat capacity - C_v vapour heat capacity - G counterflow energy flux - h flowing enthalpy - hl liquid enthalpy - h_v vapour enthalpy - k permeability - k downward vector - P pressure - S liquid saturation - T temperature - dT/dP derivative at saturation - z vertical coordinate - l liquid viscosity - v vapour viscosity - Pl liquid density - _m rock density - _v vapour density - porosity     

Density Maximum Effect on Double-diffusive Natural Convection in a Porous Cavity with Variable Wall Temperature

The effect of density maximum of water on double-diffusive natural convection in a two-dimensioned cavity filled with a water saturated isotropic porous medium is studied numerically. The horizontal walls of the cavity are insulated. The opposing vertical walls are kept at different temperatures θ _h (linearly varies with height) and θ _c (θ _c ≤ θ _h ). The concentration levels at cold wall and hot wall are, respectively, c ₁ and c ₂ with c ₁ > c ₂. Brinkman-Forchheimer extended Darcy model is used to investigate the average heat and mass transfer rates. The non-dimensional equations for momentum, energy, and concentration are solved by finite volume method with power law scheme for convection and diffusion terms. The results are presented in the form of streamlines, isotherms, and isoconcentration lines for various values of Grashof numbers, Schmidt number, porosity, and Darcy numbers. It is observed that the density maximum of water has profound effect on the thermosolutal convection. The effects of different parameters on the velocity, temperature, and species concentrations are also shown graphically.     

The flow of a non-Newtonian liquid near a rotating disk

Synopsis The flow of non-Newtonian liquid near a rotating disk has been discussed by using second order stress strain velocity relations of classical hydrodynamics. It is found that the effect of cross-viscosity depends on a non-dimensional number R _c. The boundary layer thickness decreases and the dimensionless moment coefficient increases with the increase of R _c.     

Evaporation heat transfer coefficient in single circular small tubes for flow natural refrigerants of C3H8, NH3, and CO2

An experimental study of evaporation heat transfer coefficients for single circular small tubes was conducted for the flow of C₃H₈, NH₃, and CO₂ under various flow conditions. The test matrix encompasses the entire quality range from 0.0 to 1.0, mass fluxes from 50 to 600 kg m⁻² s⁻¹, heat fluxes from 5 to 70 kW m⁻², and saturation temperatures from 0 to 10 °C. The test section was made of circular stainless steel tubes with inner diameters of 1.5 mm and 3.0 mm, and a length of 2000 mm in a horizontal orientation. The test section was uniformly heated by applying electric power directly to the tubes. The effects of mass flux, heat flux, saturation temperature, and inner tube diameter on the heat transfer coefficient are reported. Among the working refrigerants considered in this study, CO₂ has the highest heat transfer coefficient. Laminar flow was observed in the evaporative small tubes, and was considered in the modification of boiling heat transfer coefficients and pressure drop correlations.     

Collision dynamics of bubble pairs moving through a perfect liquid

Equations are derived describing the inertial motion of a bubble pair through a perfect liquid. The relative bubble motion is driven by an interactional force induced by the centre of mass motion. This force can be derived from a potential that is proportional tos ⁿ (n3) and that depends on the bubble pair orientation. The path of two bubbles passing each other is investigated. The angle of deflection of the relative velocity in a two-bubble encounter is calculated numerically as a function of the impact parameter, the relative velocityg and the ratio of the centre of mass velocity componentsc ₂/c ₁. The specific conditions necessary for two bubbles to collide are determined. Ifc ₂/c ₁>1 there is a region with irregular behaviour of the deflection angle. The collision cross-section is calculated and depends smoothly ong, approximately proportional tog ^–1, and has a weak dependence onc ₂/c ₁.