A fractional differential equation for a MEMS viscometer used in the oil industry

A mathematical model is developed for a micro-electro-mechanical system (MEMS) instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration. It is shown that, in one mode of operation, the displacement of the device satisfies a fractional differential equation (FDE). The theory of FDEs is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme. It is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised. It is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring, where the resulting FDE is nonlinear.     

The viscosity of normal butane,isobutane and their mixtures

The paper reports measurements of the viscosity of normal butane and isobutane at atmospheric pressure in the temperature range 25–195°C. In addition, the viscosity of mixtures of the two gases has been determined in the more limited temperature range 25–100°C. The absolute accuracy of the viscosity data is estimated as ±0.2%, whereas the ratio of the viscosity of the two pure components is determined with an uncertainty of ±0.1%. The viscosity data allow the evaluation of the diffusion coefficients for the n-butane-i-butane system.The experimental data demonstrate that there are discernible differences between the intermolecular potentials for the pair-interactions n-C₆H₁₀-n-C₄H₁₀; and i-C₄H₁₀-i-C₄H₁₀. All of the experimental data can be correlated within their experimental uncertainty with the aid of the extended law of corresponding states developed by Kestin, Ro and Wakeham. The optimum values of the two scaling parameters, σ_ij and ?_ij, for each interaction which secure this agreement are determined.     

Transparent,flexible and solid-state supercapacitors based on room temperature ionic liquid gel

This paper describes a novel strategy to make fully transparent, solid-state and flexible supercapacitors based on room temperature ionic liquid (RTIL) gel and ITO electrodes coated on transparent polymer substrate without a separator, which enables the roll-to-roll technique for fabrication of such supercapacitors as printable devices. This is the first type of transparent electrochemical double layer capacitor (EDLC) based on ionic liquid gel.     

Upper critical magnetic field far above the paramagnetic pair-breaking limit of superconducting one-dimensional Li0:9Mo6O17 single crystals

The upper critical field H(c2) of purple bronze Li0:9Mo6O17 is found to exhibit a large anisotropy, in quantitative agreement with that expected from the observed electrical resistivity anisotropy. With the field aligned along the most conducting axis, H(c2) increases monotonically with decreasing temperature to a value 5 times larger than the estimated paramagnetic pair-breaking field. Theories for the enhancement of H(c2) invoking spin-orbit scattering or strong-coupling superconductivity are shown to be inadequate in explaining the observed behavior, suggesting that the pairing state in Li0:9Mo6O17 is unconventional and possibly spin triplet.     

The viscosity and diffusion coefficients of the binary mixtures of xenon with the other noble gases

This paper presents new experimental data for the viscosity of binary mixtures of xenon with the remaining monatomic gases, helium, neon, argon, and krypton. The measurements have been performed in a high precision oscillating-disk viscometer at atmospheric pressure and within the temperature range 25–500°C. The data have an estimated uncertainty of ±0.1% at 25°C increasing to ±0.3% at 500°C. The collision integrals for the interactions of xenon with the other monatomic species conform to the extended law of corresponding states formulated by Kestin, Ro and Wakeham. For each binary interaction the scaling parameters σ_ij and ∈_ij have been obtained. The ensemble of experimental results can be correlated by means of the appropriate kinetic theory expressions reinforced by the extended law of corresponding states. The deviations do not exceed 0.5%. The binary diffusion coefficients were calculated from the measured mixture viscosity and compared with the available experimental results. The standard deviation was estimated as ±2% which is within the mutual uncertainty of the two sets.     

Absolute determination of the thermal conductivity of the noble gases at room temperature up to 35 MPa

The paper presents new absolute measurements of the thermal conductivity of the five noble gases at a nominal temperature of 27.5°C and over a range of pressures. Measurements were made in a transient hot-wire apparatus whose design and operational equations have been published earlier. These values replace the earlier data generated in the same instrument, which has undergone a number of improvements introduced since. The most essential difference is an improved design of the cell, the use of bare platinum wires of 7 μm in diameter, and provision for temperature calibration of electric resistance in situ.The present data enjoy an enhanced level of confidence and their accuracy is estimated at ±0.3%. It is remarkable that the present absolute values of thermal conductivity at zero density combined with earlier absolute values of viscosity, also extrapolated to zero density, satisfy the Eucken relation with an error not exceeding 0.25%.     

The theory of a vibrating-rod viscometer

The paper presents a complete theory for a viscometer based upon the principle of a circular-section rod, immersed in a fluid, performing transverse oscillations perpendicular to its axis. The theory is established as a result of a detailed analysis of the fluid flow around the rod and is subject to a number of criteria which subsequently constrain the design of an instrument. Using water as an example it is shown that a practical instrument can be designed so as to enable viscosity measurement with an accuracy of ±0.1%, although it is noted that many earlier instruments failed to satisfy one or more of the newly-established constraints.Nomenclature A, D constants in equation (46) - A _m , B _m , C _m , D _m constants in equations (50) and (51) - A _j , B _j constants in equation (14) - a _j ⁺ , a _j ^– wavenumbers given by equation (15) - C _f drag coefficient defined in equation (53) - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - f(z) initial deformation of rod - f(), F _m () functions of defined in equation (41) - F force in the rod - force per unit length near t=0 - F dimensionless force per unit length near t=0 - g _m amplitude of transient force - G modulus of rigidity - h, h* functions defined by equations (71) and (72) - H functions defined by equation (69) and (70) - I second moment of area - I _0,1, J _0,1, K _0,1 modified Bessel functions - k, k functions defined in equations (2) - L half-length of oscillator - Ma Mach number - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equations (15) and (16) - R radius of rod - R _c radius of container - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - y ₀ initial lateral displacement - y ₁, y ₂ successive maximum lateral displacement - z axial coordinate - dimensionless tension - dimensionless mass of fluid - dimensionless drag of fluid - amplification factor - logarithmic decrement in a fluid - _a , _b logarithmic decrement in fluids a and b - ₀ logarithmic decrement in vacuo - _j logarithmic decrement in mode j in a fluid - spatial resolution of amplitude - _v voltage resolution - r, , , _s, , increments in R, , , _s , , - dimensionless amplitude of oscillation - dimensionless axial coordinate - angular coordinate - _f thermal conductivity of fluid - viscosity of fluid - viscosity of fluid calculated on assumption that * - _a , _b viscosity of fluids a and b - _m constants in equation (10) - dimensionless displacement - _j j the component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - spatial component of defined in equation (11) - _j , _tm jth, mth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - streamfunction - dimensionless frequency (based on ) - angular frequency - ₀ angular frequency in absence of fluid and internal damping - _j angular frequency in mode j in a fluid - _a , _b frequencies in fluids a and b     

The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

The inelastic contributions to transport collision integrals in the infinite-order sudden approximation

The transport collision integrals of the Wang Chang and Uhlenbeck kinetic theory of polyatomic gases have been evaluated for an anisotropic intermolecular pair potential model for the He-CO₂ interaction. The calculations have made use of the infinite-order sudden approximation to evaluate the state-to-state differential cross-sections, but otherwise involve no approximations.

The results indicate that some earlier approximate methods for the evaluation of the collision integrals are quantiatively correct, although they are given a new interpretation. In particular, the simplifications introduced by Mason and Monchick are shown to yield essentially the same collision integral for viscosity as the more exact treatment. On the other hand, several long established procedures for treating the transport properties of polyatomic gases and gas mixtures are shown to be significantly in error. Specifically the diffusion coefficient for internal energy is some 10 per cent higher than the diffusion coefficient for mass, and the latter itself quite different from that obtained on the basis of the treatment of Monchick and Mason.     

Surface structure of a "non-amphiphilic" protic ionic liquid

The nanostructure of the ethanolammonium nitrate (EtAN)-air surface has been investigated using X-ray reflectometry (XRR), vibrational sum frequency spectroscopy (VSFS) and neutral impact collision ion scattering spectroscopy (NICISS). The XRR data decays more rapidly than expected for a perfectly sharp interface, indicating a diffuse electron (scattering length) density profile. Modelling of the XRR data using three different fitting routines produced consistent interfacial profiles that suggest the formation of interfacial EtAN clusters. Consistent with this, VSFS reveals that the EtAN surface is predominantly covered by -CH(2)- moieties, with the -NH(3)(+) and -OH groups of the cation buried slightly deeper in the interface. The elemental profiles determined using NICISS also show enrichment of carbon relative to nitrogen and oxygen in the outermost surface layer, which is consistent with the surface cation orientation deduced from VSFS, and with the presence of EtAN aggregates at the liquid surface.