Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

Asymptotic behaviour and boundedness of linear systems with time varying coefficients

Motivated by developments in spacecraft dynamics, the asymptotic behaviour and boundedness of solution of a special class of time varying systems in which each term appears as the sum of a constant and a time varying part, are analysed in this paper. It is not possible to apply standard textbook results to such systems, which are originally in second order. Some of the existing results are reformulated. Four theorems which explore the relations between the asymptotic behaviour/boundedness of the constant coefficient system, obtained by equating the time varying terms to zero, to the corresponding behaviour of the time varying system, are developed. The results show the behaviour of the two systems to be intimately related, provided the solutions of the constant coefficient system approach zero are bounded for large values of time, and the time varying terms are suitably restrained. Two problems are tackled using these theorems.

On the L2 stability of linear multidimensional time varying systems

This paper analyzes the L2 stability of solutions of systems with time-varying coefficients of the form [A + C(t)]x′ = [B + D(t)]x + u, where A, B, C, D are matrices. Following proof of a lemma, the main result is derived, according to which the system is L2 stable if the eigenvalues of the coefficient matrices are related in a simple way. A corollary of the theorem dealing with small periodic perturbations of constant coefficient systems is then proved. The paper concludes with two illustrative examples, both of which deal with the attitude dynamics of a rigid, axisymmetric, spinning satellite in an eccentric orbit, subject to gravity gradient torques.

Bounded-input/bounded-output stability of linear multidimensional time- varying systems

The present study of the stability of systems governed by a linear multidimensional time-varying equation, which are encountered in spacecraft dynamics, economics, demographics, and biological systems, gives attention the lemma dealing with L(inf) stability of an integral equation that results from the differential equation of the system under consideration. Using the proof of this lemma, the main result on L(inf) stability is derived according; a corollary of the theorem deals with constant coefficient systems perturbed by small periodic terms. (O.C.)

Design of Constant Accuracy Linear Proportional Weir

Five different shaped weirs were designed and pertinent data for their use are given. One of these weir shapes had the least “sharp edge” at the junction of the base weir and “complementary weir.” Two other types of weirs had equal slopes at the junction of the base weir and complementary weir. Another shape, for which neither the indication accuracy was constant nor the slope was equal at the junction of the base weir and complementary weir, was also tested. The results of the four weir shapes hydraulically tested give consistent values for the coefficient of discharge varying between 0.625 to 0.631. The indication accuracies of all the previously designed linear proportional weirs (includig Sutro weir) are neither constant nor unity, as is believed.

Geometrically simple linear weirs using circular quadrants

A detailed theoretical analysis of flow through a quadrant plate weir is made in the light of the generalized theory of proportional weirs, using a numerical optimization procedure. It is shown that the flow through the quadrant plate weir has a linear discharge-head relationship valid for certain ranges of head. It is shown that the weir is associated with a reference plane or datum from which all heads are reckoned.Further, it is shown that the measuring range of the quadrant plate weir can be considerably enhanced by extending the tangents to the quadrants at the terminals of the quadrant plate weir. The importance of this weir (when the datum of the weir lies below its crest) as an outlet weir for grit chambers is highlighted. Experiments show excellent agreement with the theory by giving a constant average coefficient of discharge.

On some results of A. E. Livingston and coefficient problems for concave univalent functions

We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions.

Data compression by linear prediction for storage and transmission of EEG signals

The EEG time series has been subjected to various formalisms of analysis to extract meaningful information regarding the underlying neural events. In this paper the linear prediction (LP) method has been used for analysis and presentation of spectral array data for the better visualisation of background EEG activity. It has also been used for signal generation, efficient data storage and transmission of EEG. The LP method is compared with the standard Fourier method of compressed spectral array (CSA) of the multichannel EEG data. The autocorrelation autoregressive (AR) technique is used for obtaining the LP coefficients with a model order of 15. While the Fourier method reduces the data only by half, the LP method just requires the storage of signal variance and LP coefficients. The signal generated using white Gaussian noise as the input to the LP filter has a high correlation coefficient of 0.97 with that of original signal, thus making LP as a useful tool for storage and transmission of EEG. The biological significance of Fourier method and the LP method in respect to the microstructure of neuronal events in the generation of EEG is discussed.

Linear electro‐optic dispersion in (‐)‐2‐(α‐methylbenzylamino)‐5‐nitropyridine single crystals

The variation of the linear electro-optic effect in (-)-2-(alpha-methylbenzylamino)-5-nitropyridine with the wavelength of the incident light at room temperature has been measured. The reduced half-wave voltages have been found to have the values 2.1, 2.8, and 6.0 kV at 488, 514.5, and 632.8 nm respectively and the corresponding values of the linear electro-optic coefficient have been evaluated.;The interpretation of the results in terms of the structures of the molecule and the crystal is discussed. The thermal variation of the birefringence has also been investigated and the coefficient for the temperature variation of the refractive index difference is found to have the value (d Delta n/dT)=9.3X10(-5) K-1.

Non-Linear effects of membrane fluctuations in the dilute lamellar phase

The static structure factor of the dilute sterically stabilised lamellar phase is calculated and found to have an Ornstein-Zernike form with a correlation length that diverges at infinite dilution. The relaxation time for concentration fluctuations at large wave number q is shown to go as q-3 with a coefficient independent of the membrane bending rigidity. The membrane fluctuations also give rise to strongly frequency-dependent viscosities at high frequencies.

Practical constant-accuracy linear weir

This paper is concerned with the modifications of the Extended Bellmouth Weir (EBM weir) earlier designed by Keshava Murthy. It is shown that by providing inclined sides (equivalent to providing an inward-trapezoidal weir) over a sector of a circle of radius R, separated by a distance 2t, and depth d, the measurable range of EBM can be considerably enhanced (over 375%). Simultaneously, the other parameters of the weir are optimized such that the reference plane of the weir coincides with its crest making it a constant-accuracy linear weir. Discharge through the aforementioned weir is proportional to the depths of flow measured above the crest of the weir for all heads in the range of 0.5R less-than-or-equal-to h less-than-or-equal-to 7.9R, within a maximum deviation of +/-1% from the theoretical discharge. Experiments with two typical weirs show excellent agreement with the theory by giving a constant-average coefficient of discharge of 0.619

Thermal expansion of irradiated polyvinyl chloride from 10 K to 340 K

The coefficient of thermal expansion is measured for irradiated Polyvinyl Chloride (PVC) from 10K to 340K. The samples of PVC are irradiated, up to 500 Mrad in steps of 100 Mrad, in air at room temperature by using Co gamma rays with a dose rate of 0.3 Mrad/h. The PVC is an amorphous sample which is confirmed by X-ray diffraction. The coefficient of thermal expansion is found to decrease with radiation dose from 10K to 110K and it increaseswith radiation dose from 110K to 340K. The results are explained on the basis of radiation induced degradation of the sample.

Thermal expansion studies of substituted CTP derivatives

Several new Na, Y and Zr substituted derivatives of Ca-0.5 Ti-2(PO4)(3) (CTP) have been synthesized. These derivatives retain the hexagonal structure of the parent (CTP) compound with minor changes in lattice parameters. Linear thermal expansion coefficients (alpha) have been obtained using a high sensitivity dilatometer.

The log delta log t method for the determination of the coefficient of consolidation

The behaviour of saturated soils undergoing consolidation is very complex, It may not follow Terzaghi's theory over the entire consolidation process, Different soils may behave in such a way as to fit into Terzaghi's theory over some specific stages of the consolidation process (percentage of consolidation), This may be one of the reasons for the difficulties faced by the existing curve-fitting procedures in obtaining the coefficient of consolidation, c(v). It has been shown that the slope of the initial linear portion of the theoretical log U-log T curve is constant over a wider range of degree of consolidation, U, when compared with the other methods in use, This initial well-defined straight line in the log U-log T plot intersects the U = 100% line at T = pi/4, which corresponds to U = 88.3%, The proposed log delta-log t method is based on this observation, which gives the value of c(v) through simple graphical construction, In the proposed method, which is more versatile, identification of the characteristic straight lines is very clear; the intersection of these lines is more precise and the method does not depend upon the initial compression for the determination of c(v).

Thermal expansion of LaBa2Cu2CoO7+delta

In this paper, we report high-temperature X-ray diffraction (HTXRD) and dilatometric studies on LaBa2Cu2CoO7+delta. Bulk and volume thermal expansion studies, along with a study of its phase transition, were carried out. The linear and volume thermal expansion coefficients were found to be 11.7 X 10(-6) K-1 and 42.3 X 10(-6) K-1, respectively. (C) 2000 Elsevier Science Ltd.