Investigation of laminar flow in microtubes with random rough surfaces

A new approach of numerically generating a microtube with three-dimensional random surface roughness is presented. In this approach, we combined a bi-cubic Coons patch with Gaussian distributed roughness heights. Two random roughness generation methods are studied. A computational fluid dynamic solver is used to solve the 3-D N–S equations for the flow through the generated rough microtubes with D = 50 μm and L = 100 μm. The effects of the peak roughness height, H, asperities spacing in the θ direction, S _θ , and Z direction, S _Z , standard deviation of the Gaussian distribution, σ, arithmetical mean roughness, R _a, on the Poiseuille number, Po are investigated. It is found that when H/D < 5% the Po number can still be predicted by the conventional flow theory if the mean diameter of rough microtubes, D _m, is used to be the hydraulic diameter D _h. When H/D = 10%, the main flow is strongly affected by the roughness at Reynolds number Re = 1,500. The Po number increases with Re and deviates from the prediction up to 11.9%. The Po number does not change a lot with S _θ and S _Z because D _m almost keeps constant when the spacing is changed. For the rough microtubes with different R _a values, the Po numbers can be almost the same, which prove that only with the R _a value we can not determine the friction in the rough microtube. The mean value μ, the maximum and minimum values of the random roughness are found to be critical to determine the Po number.     

Effect of dispersed phase viscosity on maximum droplet generation frequency in microchannel emulsification using asymmetric straight-through channels

Uniformly sized droplets of soybean oil, MCT (medium-chain fatty acid triglyceride) oil and n-tetradecane with a Sauter mean diameter of d _3,2 = 26–35 μm and a distribution span of 0.21–0.25 have been produced at high throughputs using a 24 × 24 mm silicon microchannel plate consisting of 23,348 asymmetric channels fabricated by photolithography and deep reactive ion etching. Each channel consisted of a 10-μm diameter straight-through micro-hole with a length of 70 μm and a 50 × 10 μm micro-slot with a depth of 30 μm at the outlet of each channel. The maximum dispersed phase flux for monodisperse emulsion generation increased with decreasing dispersed phase viscosity and ranged from over 120 L m⁻² h⁻¹ for soybean oil to 2,700 L m⁻² h⁻¹ for n-tetradecane. The droplet generation frequency showed significant channel to channel variations and increased with decreasing viscosity of the dispersed phase. For n-tetradecane, the maximum mean droplet generation frequency was 250 Hz per single active channel, corresponding to the overall throughput in the device of 3.2 million droplets per second. The proportion of active channels at high throughputs approached 100% for soybean oil and MCT oil, and 50% for n-tetradecane. The agreement between the experimental and CFD (Computational Fluid Dynamics) results was excellent for soybean oil and the poorest for n-tetradecane.     

Complexities of Homomorphism and Isomorphism for Definite Logic Programs

A homomorphism (?) of logic programs from P to P' is a function mapping Atoms(P) to Atoms(P') and it preserves complements and program clauses. For each definite program clause a←a1,...,an∈P it implies that (?)(a)←(?)(a1),...,(?)(an) is a program clause of P'. A homomorphism (?) is an isomorphism if (?) is a bijection. In this paper, the complexity of the decision problems on homomorphism and isomorphism for definite logic programs is studied. It is shown that the homomorphism problem (HOM-LP) for definite logic programs is NP-complete, and the isomorphism problem (ISO-LP) is equivalent to the graph isomorphism problem (GI).     

A study of capillary flow from a pendant droplet

Surface tension driven capillary flow from a pendant droplet into a horizontal glass capillary is investigated in this paper. Effect of the droplet surface on dynamic behavior of such capillary flow is examined and compared with surface tension driven capillary flow from an infinite reservoir. In the experiment, capillaries of 300–700 μm in diameter were used with glycerol–DI water mixture solutions having viscosities ranging from 80 to 934 mPa s. It is observed that compared to the capillary flow from an infinite reservoir, the capillary flow from a droplet exhibits higher rates of meniscus displacement. This is due to an additional driving force resulted from change in droplet surface area (or curvature). The two main parameters influencing the flow are the dimensionless droplet geometry parameter (k) and the dynamic contact angle (θ _D). The molecular kinetics theory of Blake and De Coninck’s model [Adv Colloid Interface Sci 96(1–3):21–36, 2002] is used to interpret the dynamic contact angle. This theory considers a molecular friction coefficient (ζ) at the liquid front flowing over a solid surface. Moreover, three models are proposed to describe the shape of the pendant droplet during capillary action. It is found that the egg-shaped model provides a more realistic model to compute the shape of the pendant droplet deformed during the capillary action. Thus the predictions by the egg-shaped model are in good agreement with the experimental data.     

On a cosmological invariant as an observational probe in the early universe

k-essence scalar field models are usually taken to have Lagrangians of the form L = −V (φ)F(X) with F some general function of X = ▿ _μ φ▿ ^μ ϕ. Under certain conditions, this Lagrangian can take the form of that of an oscillator with time-dependent frequency. The Ermakov invariant for a time-dependent oscillator in a cosmological scenario then leads to an invariant quadratic form involving the Hubble parameter and a logarithm of the scale factor. In principle, this invariant can lead to further observational probes for the early Universe. Moreover, if such an invariant can be observationally verified, then the presence of dark energy will also be indirectly confirmed.     

Theoretical and experimental investigation on electrochemical micromachining

Based on the principle of electrochemical reactions, the mechanism of electrochemical micromachining with ultra short voltage pulses was investigated and the theoretical model was developed. The effects of electrode gap and pulse parameters on the result of electrochemical micromachining were discussed. Using the self-developed experimental system, microtool-electrode and the complex microstructures were sequentially machined. Upon the application of ultrashort voltage pulses, the troughs with 20 μm in width were fabricated by φ 10 μm tool electrode produced.     

Superpositions of continuous functions defined on a Baire space

We consider uniformly continuous functions on a Baire space and introduce the notion of a continuity modulus of a function. We formulate a condition on the growth of the continuity modulus φ guaranteeing that superpositions of n-ary functions with continuity modulus φ do not exhaust all (n + 1)-ary functions with continuity modulus φ for any n. Moreover, negating this property leads to the inverse effect.     

Generating Combinations by Three Basic Operations

We investigate the problem of listing combinations using a special class of operations, prefix shifts. Combinations are represented as bitstrings of O＇s and l＇s, and prefix shifts are the operations of rotating some prefix of a bitstring by one position to left or right. We give a negative answer to an open problem asked by F. Ruskey and A. Williams （Generating combinations by prefix shifts, In Proc. llth Annual International Computing and Combinatorics Conference 2005, LNCS 3595, Springer, 2005, pp.570-576）, that is whether we can generate combinations by only using three very basic prefix shifts on bitstrings, which are transposition of the first two bits and the rotation of the entire bitstring by one position in either direction （i.e., applying the permutations σ2, σn and σn^-1 to the indices of the bitstrings）.     

Droplet coalescence by geometrically mediated flow in microfluidic channels

Microfluidic flow is geometrically mediated at a trifurcating junction allowing periodically formed, equally spaced out emulsion droplets to redistribute and fuse consistently. This is achieved by controlling the ratio between the droplet transport time across the trifurcating junction and the drainage time of the fluid volume separating the droplets t _r/t _d. Three different microfluidic trifurcation geometries have been designed and compared for their droplet fusion efficiencies. Fusion of up to six droplets has been observed in these devices. The fusion of two droplets occurs when t _r/t _d is equal to 1.25 and the number of fused droplets increases with t _r/t _d. When the junction length (d) is 216 μm fusion of 2–6 six droplets are possible however when the junction length is increased to 360 μm fusion of only two droplets is observed.     

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝ^n×n for the general case where the matrix elements a_ij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λ_μ, x^(μ)), μ = 1, ..., n, is suggested for the case where λ_μ is a simple eigenvalue. It is based on the use of an affine interval approximation of a_ij(p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.     

Structure-induced spreading of liquid in micropillar arrays

Contact angle measurements on micropillar arrays were used to determine the conditions that trigger spontaneous penetration of liquids into surface structures. Square micropillars (20 μm) were fabricated in photoresist or quartz and modified chemically to alter the inherent contact angle (i.e., for a flat surface). The lattice spacing of the pillar array and pillar height was also adjusted to investigate the influence of geometry on the wetting behavior. A critical inherent contact angle, θ ₀, was observed below 90°, at which enhanced hydrophobicity switches to enhanced hydrophilicity. This differs from Wenzel’s prediction of θ ₀ = 90°. The transition is not a Cassie-Wenzel state transition. Above the critical angle, the static advancing contact angle increased with pillar coverage due to pinning. Below the critical angle, liquid spreads ahead of the droplet between the pillars to form a stable film. An example of chemical detection and the implications for multiphase microfluidics is discussed.     

Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets

The present paper is the first part of the four-part work on Michell cantilevers transmitting a given point load to a given segment of a straight-line support, the feasible domain being a part of the half-plane contained between two fixed half-lines. The axial stress σ in the optimal cantilevers is assumed to be bounded by −σ _C ≤σ≤σ _T , where σ _C and σ _T represent the allowable compressive and tensile stresses, respectively. The work provides generalization of the results of the article of Lewiński et al. (Int J Mech Sci 36:375–398, 1994a) to the case of σ _T ≠σ _C . The present, first part of the work concerns the analytical formation of the Hencky nets or the lines of fibres filling up the interior of the optimal cantilevers corresponding to an arbitrary position of the point of application of the given concentrated force.     

Probability densities for conditional statistics in the cubic sensor problem

This paper applies the techniques of Malliavin’s stochastic calculus of variations to Zakai’s equation for the one-dimensional cubic sensor problem in order to study the existence of densities of conditional statistics. Let {X _t} be a Brownian motion observed by a cubic sensor corrupted by white noise, and let denote the unnormalized conditional estimate of φ(X _i ). If φ₁,...,φ _n are linearly independent, and if , it is shown that the probability distribution of admits a density with respect to Lebesgue measure for anyn. This implies that, at any fixed time, the unnormalized conditional density cannot be characterized by a finite set of sufficient statistics. Research supported in part by NSF Grant No. MCS-8301880 and by the Institute for Mathematics and It Applications, Minneapolis, Minnesota.     

Communication Complexity Under Product and Nonproduct Distributions

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R_∈(f) and $D^\mu_\in (f)$D^\mu_\in (f) denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that $R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}$R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function f : {0, 1}ⁿ ×{0, 1}ⁿ ? {0, 1}f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\} for which max_{μ product} {D^m _? (f)} = Q(1)  but R _? (f) = Q(n)\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n). We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).     

Measures on MV-algebras

 We study sequentially continuous measures on semisimple M V-algebras. Let A be a semisimple M V-algebra and let I be the interval [0,1] carrying the usual Łukasiewicz M V-algebra structure and the natural sequential convergence. Each separating set H of M V-algebra homomorphisms of A into I induces on A an initial sequential convergence. Semisimple M V-algebras carrying an initial sequential convergence induced by a separating set of M V-algebra homomorphisms into I are called I-sequential and, together with sequentially continuous M V-algebra homomorphisms, they form a category SM(I). We describe its epireflective subcategory ASM(I) consisting of absolutely sequentially closed objects and we prove that the epireflection sends A into its distinguished σ-completion σ _H (A). The epireflection is the maximal object in SM(I) which contains A as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of σ _H (A) depending on the choice of H. We show that the coproducts in the category of D-posets [9] of suitable families of I-sequential M V-algebras yield a natural model of probability spaces having a quantum nature. The motivation comes from probability: H plays the role of elementary events, the embedding of A into σ _H (A) generalizes the embedding of a field of events A into the generated σ-field σ(A), and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras in [8, 11, 14]. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras [13] and, unlike in the Loomis–Sikorski Theorem, objects in ASM(I) correspond to the generated tribes (no quotient is needed, no information about the elementary events is lost). Finally, D-poset coproducts lift fuzzy events, random functions and probability measures to events, random functions and probability measures of a quantum nature. Supported by VEGA Grant 2/7193/01     

Michell cantilevers constructed within a half strip. Tabulation of selected benchmark results

The paper delivers the benchmark results for the Michell cantilevers constructed within a half strip, for selected values of the σ _T /σ _C ratio, σ _T , σ _C being the admissible stresses in tension and compression, respectively.     

Steiner Trees in General Nonuniform Orientations

In this paper we consider Steiner minimum trees (SMT) in the plane, where the connections can only be along a given set of fixed but arbitrary (not necessarily uniform) orientations. The orientations define a metric, called the general orientation metric, A _σ, where σ is the number of orientations. We prove that in A _σ metric, there exists an SMT whose Steiner points belong to an (n−2)-level grid. This result generalizes a result by Lee and Shen [11], and a result by Du and Hwang [5]. In the former case, the same result was obtained for the special case when all orientations are uniform, while in the latter case the same result was proven for the special case when there are only three arbitrary orientations. We then modify the proof used in the main result for the special case when σ=3, i.e., only three arbitrary orientations are considered, and obtain a better result, which states that there exists an SMT whose Steiner points belong to an -level grid. The result has also been obtained by Lin and Xue [9] using a different approach. Received September 27, 1999; revised August 14, 2000     

Sulla costruzione di un sistema di polinomi ortonormali a partire da tre suoi polinomi consecutivi

The above mentioned construction requires that the three assigned polynomials (with degreesn, n+1, n+2) verify conditions stated in Theorem II. In this case the three polynomials belong to infinite many sequences of orthogonal polynomials wich correspond to non negative measuresdϕ(x) with fixed moments μ₀, μ₁,⋯, μ_2n+4.

---

---

Dedicato al Professor S. Faedo in occasione del suo settantesimo compleanno.     

Single-machine scheduling to stochastically minimize maximum lateness

We study the problem of scheduling a set of jobs on a single machine, to minimize the maximum lateness ML or the maximum weighted lateness MWL under stochastic order. The processing time P _i , the due date D _i , and the weight W _i of each job i may all be random variables. We obtain the optimal sequences in the following situations: (i) For ML, the {P _i } can be likelihood-ratio ordered, the {D _i } can be hazard-rate ordered, and the orders are agreeable; (ii) For MWL, {D _i } are exponentially distributed, {P _i } and {W _i } can be likelihood-ratio ordered and the orders are agreeable with the rates of {D _i }; and (iii) For ML, P _i and D _i are exponentially distributed with rates μ _i and ν _i , respectively, and the sequence {ν _i (ν _i +μ _i )} has the same order as {ν _i (ν _i +μ _i +A ₀)} for some sufficiently large A ₀. Some related results are also discussed. This work was partially supported by the Research Grants Council of Hong Kong under Earmarked Grants No. PolyU 5146/02E, CUHK 4170/03E, and NSFC Research Funds No. 70329001, 70518002.     

On estimation of the probability of absence of collisions of some random mappings

Let X and Y be finite sets and φ: (X,Y) →Y be a mapping. Consider a random mapping i → φ(x_i,y_i), where x_i are arbitrarily chosen from the set X, whereas (y_i) is a random sample from Y without replacement. A two-sided bound is derived for the probability of absence of collisions of this mapping. A case of mapping, defined as φ(x, y)=x+ y modulo n, is considered in particular. The results may be used in the selection of identification codes. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 132–137, January–February, 2000.