Effective rheology of bubbles moving in a capillary tube

Abstract

The ALICE Collaboration has measured inclusive J/ψ production in pp collisions at a center of mass energy √ s = 2.76 TeV at the LHC. The results presented in this Letter refer to the rapidity ranges |y| < 0.9 and 2.5 < y < 4 and have been obtained by measuring the electron and muon pair decay channels, respectively. The integrated luminosities for the two channels are L e int = 1.1 nb −1 and L µ int = 19.9 nb −1 , and the corresponding signal statistics are N e + e − J/ψ = 59 ± 14 and N= 1364 ± 53. We present dσ J/ψ /dy for the two rapidity regions under study and, for the forward-y range, d 2 σ J/ψ /dydp t in the transverse momentum domain 0 < p t < 8 GeV/c. The results are compared with previously published results at √ s = 7 TeV and with theoretical calculations.Inclusive J/ψ production in pp collisions at √ s = 2.76 TeV 3 IntroductionAlmost forty years after the discovery of charmonium, its production in hadronic collisions still remains not completely understood, and charmonium production data represent a complex and severe test for QCD-inspired models [1].Recently, first results from the Large Hadron Collider (LHC) on J/ψ production in pp collisions at √ s = 7 TeV became available [2][3][4][5][6], significantly extending the energy reach beyond that of the Tevatron and RHIC hadron colliders [7][8][9]. A reasonable description of the transverse momentum spectra has been obtained by theoretical models [10][11][12][13], and first results on J/ψ polarization, a crucial testing ground for theory [14][15][16], are also available [17] at LHC energy.At the beginning of 2011, the LHC delivered pp collisions at √ s = 2.76 TeV. The main goal of this short run was to provide a reference for the Pb-Pb data which were taken at the same √ s per nucleon-nucleon collision. On the other hand, these data offer the possibility of studying J/ψ production at an intermediate energy between Tevatron and the present LHC top energy, and represent therefore an interesting test for models.In this Letter, we present results on inclusive J/ψ production at √ s = 2.76 TeV as obtained by the ALICE experiment [18]. J/ψ particles were measured, down to zero transverse momentum, via their decay into e + e − at mid-rapidity (|y| < 0.9) and into µ + µ − at forward rapidity (2.5 < y < 4). Results from ALICE on J/ψ production at √ s = 7 TeV were recently published [5,17]. Since the experimental apparatus and the data analysis procedure are basically the same for the two data samples, they will be concisely described, referring where necessary to our previous publications. Results will then be shown for dσ J/ψ /dy at central and at forward rapidity. For the region 2.5 < y < 4 the differential cross section d 2 σ J/ψ /dydp t will also be given, for the transverse momentum range 0 < p t < 8 GeV/c. A comparison with the previous results at √ s = 7 TeV will be carried out and next-to-leading order Non-Relativistic QCD (NLO NRQCD) theoretical calculations will be compared to the experimental data. Experimental apparatus and data analysisThe ma...

Ensemble distribution for immiscible two-phase flow in porous media

“…The system characterized by ensemble distribution in equation (7) is by construction ergodic [46]. The time average of a function g = g(x b ) is g, and we have…”

“…Working towards the goal to determine the ensemble distribution beyond one dimension, we start with conclusions from a one-dimensional sequence of links [46]. We have reported earlier that the probability that a bubble has a certain position x b,ij in the link, is inversely proportional to the velocity dx b /dt of the fluid in that link…”

“…For a link with a single bubble with centerof-mass position x b,ij and saturation s ij , and surface tension γ, the capillary pressure on the bubble is given by [46] p c,ij (x b,ij , s ij , r 0,ij ) = 4γ r 0,ij sin(πs ij ) sin( 2πx b,ij l ) ,…”

“…Sinha et al [46] derived a statistical description of steady state two-phase flow in a single capillary tube. They showed that the well-known Washburn equation could be derived from the entropy production in the tube.…”

Effective Rheology of Two-Phase Flow in a Capillary Fiber Bundle Model

“…The reason behind the discrepancy between the results of Rassi et al and those of Sinha et al is the possibility of a non-zero threshold pressure that observed in the later study, under which there would be no flow, which was assumed to be zero in the former study. The reciprocals in the brackets are provided in order to compare the exponent values reported in the literature [14-17] with those we present here in this article, as we express our results as Q as a power law in ∆P , whereas the cited papers write ∆P as a power law in Q .This power law behavior is in contrast to the assumption of linearity in the relation between flow rate and pressure drop that is generally assumed in the relative permeability approach dominating reservoir simulations [18].For a single capillary tube with varying diameter, Sinha et al [19] showed that the average volumetric flow rate q in the steady state has a non-linear squareroot type relationship with the pressure drop ∆P as q ∼ ∆P 2 − P 2 c . This was shown analytically by integrating the instantaneous linear two-phase flow equation over the whole capillary tube.…”