Three-phase boundary motions under constant velocities. I: The vanishing surface tension limit

1996 ◽  
Vol 126 (4) ◽  
pp. 837-865 ◽  
Author(s):  
Fernando Reitich ◽  
H. Mete Soner
Keyword(s):  
Surface Tension ◽  
Formal Analysis ◽  
Local Equilibrium ◽  
Convergence Result ◽  
Normal Velocity ◽  
Infinitely Many Solutions ◽  
Weak Formulation ◽  
Material Interfaces ◽  
Internal Surfaces ◽  
Image Position

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equationHere Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires thatIn case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.

On the logic of incomplete answers

1965 ◽  
Vol 30 (1) ◽  
pp. 65-68 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Formal Analysis ◽  
Complete List ◽  
Strict Implication ◽  
Complete Answer ◽  
Questions And Answers ◽  
Image Position

I have argued in [1] that a concept bearing some resemblance to ‘p is the answer to d’ (p a proposition and d a question) can be defined wherever d has the form,‘For which a's is it the case that A (a)?’ (Qa)A(a)where a is a variable and A a wff containing a. To say that p is the true and complete answer to (Qa)A(a) is expressed as saying that p is logically equivalent to the true conjunction of A(a) or ~A(a) for each a. It is defined as;Such a concept of answer is like Belnap's [2] direct true answer to a complete list question, or like Harrah's use [3] (p. 43) of the notion of a state description. The main difference between my approach and that of Belnap and Harrah is that while they are concerned to develop a formal metalanguage for discussion of questions and answers I am concerned to express, as far as possible in existing systems, certain interrogative statements; in particular statements of the form ‘— is the (an) answer to —’.While the account in [1] does give a formal analysis of one ‘answer’ concept there are respects in which it is inadequate.1. Since it uses entailment (or strict implication) to define the relation between p the answer and d the question we can shew that if p is the answer to d and q is logically equivalent to p then q is the answer to d.

Investigations on condensation phenomena in mercury vapour

1934 ◽  
Vol 30 (2) ◽  
pp. 216-224
Author(s):  
P. C. Ho
Keyword(s):  
Surface Tension ◽  
Chemical Properties ◽  
Mercury Vapour ◽  
Physical And Chemical Properties ◽  
Critical Supersaturation ◽  
Different Temperatures ◽  
Image Position ◽  
Physical And Chemical ◽  
Theoretical Considerations ◽  
And Chemical Properties

Owing to its physical and chemical properties being greatly different from those of any of the liquids which have hitherto been used in the Wilson cloud chamber, mercury has been used in the experiments described in this paper and the condensation phenomena of its vapour at different temperatures observed. Before constructing the apparatus it was considered necessary to get from theoretical considerations some idea about the magnitude of the critical supersaturation for mercury vapour in equilibrium with a drop carrying unit charge. Assuming that J. J. Thomson's formula.where s is the supersaturation of mercury vapour in equilibrium with a drop of mercury of radius a, charge e, density σ and surface tension T, the value of which is assumed here to be independent of the radius of the drop, K the specific inductive capacity of the dielectric surrounding the drop, and R the gas constant for one gramme of weight, all at temperature θ, can be applied to the present problem, this critical supersaturation sm is given by the formula

scholarly journals Liquid toroidal drop under uniform electric field

2017 ◽  
Vol 473 (2202) ◽  
pp. 20160633 ◽  
Author(s):  
Michael Zabarankin
Keyword(s):  
Surface Tension ◽  
Electric Field ◽  
Dielectric Constants ◽  
Uniform Electric Field ◽  
Normal Velocity ◽  
Spherical Drop ◽  
Electric Stress ◽  
Freely Suspended ◽  
Major Radius ◽  
External Forces

The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary when Q =(2 R 2 +3 R +2)/(7 R 2 ), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca E , that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca E =0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca E >0. This work finds Q and Ca E such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius ρ and volume 4 π /3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca E depend on R and ρ , and for large ρ , e.g. ρ ≥3, they have simple approximations: Q ∼( R 2 + R +1)/(3 R 2 ) and Ca E ∼ 3 3 π ρ / 2   ( 6  ln  ⁡ ρ + 2  ln ⁡ [ 96 π ] − 9 ) / ( 12  ln  ⁡ ρ + 4  ln ⁡ [ 96 π ] − 17 )   ( R + 1 ) 2 / ( R − 1 ) 2 .

Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

Approximation by a sum of polynomials of different degrees involving primes

1979 ◽  
Vol 27 (4) ◽  
pp. 454-466
Author(s):  
Ming-Chit Liu
Keyword(s):  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Coefficients ◽  
Image Position

AbstractLet λj (1 ≦ j ≦ 4) be any nonzero real numbers which are not all of the same sign and not all in rational ratio and let pj be polynomials of degree one or two with integer coefficients and positive leading coefficients. The author proves that if exactly two pj are of degree two then for any real n there are infinitely many solutions in primes pj of the inequality . where 0 <β < (√(21)–1)∖5760.

scholarly journals THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

2020 ◽  
Vol 8 ◽  
Author(s):  
BRAD RODGERS ◽  
TERENCE TAO
Keyword(s):  
Pair Correlation ◽  
Local Equilibrium ◽  
Local Distribution ◽  
Distribution Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
De Bruijn ◽  
Image Position ◽  
Argument Proceeds ◽  
Strong Control

For each $t\in \mathbb{R}$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $\unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant) such that the zeros of $H_{t}$ are all real precisely when $t\geqslant \unicode[STIX]{x1D6EC}$ . The Riemann hypothesis is equivalent to the assertion $\unicode[STIX]{x1D6EC}\leqslant 0$ , and Newman conjectured the complementary bound $\unicode[STIX]{x1D6EC}\geqslant 0$ . In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $\unicode[STIX]{x1D6EC}<t\leqslant 0$ , until one establishes that the zeros of $H_{0}$ are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

Action of a force near the planar surface between semi-infinite immiscible liquids at very low Reynolds numbers: Addendum

1978 ◽  
Vol 19 (2) ◽  
pp. 309-318 ◽  
Author(s):  
K. Aderogba ◽  
J.R. Blake
Keyword(s):  
Surface Tension ◽  
Normal Force ◽  
Order Approximation ◽  
Logarithmic Singularity ◽  
Reynolds Numbers ◽  
Point Force ◽  
Immiscible Liquids ◽  
Two Fluids ◽  
First Order Approximation ◽  
Image Position

In the paper by Aderogba and Blake [1], the first order approximation to the shape of the interface between two immiscible liquids, at which surface tension acted, was obtained in the case of a point force directed parallel to the interface. In the case of the normal force no solution was obtained due to the logarithmic singularity associated with problems of this type. However, the problem is not physically well-posed in the case of the normal force. Surface tension is not suitable because it cannot alone balance the induced stress on the interface due to a point force. We need an additional force to balance the action of the point force on the interface. The obvious solution is to include a density difference Δρ* between the two fluidswhere and are the densities of the lower and upper fluid respectively.

scholarly journals Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

2017 ◽  
Vol 148 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Eun Bee Choi ◽  
Jae-Myoung Kim ◽  
Yun-Ho Kim
Keyword(s):  
Weak Solutions ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Neumann Problems ◽  
Neumann Boundary Value Problem ◽  
Cerami Condition ◽  
Image Position ◽  
Laplace Type

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operatorwhere the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

scholarly journals Pairwise relatively prime solutions of linear diophantine Equations

1984 ◽  
Vol 37 (1) ◽  
pp. 39-44 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Diophantine Equations ◽  
Infinitely Many Solutions ◽  
Linear Diophantine Equations ◽  
Image Position

AbstractIt is shown that if a1,…,am are relatively prime integers then for every integer n the equation has infinitely many solutions in pairwise relatively prime integers x1,…,xm.

Diophantine Approximation and Horocyclic Groups

1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Modular Group ◽  
Positive Real Number ◽  
Irrational Number ◽  
Infinitely Many Solutions ◽  
Positive Real ◽  
Closed Set ◽  
Coprime Integers ◽  
Image Position

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

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