Nearly free molecular flow through an orifice

The problem of the flow through an orifice is a very interesting one in fluid mechanics, as it promises to be one of the few configurations which can be investigated over virtually the whole range of possible motions. For this reason, Liepmann(1) has recently made measurements of the mass flow through an orifice at what are practically infinite pressure rations, through a range of Knudsen numbers covering the transition from continuum to free molecule flow. The mass flow rate per unit area in the Knudsen limit (i.e., at high K = λ1/R where λ1 is the mean free path at upstream infinity and R is the radius of the hole) is well known from kinetic theory to be m = 1/4p1c1 where p1 is the density and c1 the mean molecular speed at upstream infinity. The purpose of this note is to estimate the effect on m of a Knudsen number K that is not so large

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k > 2$ such that $i_f(n) \ll n^{\frac{1}{4}}$ for all $n \gg 0$.Comment: To appear in The Ramanujan Journa

On sign changes of q-exponents of generalized modular functions

Let f be a generalized modular function of weight 0 of level N such that its q-exponents c(n)(n>0) are all real, and div(f) is zero. In this note, we show the equidistribution of signs for c(p)(p prime) by using equidistribution theorems for normalized cuspidal eigenforms of integral weight.Comment: to appear in the Journal of Number Theor

On Mori cone of Bott towers

A Bott tower of height $r$ is a sequence of projective bundles $$X_r \overset{{\pi_r}}\longrightarrow X_{r-1} \overset{\pi_{r-1}}\longrightarrow \cdots \overset{\pi_2}\longrightarrow X_1=\mathbb P^1 \overset{\pi_1} \longrightarrow X_0=\{pt\},$$ where $X_i=\mathbb P (\mathcal O_{X_{i-1}}\oplus \mathcal L_{i-1})$ for a line bundle $\mathcal L_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $\mathbb P(-)$ denotes the projectivization. These are smooth projective toric varieties and we refer to the top object $X_{r}$ also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.Comment: The conditions in Theorem 6.3 have been correcte

Modelling the transitional boundary layer

Recent developments in the modelling of the transition zone in the boundary layer are reviewed (the zone being defined as extending from the station where intermittency begins to depart from zero to that where it is nearly unity). The value of using a new non-dimensional spot formation rate parameter, and the importance of allowing for so-called subtransitions within the transition zone, are both stressed. Models do reasonably well in constant pressure 2-dimensional flows, but in the presence of strong pressure gradients further improvements are needed. The linear combination approach works surprisingly well in most cases, but would not be so successful in situations where a purely laminar boundary layer would separate but a transitional one would not. Intermittency-weighted eddy viscosity methods do not predict peak surface parameters well without the introduction of an overshooting transition function whose connection with the spot theory of transition is obscure. Suggestions are made for further work that now appears necessary for developing improved models of the transition zone