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The article offers information regarding the number pi, a mathematical constant. It highlights the application of computers in the search for more digits of pi as well as the development of the formulas for the total number of collisions. It also highlights information concerning the dynamic equations.

The article offers information regarding the number pi, a mathematical constant. It highlights the application of computers in the search for more digits of pi as well as the development of the formulas for the total number of collisions. It also highlights information concerning the dynamic equations.

Natural convection in a vertical slot filled with a fluid-saturated porous medium modeled by the Brinkman equation is considered. A continuum of solutions are found from capped to open ends in fully-developed two-dimensional flow. This brings to flow in a porous media some aspects of the Newtonian conduction regime results of Bühler (Heat Mass Transf 39: 631, 2003) and the convection regime results of Weidman (Heat Mass Transf 43: 103, 2006). The Brinkman model, valid for sufficiently large porosity and governed by parameter α, allows one to connect Bühler’s conduction regime flow at low α to a near-Darcy convection regime boundary-layer flow at high α. [ABSTRACT FROM AUTHOR]

The linear stability of an axisymmetric rotating two-fluid column in zero gravity is investigated. The inner core fluid of radius a and density $\rho_1$ is surrounded by fluid of density $\rho_2$ itself bounded by an outer impermeable cylinder of radius b. The surface tension coefficient at the immiscible interface is $\gamma$ . The parameters governing instability are the density ratio $\lambda = \rho _1/ \rho _2$ , the radius ratio $\kappa = b/a$ and the generalized Hocking parameter $L_i = \gamma / \rho _i\Omega^2 a^3$ . While the necessary and sufficient condition for stability $L_2 \le (1 - \lambda )$ is known, the preferred modes and wavenumbers at onset of instability have not been determined over the space of dimensionless parameters. Analytically and numerically determined maximum growth rates and corresponding preferred instability wavelengths for a rotating liquid column show that axisymmetric disturbances are most unstable for $L_1 > 0.1053$ while planar disturbances determine instability when $L_1 < 0.1053$ . The rotating two-fluid system bounded by an outer cylinder with statically stable density ratios $0 \le \lambda \le 1$ is unstable only to axisymmetric disturbances. Maximum growth rates and preferred wavelengths are computed as a function of $\lambda$ and $L_2$ at radius ratios $\kappa = 1.2$ and $\kappa = 5.0$ , respectively corresponding to relatively large and relatively small fluid cores. Complete results for the hollow-core vortex $(\lambda = 0)$ and a stationary two-fluid system $(L_2 \to \infty)$ are also presented. [ABSTRACT FROM AUTHOR]

In some three-dimensional laminar boundary layer problems a coordinate decomposition reduces the governing equations to a primary nonlinear ordinary differential equation describing the streamwise flow in a semi-infinite domain and a secondary linear equation coupled to the primary solution describing the cross flow of infinite spanwise extent. Five new cross-flow problems of this type are investigated within the confines of laminar boundary layer theory. First, the equation for uniform flow transverse to a planar laminar wall jet is found and solved exactly. Second, two solutions for motion transverse to a uniform shear flow along a flat plate are given. A third problem considered is the transverse motion over a flat plate aligned with the uniform mainstream and advancing toward or receding from the mainstream. In the fourth problem, a family of solutions for transverse uniform streams above and below a planar laminar jet is given in closed form. This solution depends on the momentum J of the planar jet and the velocity ratio $\kappa\$ of the transverse streams. The last problem addresses the motion of transverse uniform streams above and below a planar laminar wake. At leading order the cross flow depends only on the velocity ratio $\kappa\$ and not on the drag D produced by the body forming the wake. The influence of the drag first appears at $O(x^{-1} \ln x)$ in the streamwise coordinate expansion of the cross-flow solution. [ABSTRACT FROM AUTHOR]