A NOTE ON THE NUMBER OF BIPARTITE PLANAR MAPS

Let H(n,m) be the number of rooted non-isomorphic bipartite planar maps with m edges and the valency of the rooted face being 2n. This note provides the following results: [GRAPHICS] for n greater-than-or-equal-to 2, where \DELTA(i)\m = (2i - 2)! divided-by (i - 1)!i! {(4i - 2)alpha(i + 1, m - i) + alpha(i, m - i) - i-alpha(i, m - i + 1) + (i + 1)beta(i, m - i)}, m > i, (2i - 2)! divided-by (i - 1) !i! (4i - 1 - i2), m = i;0, m < i. [GRAPHICS] Meanwhile, the combinatorial identity [GRAPHICS] is also found. In what mentioned above, alpha(s, t) and beta(s, t) are expressed by the following finite sums with all the terms positive:(~)[GRAPHICS]

THE RELATION OF THE THRESHOLD VELOCITY OF COAL DUST TO ITS SIZE AND HUMIDITY

On the ground of wharf and a station where coal is piled up, the sources polluting the atmosphere are mainly the raised coal dust. According to the principle of mechanics, with the consideration of the gravities of coal dust particles and liquid droplets, coal dust size and humidity, under the condition of force balance when the coal dust was raised, the authors obtaineda theoretical formula for the threshold velocity of coal dust with humidity,Finally, the theoretical values from the formula gave a good agreement with the experimental data from some wind tunnels.

THE SEPARATION OF LARGE VORTEXES AND THE CLOSED EQUATIONS OF TURBULENCE MODEL

This paper presents a new kind of everage for the loeally-generated large vortexes so that the physieal quantities of the locally-generated large vortexes and the external large vortexes canberigorously separated from the equal ions for the large vortexes proposed in a previous paper[1] To the equations for the two kinds of large vortexes, some auxiliary relations are introduced, and the value, of the length-scale lN of energy dissipation of the external large vortexes may he determined according to the actual circumstances of the disturbance of external sources. Thus the resulting equations of the second moments of turbulent velocity fluctuations for the two kinds of large vortexes can be made closed. Meanwhile, the corresponding coefficients of diffusion in the previous paper[1] are improved,Finally, a closed set of numerically-solvable equations of turbulence model are obtained.

CONTAMINANT CONTRACTION IN TWO-DIMENSIONAL OSCILLATORY FLOWS

If the vertically-mixing time is comparable with the period of oscillatory current, the contaminant contraction may occur, then the coefficient of shear dispersion will be negative (singularity). According to the two-dimensional delay-diffusion equation derived by Liu in another paper, The equations for X(t), Y(t), central displacement as well as dispersion tensor were derived. Generally speaking, δDij/δτ is positive when τ is small; but when τ is large, the memory functions may be negative. The analytical expressions for Dij and X,Y were also given.

COORELATION ANALYSIS OF SURFACE PRESSURE FLUCTUATIONS ON THREE EQUISPACED CYLINDERS

The fluctuating pressures on the surface of three circular cylinders in equispaced arrangement were measured simultaneously in the medium subcritical regime (Re=3×104). The distributions of their rms values and circumferential correlation were obtained in the spacing range from 1.25 to 6.0. The spatial correlation of pressures of two pressure taps on two different cylinders at the same azimuth was investigated as well. Owing to different flow patterns in various arrangements and spacing, these correlation curves of pressure show different distributions. The lift fluctuation can be estimated qualitatively from the circumferential coefficient curve with the knowledge of the rms pressure distribution. Some critical spacing for each arrangement were found that the flow regimes and the corresponding pressure correlation take a sudden change. Flow interference between cylinders in each arrangement was also discussed with various flow regimes.

CONJUGATE FLOWS OVER A STEP IN A THREE-LAYER FLUID

The conjugate flows over a step with the height h_0, which might be positive or negative, was studied in a three-layer fluid and the coupled nonlinear equations were derived, with which the effects of varying height h_0 on the existence and evolution of conjugate flows were examined. It is concluded that the conjugate flow is sharply sensitive to the thickness of fluid layers and its characteristics alters remarkably due to the existence of the step. As the flow climbs up a step (h_0>0), the conjugate flow with a convex lower interface and a concave upper interface is allowed to appear, while the flow with the concave lower interface or the simultaneous concave interfaces will be depressed. As the flow goes down a step (i.e., h_0<0), on most occasions only one kind of conjugate flow could exist, which prossesses the form with the simultaneous convex interfaces and will disappear rapidly with the increase of the step depth.

NONLINEAR WAVE INDUCED BY AN ACCELERATING CYLINDRICAL TANK

This paper presents a method for solving Lagrangian version of governing equations that allows boundary conditions at the free surface to be satisfied exactly, which is a three-dimensional generalization of a method first put forward by Stoker. Analytical expressions of nonlinear hydrodynamic pressure up to the third order and of free surface displacement up to the fourth order of an accelerating cylindrical tank were obtained. Here only the motions of objects in their early stage after initial impulses was considered. As a justification of the method, the important special case when the ratio of tank diameter to fluid depth tends to infinity was taken as an example, which shows that the linear hydrodynamic pressure obtained agrees completely with Westergaard or von Karman's classical result.