Approximation by New Family of Modified Bernstein Type Polynomials

In the present paper our main aim is to use the approximation methods to express the Laplace formula of theory of probability by new family of modified Bernstein Type Polynomials defined for the function f(u) of .

Contact problem for wavy surfaces in the presence of an incompressible liquid and a gas in interface gaps

This paper presents a study on smooth elastic contact between two semi-infinite elastic bodies, one of which has a wavy surface, for the case when there are an incompressible liquid, not wetting the surfaces of the bodies, at the central region of each interface gap and a gas under constant pressure at the edges of each gap. Due to the surface tension of the liquid, a pressure drop occurs in the liquid and the gas, which is described by the Laplace formula. The formulated contact problem is reduced to a singular integral equation (SIE) with the Hilbert kernel, which is transformed into a SIE with the Cauchy kernel for a derivative of a height of the gaps. A system of transcendental equations for a width of each gap and a width of the gap region filled with the liquid is obtained from the condition of boundedness of the contact stresses at the gap ends and the condition of liquid amount conservation. It is solved numerically, and the dependences of the width and shape of the gaps, the width of the gap regions filled with the liquid and the contact approach of the bodies on the applied load and the surface tension of the liquid are analyzed.

Application of Laplace Formula in Additional Pressure Deduction of Curved Surface

By adopting Laplace formula in arbitrary minimal size to deduct the additional pressure of curved surface, this article aims to conclude additional pressure expressions of arbitrary curved shape. Then, the origin of additional pressure of each curved surface can be clearly comprehended.

Analysis and Simulation of Micro-Fluidic Inertial Switch

This paper presents a micro-fluidic inertial switch with varying rectangular cross section, which employs the moving mercury droplet in the micro-channel to close a switch. Combining the Young-Laplace formula with the structure, the formula of the threshold g-value is derived. The influence of design parameters of microchannel on the threshold g-value is analyzed. Based on the VOF approach containing contact angle effects, the dynamic behavior of mercury droplet in the microchannel is simulated using Fluent. The response time is predicted through simulation with the contact angles ranging from 130° to 170°. In addition, the dynamic process of inertial switch is simulated, and the result indicates that the selected design parameters can achieve reliable switching under given threshold g-value.

SPHERICAL HARMONICS ASSOCIATED TO THE LAPLACE–BESSEL OPERATOR AND GENERALIZED SPHERICAL CONVOLUTIONS

A theory of spherical harmonics associated to the Laplace–Bessel differential operator is developed. Natural analogs of the Plancherel theory, the Laplace formula, the Funk–Hecke formula, the product formula, and the addition theorem are obtained. Symmetry properties of the Fourier–Bessel transform, decompositions of smooth functions, and convolution operators are studied.