Although many gas-phase microfluidic devices contain curved surfaces, relatively little research has been conducted on the degree of slip over nonplanar surfaces. The present study demonstrates the influence of the surface shape (i.e., convex/concave) on the velocity slip and formation of the Knudsen layer. In addition, the study reveals that there is a simple relationship between the shear stress exerted on the surface and the velocity defect in the Knudsen layer.
In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coefficients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.
In this work, we investigate a linear completely nonhomogeneous nonlocal multipoint problem for an m-order ordinary differential equation with generally variable nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness. A system of m + 1 integro-algebraic equations called the special adjoint system is constructed for this problem. Green's functional is a solution of this special adjoint system. Its first component corresponds to Green's function for the problem. The other components correspond to the unit effects of the conditions. A solution to the problem is an integral representation which is based on using this new Green's functional. Some illustrative implementations and comparisons are provided with some known results in order to demonstrate the advantages of the proposed approach.
In the present work, the problem of two collinear cracks in an isotropic, homogeneous elastic medium which is subjected to uniform anti-plane shear loading at infinity is investigated in the context of nonlocal theory of elasticity. Governing equations of the problem are obtained by employing the field equations of the nonlocal elasticity. By use of the boundary conditions, the solution of the problem is reduced first to the investigation of the triple integral equations and then to a singular Fredholm integral equation. Numerical calculations are carded out for different values of the chosen parameter. Then stress analyses are given for single and double cracks.
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