Asymptotics of the Velocity Potential of an Ideal Fluid Flowing around a Thin Body

作者：Ershov A A^*; Krutova Yu A

来源：Proceedings of the Steklov Institute of Mathematics, 2018, 301(S1): S15-S31.

DOI：10.1134/S0081543818050024

• boundary value problem
• Laplace equation
• asymptotic expansion
• matching method
• thin body
• laminar flow
• ideal fluid

We consider the Neumann problem outside a small neighborhood of a planar disk in the three-dimensional space. The surface of this neighborhood is assumed to be smooth, and its thickness is characterized by a small parameter epsilon. A uniform asymptotic expansion of the solution of this problem with respect to epsilon is constructed by the matching method. Since the problem turned out to be bisingular, an additional inner asymptotic expansion in the so-called stretched variables is constructed near the edge of the disk. A physical interpretation of the solution of this boundary value problem is the velocity potential of a laminar flow of an ideal fluid around a thin body, which is the neighborhood of the disk. It is assumed that this flow has unit velocity at a large distance from the disk, which is equivalent to the following condition for the potential: u(x (1), x (2), x (3), epsilon) = x (3)+O(r (-2)) as r -> a, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: a,u/a,n = 0 at the boundary. After subtracting x (3) from the solution u(x (1), x (2), x (3), epsilon), we get a boundary value problem for the potential (x (1), x (2), x (3), epsilon) of the perturbed motion. Since the integral of the function a,/a,n over the surface of the body is zero, we have (x (1), x (2), x (3), epsilon) = O(r (-2)) as r -> a. Hence, all the coefficients of the outer asymptotic expansion with respect to epsilon have the same behavior at infinity. However, these coefficients have growing singularities at the approach to the edge of the disk, which implies the bisingularity of the problem.

• 出版日期2018-7