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

Resumo

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 ε. A uniform asymptotic expansion of the solution of this problem with respect to ε 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(x₁, x₂, x₃, ε) = x₃+O(r⁻²) as r → ∞, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: ∂u/∂n = 0 at the boundary. After subtracting x₃ from the solution u(x₁, x₂, x₃, ε), we get a boundary value problem for the potential ũ(x₁, x₂, x₃, ε) of the perturbed motion. Since the integral of the function ∂ũ/∂n over the surface of the body is zero, we have ũ(x₁, x₂, x₃, ε) = O(r⁻²) as r → ∞. Hence, all the coefficients of the outer asymptotic expansion with respect to ε 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.