Applications of potential theory in mechanics. Selection of by V.I. Fabrikani

By V.I. Fabrikani

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Solve the problem above in the case when σ=σ /ρn. 0 2σ √π Γ[( n − 1)/2] Answer: Q T = 0 ( n − 2) Γ( n /2) a n-2 . 6 Some fundamental integrals The integrals are called fundamental because of their primary importance to the new method, and also because almost all the integral representations, derived earlier, are just particular cases of the fundamental ones to be evaluated here. Consider three points in the system of cylindrical coordinates, namely, M (ρ,φ, z ), M 0(ρ0,φ0, z 0), and N ( r ,ψ,0).

5 External mixed boundary value problem for a half-space The material in this section follows the paper (Fabrikant, 1986f). The problem is called external when non-zero boundary conditions are prescribed outside the disc. As in the previous section, we consider two types of problem. Problem 1. It is necessary to find a function, harmonic in the half-space z ≥0, vanishing at infinity, and subject to the mixed boundary conditions on the plane z =0, namely, 43 44 CHAPTER 1, ∂V = 0, ∂ z z=0 for ρ< a , 0≤φ<2π; V = v (ρ,φ), for ρ≥ a , 0≤φ<2π.

The following boundary conditions are prescribed at z =0 V = ( v /ρ3) e3 i φ, 3 for ρ≥ a , 0≤φ<2π; ∂V = 0, for ρ< a , 0≤φ<2π. ∂z Find the potential function. 15 v ( a 2 − l 21)1/2  3 1  e3 i φ  1 − Answer: V (ρ,φ, z ) = 2 a  4ρ3  ( a 2 − l 21)1/2 3 ( a 2 − l 21)1/2 5  1 1   + 1 −   . 1 −  − 3 10 a a      8. Let the following boundary conditions be prescribed at z =0: for ρ≤ a , V = 0, 0≤φ<2π; ∂V = − 2πσ /ρ2, σ =const, for ρ> a , 0≤φ<2π. 0 0 ∂z Find the potential function and the charge distribution.

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