Minimum-drag shapes in magnetohydrodynamics

Abstract

A necessary optimality condition for the minimum-drag shape for a non-magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained. It is assumed that the flow and magnetic field are uniform and parallel at infinity, and that the body and fluid have the same magnetic permeability. The condition is derived based on the linearized magnetohydrodynamic (MHD) equations subject to a constraint on the body's volume, and generalizes the existing optimality conditions for the minimum-drag shapes for the body in the Stokes and Oseen flows of a non-conducting fluid. It is shown that for any Hartmann number M , Reynolds number Re and magnetic Reynolds number Re m , the minimum-drag shapes are fore-and-aft symmetric and have conic vertices with an angle of 2p/3. The minimum-drag shapes are represented in a function-series form, and the series coefficients are found iteratively with the derived optimality condition. At each iteration, the MHD problem is solved via the boundary integral equations obtained based on the Cauchy integral formula for generalized analytic functions. With respect to the equal-volume sphere, drag reduction as a function of the Cowling number S = M 2 /(Re m Re) is smallest at S = 1. Also, in the considered examples, the drag values for the minimum-drag shapes and equal-volume minimum-drag spheroids are sufficiently close.

Minimum-resistance shapes in linear continuum mechanics

“…Let r be the radius vector representing the surface of the minimum-resistance shape, S, and let r = r + f (r)n be the radius vector corresponding to the surface of a shape variation, S , where is a positive small number and f (r) is any continuous bounded scalar function satisfying the condition S f dS = 0, which follows from the fact that the inclusion has fixed volume (see [11,14]). Let u be the displacement field for the elastic medium with the inclusion having the minimum-resistance shape S, and let u = u + ũ + o( ) be a variation of u, whereũ satisfies the Navier equations (1.1) and vanishes at infinity.…”

“…For the minimum-resistance (fore-and-aft symmetric) shape, is approximated in the rz-half plane for z ≥ 0 by [14,41]). For z ≤ 0, is determined by ζ = ζ (t), t ∈ [0, 1].…”

“…As shown in [14,23,24], similar derivative-free representations provide a simple coordination of all the parts of the optimization scheme: for axisymmetric Stokes and Oseen flows, the drag of the body, the optimality conditions for the minimum-drag shape and the corresponding BIEs are all formulated in terms of the same generalized analytic function. The goal of this work is twofold: (i) to obtain the necessary optimality condition for the minimum-resistance shape for a fixed-volume rigid inclusion translated in an unbounded isotropic elastic medium and to analyse the optimal shape's dependence on the Poisson ratio ν and (ii) to demonstrate the approach of generalized analytic functions in finding the optimal shape, which includes reformulating the optimality condition in terms of generalized analytic functions and obtaining the BIEs based on the generalized Cauchy integral formula.…”

Cauchy integral formula for generalized analytic functions in hydrodynamics

“…With the generalized Cauchy integral formulae, the MHD problem is reduced to boundaryintegral equations, which are then employed in an iterative procedure for finding minimum-drag shapes for solid nonconducting bodies in the MHD flow under the assumption of small R, R m and M; see Zabarankin (2011b).…”

Generalized analytic functions in magnetohydrodynamics

“…How does drag reduction depend on S? Answering these questions is the subject of this paper and the follow-up work (Zabarankin 2011).…”