• CARMA SEMINAR
  • Speaker: Dr Alexander Plakhov, Center for Research and Development in Mathematics and Applications, University of Aveiro
  • Title: Problems of optimal resistance in Newtonian aerodynamics
  • Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Time and Date: 3:00 pm, Thu, 5th Jul 2012
  • Abstract:

    A body moves in a rarefied medium of resting particles and at the same time very slowly rotates (somersaults). Each particle of the medium is reflected elastically when hitting the body boundary (multiple reflections are possible). The resulting resistance force acting on the body depends on the time; we are interested in minimizing the time-averaged value of resistance (which is called $R$). The value $R(B)$ is well defined in terms of billiard in the complement of $B$, for any bounded body $B \subset \mathbb{R}^d$, $d\geq 2$ with piecewise smooth boundary.

    Let $C\subset\mathbb{R}^d$ be a bounded convex body and $C_1\subset C$ be another convex body with $\partial C_1 \cap C=\varnothing$. It would be interesting to get an estimate for $$R(C1_,C)= \inf_{C_1\subset B \subset C} R(B) .................. (1)$$ If $\partial C_1$ is close to $\partial C$, problem (1) can be referred to as minimizing the resistance of the convex body $C$ by "roughening" its surface. We cannot solve problem (1); however we can find the limit $$\lim_{\text{dist}(\partial C_1,\partial C)\rightarrow 0} \frac{R(C_1,C)}{R(C)}. .................. (2) $$

    It will be explained that problem (2) can be solved by reduction to a special problem of optimal mass transportation, where the initial and final measurable spaces are complementary hemispheres, $X=\{x=(x_1,...,x_d)\in S^{d-1}: x_1\geq 0\}$ and $Y=\{x\in S^{d-1}:x_1\leq 0\}$. The transportation cost is the squared distance, $c(x,y)=\frac{1}{2}|x-y|^2$, and the measures in $X$ and $Y$ are obtained from the $(d-1)$-dimensional Lebesgue measure on the equatorial circle $\{x=(x_1,...,x_d):|x|\leq 1,x_1=0\}$ by parallel translation along the vector $e_1=(1,0,...,0)$. Let $C(\nu)$ be the total cost corresponding to the transport plan $\nu$ and let $\nu_0$ be the transport plan generated by parallel translation along $e_1$; then the value $\frac{\inf C(\nu)}{C(\nu_0)}$ coincides with the limit in (2).

    Surprisingly, this limit does not depend on the body $C$ and depends only on the dimension $d$.

    In particular, if $d=3$ ($d=2$), it equals (approximately) 0.96945 (0.98782). In other words, the resistance of a 3-dimensional (2-dimensional) convex body can be decreased by 3.05% (correspondingly, 1.22%) at most by roughening its surface.


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