Appl Math Optim 47:59–78 (2003)
2002 Springer-Verlag New York Inc.
Shape Optimization in Viscous Compressible Fluids
Mathematical Institute AV
Zitn´a 25, 115 67 Praha 1, Czech Republic
Abstract. We present a method for solving the optimal shape problems for proﬁles
surrounded by viscous compressible ﬂuids. The class of admissible proﬁles is quite
general including the minimal volume condition and a constraint on the thickness of
the boundary. The ﬂuid ﬂow is modelled by the Navier–Stokes system for a general
viscous barotropic ﬂuid.
Key Words. Optimal shape design, Compressible ﬂow, Navier–Stokes equations.
AMS Classiﬁcation. 35B30, 35Q30.
An important industrial problem is to ﬁnd the shape of a wing which would minimize
the drag—the reaction of the surrounding ﬂuid on the wing. The force
F acting on a
wing S moving at a constant speed
is given by the formula
Σ ·n − pn dσ,
where Σ is the viscous stress tensor, p is the pressure, and n stands for the outer normal
vector to ∂ S. With a uniform ﬂow at inﬁnity, the drag J (S) is the component of
to the velocity
J (S) =
( ·n) ·
− pn ·
This work was supported by Grant A1019002 GA AV