Similar solutions for viscous hypersonic flow over a slender three-fourths-power body of revolution

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NASA Technical Memorandum 100205

Similar Solutions for Viscous Hypersonic
Flow Over a Slender Three-Fourths-Power
Body of Resolution




(l.ASA^ 23 n CSCL 20D Unclas

‘^*^^’ P G3/34 0110964

Chin-Shun Lin

Institute for Computational Mechanics in Propulsion

Lewis Research Center

Cleveland, Ohio

December 1987









C.S. Lin

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135


For hypersonic flow with a shock wave, there Is a similar solution con-
sistent throughout the viscous and Invlscid layers along a very slender three-
fourths-power body of revolution. The strong pressure Interaction problem can
then be treated by the method of similarity. In the present study, numerical
calculations are performed In the viscous region with the edge pressure dis-
tribution known from the Invlscid similar solutions. The compressible laminar
g boundary-layer equations are transformed Into a system of ordinary differential
00 equations. The resulting two-point boundary value problem Is then solved by
^ the Runge-Kutta method with a modified Newton’s method for the corresponding
boundary conditions. The effects of wall temperature, mass bleeding, and body
transverse curvature are Investigated. The Induced pressure, displacement
thickness, skin friction, and heat transfer due to the previously mentioned
parameters are estimated and analyzed.


At hypersonic speeds, flow Is decelerated by the work of compression and
viscous dissipation, therefore a high- temperature gas is produced In the bound-
ary layer. The density of the hot gas Is very low, so the mass flux in this
boundary layer is small. Because of this high temperature, the thickness of
the boundary layer on the body surface Increases and the streamlines in the
flow external to the boundary layer are displaced outward. The displacement
thickness may be comparable to or may even exceed the body thickness, so that
the effect of body transverse curvature is significant. The effective thick-
ening of the body can also Induce a large pressure, which Is transmitted Into
the external invlscid field along the Mach lines. These pressures are then
transmitted essentially without change through the boundary layer and. In turn,
govern the growth of the boundary layer. Along with the pressure interaction,
the vorticity Interaction may also occur because of the curved shock wave.
Thus, the boundary-layer structure will be governed not only by the pressure
gradient, but also by the vorticity at the edge of the boundary layer (ref. 1).

Another Important fact Is that high temperature can also cause the gas to
depart from the perfect-gas behavior. Thus the high-temperature gasdynamics
needs to be taken Into account. The viscous-lnviscid interaction and the
physical-chemical phenomena are more or less dependent on each other – a fact
that makes the theoretical investigation much more difficult. In the present
study, we assume that the perfect-gas relation holds and that the vorticity
interaction Is negligible (i.e., only the pressure Interaction is considered).

For hypersonic flow with a shock wave, a similar solution Is found to be
consistent throughout the viscous and invlscid layers along a very slender

three-fourths-power body of revolution. The strong pressure interaction prob-
lem can then be treated by the method of similarity. In the present study,
numerical calculations are performed in the viscous region with the edge pres-
sure distribution known from the inviscid similar solutions. The compressible
laminar boundary-layer equations are transformed into a set of ordinary differ-
ential equations, and a two-point boundary value problem results. The Runge-
Kutta method is then used with a modified Newton’s method to solve the
resulting simultaneous nonlinear equations for the corresponding boundary

Although the thermodynamic and fluid dynamic phenomena associated with
flight at hypersonic speeds have been the subject of intensive research for the
past decades, only a few studies of the effects of body transverse curvature
and mass bleeding have been carried out (refs. 2 to 7). The purpose of this
study is to contribute to the investigation of the effects of wall temperature,
mass bleeding, and body transverse curvature on the strong pressure interaction
region during hypersonic flights. The induced pressure, displacement thick-
ness, skin friction, and heat transfer are then analyzed based on these


For axial ly symmetric flow with body forces neglected, the compress-
ible laminar boundary-layer equations can be written as follows:




The problem of the body of revolution of minimal resistance

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Newton’s problem of the body of minimal aerodynamic resistance is traditionally stated in the class of {\it convex} axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but {\it generally nonconvex} bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton’s problem and for the problem in the intermediate class of axisymmetric bodies satisfying the {\it single impact} assumption \cite{CL1}. In particular, the minimal resistance in our class is smaller than in Newton’s problem; the ratio goes to 1/2 as (length)/(width of the body) $\to 0$, and to 1/4 as (length)/(width) $\to +\infty$.
Mathematics subject classifications: 49K30, 49Q10

Key words and phrases: Newton’s problem, bodies of minimal re-
sistance, calculus of variations, billiards

Running title: Problem of minimal resistance

* Aberystwyth University, Aberystwyth SY23 3BZ, UK, on leave from Department of

Mathematics, University of Aveiro, Aveiro 3810-193, Portugal

^Department of Mathematics, Aveiro University, Aveiro 3810, Portugal

■’■This work was supported by Centre for Research on Optimization and Control

(CEOC) from the “Fundacao para a Ciencia e a Tecnologia” (FCT), cofinanced by

the European Community Fund FEDER/POCTI, and by the FCT research project

PTDC/M AT /72840 /2006.


Figure 1: The Newton solution for h = 2.

1 Introduction

In 1687, I. Newton in his Principia [I] considered a problem of minimal
resistance for a body moving in a homogeneous rarefied medium. In slightly
modified terms, the problem can be expressed as follows.

A convex body is placed in a parallel flow of point particles. The density
of the flow is constant, and velocities of all particles are identical. Each
particle incident on the body makes an elastic reflection from its boundary
and then moves freely again. The flow is very rare, so that the particles do not
interact with each other. Each incident particle transmits some momentum
to the body; thus, there is created a force of pressure on the body; it is called
aerodynamic resistance force, or just resistance.

Newton described (without proof) the body of minimal resistance in the
class of convex and axially symmetric bodies of fixed length and maximal
width, where the symmetry axis is parallel to the flow velocity. That is,
any body from the class is inscribed in a right circular cylinder with fixed
height and radius. The rigorous proof of the fact that the body described
by Newton is indeed the minimizer was given two centuries later. From now
on, we suppose that the radius of the cylinder equals 1 and the height equals
h, with h being a fixed positive number. The cylinder axis is vertical, and
the flow falls vertically downwards. The body of least resistance for h = 2 is
shown on fig.ED

Since the early 1990s, there have been obtained new interesting results
related to the problem of minimal resistance in various classes of admissible
bodies [2]-|12|. In particular, there has been considered the wider class of
convex (generally non-symmetric) bodies inscribed in a given cylinder [2]-
[i].[7|.[T0]. It was shown that the solution in this class exists and does not
coincide with the Newton one. The problem is not completely solved till


Figure 2: The non-symmetric solution for h = 1.5.

now. The numerical solution for h = 1.5 is shown on fig. 20

By removing both assumptions of symmetry and convexity, one gets the
(even wider) class of bodies inscribed in a given cylinder. More precisely, a
generic body from the class is a connected set with piecewise smooth bound-
ary which is contained in the cylinder, contains an orthogonal cross section of
the cylinder, and satisfies a regularity condition to be specified below. Notice
that there may occur multiple reflections of particles from the surface of a
non-convex body, while reflections from convex bodies are always single. The
problem of minimal resistance in this class was solved in [HI [12]. In contrast
to the class of convex & axisymmetric bodies and the class of convex bod-
ies, the infimum of resistance here equals zero, and we believe the infimum
cannot be attained.

In addition to the classes of admissible bodies discussed above:

(i) convex & axisymmetric (the classical Newton problem);

(ii) convex but generally non-symmetric;

(iii) generally nonconvex and non-symmetric,
there remains a class that has not been studied as yet:

(iv) axisymmetric but generally nonconvex bodies.

The aim of this paper is to fill this gap: we shall solve the minimal resistance
problem for the fourth class.

Note that in the paper [8] there was considered the intermediate class of

(v) axially symmetric nonconvex bodies, under the additional so-called
“single impact assumption”.

This geometric assumption on the body’s shape means that each particle
hits the body at most once; multiple reflections are not allowed. On the

1 This figure is reproduced with kind permission of E. Oudet.


contrary, multiple reflections are allowed in our setting; we only assume that
the body’s boundary is piecewise smooth and satisfies the regularity condition
stated below.

The class (v) is intermediate between the classes (i) and (iv); it contains
the former one and is contained in the latter one. We shall determine the
minimal resistance and the minimizing sequence of bodies for the class (iv)
(which will be referred to as nonconvex case), and compare them with the
corresponding results for the class (i) (Newton case) and for the class (v)
(single impact case)H

Consider a compact connected set B C M 3 and choose an orthogonal
reference system Oxyz in such a way that the axis Oz is parallel to the flow
direction; that is, the particles move vertically downwards with the velocity
(0,0,-1). Suppose that a flow particle (or, equivalently, a billiard particle
in M 3 \ B) with coordinates x(t) = x, y(t) = y, z(t) = —t makes a finite
number of reflections at regular points of the boundary OB and moves freely
afterwards. Denote by v B (x,y) the final velocity. If there are no reflections,
put v B (x,y) = (0,0, -1).

Thus, one gets the function v B = (v B ,v B ,v B ) taking values in S 2 and
defined on a subset of M 2 . We impose the regularity condition requiring that
v B is defined on a full measure subset o/M 2 . All convex sets B satisfy this
condition; examples of non-convex sets violating it are given on figure 3.
Both sets are of the form B = Gx[0, l]cK^ z x i?*, with G being shown
on the figure. On fig. 3a, a part of the boundary is an arc of parabola with
the focus F and with the vertical axis. Incident particles, after making a
reflection from the arc, get into the singular point F of the boundary. On
fig. 3b, one part of the boundary belongs to an ellipse with foci F\ and F 2 ,
and another part, AE>, belongs to a parabola with the focus F 1 and with the
vertical axis. After reflecting from AB, particles of the flow get trapped in the
ellipse, making infinite number of reflections and approaching the line F1F2
as time goes to +00. In both cases, v$ is not defined on the corresponding
positive-measure subsets of K 2 .

Each particle interacting with the body B transmits to it the momentum
equal to the particle mass times ((0,0,-1) — u B (x,y)). Summing up over
all momenta transmitted per unit time, one obtains that the resistance of B
equals —pR(B), where

and p is the flow density. One is usually interested in minimizing the third

2 Note that Newton himself did not state explicitly the assumption of convexity; in this
sense, the cases (iv) and (v) can be regarded as “relaxed versions'” of the Newton problem.


(a) (b)

Figure 3: (a) After reflecting from the arc of parabola, the particles get into
the singular point F. (b) After reflecting from the arc of parabola AB, the
particles get trapped in the ellipse.

component of

R Z (B)= [[ (l + v* B (x,y))dxdy. (1)

If B is convex then the upper part of the boundary dB is the graph of
a concave function w(x,y). Besides, there is at most one reflection from
the boundary, and the velocity of the reflected particle equals u B (x,y) =
(1 + \Vw\ 2 y l (—2w x , —2w y , 1 — \Vw\ 2 ). Therefore, the formula JTJ) takes the

the integral being taken over the domain of w.

Further, if B is a convex axially symmetric body then (in a suitable
reference system) the function w is radial: w(x,y) = f(^/x 2 + y 2 ), therefore
one has


the integral being taken over the domain of /.

Thus, in the cases (i), (ii), and (v) the problem of minimal resistance
reads as follows:

minimize J – – dr (4)

3 Note that in the axisymmetric cases (i), (iv), and (v), the first and second components
of R(5) are zeros, due to radial symmetry of the functions v% and v v B : K X (B) = =


over all concave monotone non-increasing functions / : [0, 1] — ► [0, h];

(ii) minimize

n 1 + \Vw(x,y)\ 2


dx dy

over all concave functions w : ft — > [0, h], where ft = {x 2 + y 2 < 1} is the
unit circle;

(v) minimize the functional pj over the set Ch of functions / : [0, 1] — > [0, h]
satisfying the single impact condition (see [8], formulas (3) and (1)).

In the nonconvex cases (iii) and (iv) the functional to be minimized ([TJ)
cannot be written down explicitly in terms of the body’s shape. Still, in the
radial case (iv) it can be simplified in the following way.

Let B be a compact connected set inscribed in the cylinder x 2 + y 2 < 1,
< z < h and possessing rotational symmetry with respect to the axis Oz.
This set is uniquely defined by its vertical central cross section G = {(x, z) :
(x, 0, z) G B}. It is convenient to reformulate the problem in terms of the
set G.

Consider the billiard in IR 2 \ G and suppose that a billiard particle ini-
tially moves according to x(t) = x, z{t) = —t, then makes a finite num-
ber of reflections (maybe none) at regular points of dG, and finally moves
freely with the velocity Vq(x) = (vq(x),Vq(x)). The regularity condition
now means that that the so determined function Vq is defined for almost ev-
ery x. One can see that Ug(x,y) = (x/^/x 2 + y 2 )vQ(\J x 2 + y 2 ), v y B [x,y) =
(y/^/x 2 + y 2 )v v G {\/ x 2 + y 2 ), and v z B [x,y) = Vq(\Jx 2 + y 2 ). It follows that

R X (B) = = Ry(B) and R Z (B) = 2tt J Q (l + v z G (x))xdx. Thus, our mini-
mization problem takes the form

and Qh is the class of compact connected sets GcK 2 with piecewise smooth
boundary that are inscribed in the rectangle — 1 < £ < 1, < z are
symmetric with respect to the axis Oz, and satisfy the regularity condition
(see fig. Sj).

The main results are stated in section 2: the minimization problem is
solved and the solution is compared with the Newton solution (case (i)) and
the single-impact solution (case (v)). Details of all proofs are put in section

4 That is, belong to the rectangle and have nonempty intersection with each of its sides.

inf R(G), where



Figure 4: A set G G Q h .

Figure 5: Modified reflection law.

2 Statement of the results

Denote by Ql° nv the class of convex sets from Qh- One can easily see that if
G E Qh then convG G Q c h onv . For G C Q c h onv define the modified law of reflec-
tion as follows. A particle initially moves vertically downwards according to
x(t) = x, z(t) = —t and reflects at a regular point of the boundary dG; at this
point the velocity instantaneously changes to Vg(%) = (vq(x), Vq(x)), where
vg{ x ) is the unit vector tangent to dG such that v G (x) < and x-v G (x) >
(see fig. [5]).

The set G G Qh is bounded above by the graph of a concave even function
z = fdx). For x > 0, one has

m«) (U ‘ a(x)) (6)

The resistance of G under the modified reflection law equals (0, — R(G)),



R(G) = [ (l+v z G (x))xdx. (7)

Taking into account ([6]), one gets

R(G) = I | 1 + ; &^ ! .r<lr: (H)

Jo { VTTMxj 1

the function f G is concave, nonnegative, and monotone non-increasing, with
/(0) = /».

Theorem 1.

inf R(G) = inf R(G). (9)

This theorem follows from the following lemmas Q] and [2] which will be
proved in the next section.

Lemma 1. For any G G Qh one has

R[G) > R(convG).

Lemma 2. Let G G Q™ nv . Then there exists a sequence of sets G n G Qh
such that

lim R(G n ) = R(G).

Indeed, lemma Q] implies that inf Gg g h R(G) > inf^ggcon!, R(G), and lemma
[2] implies that mf G£ g h R(G) < inf Ggg c n« R(G).

Theorem [Hallows one to state the minimization problem ([5]) in an explicit
form. Namely, taking into account (E|) and putting / = h — fa, one rewrites
the right hand side of j9]) as


inf / 1 . J w \xdx, (10)

where Th is the set of convex monotone non-decreasing functions / : [0, 1] — >
[0, h] such that /(0) = 0. The solution of f fTOl l is provided by the following
general theorem.

Consider a positive piecewise continuous function p defined on M. + : =
[0, +oo) and converging to zero as u — > +oo, and consider the problem

inf K[f], where K[f] = [ p(f'(x))xdx. (11)

Denote by p(u), u G M + the greatest convex function that does not exceed
p(u). Put £ = — l/p'(0) and u = inf{u > : p(u) = p(u)}. One always
has £o > 0; if Mo = and there exists p'(0) then £ = — 1/V(0)? an d if «o >
then £ = wo/(p(0) — p(«o))- Denote by u = v(z), z > £ the generalized
inverse of the function z = —l/p'(u), that is, v(z) = sup{w : —l/p'(u) < z}.
By T, denote the primitive of v: T(z) = v(£)<i£, z > £ . Finally, put

n(h)-.= M f ^ h n[f].

Theorem 2. For any h > the solution fh of the problem ( TO]] exists and
is uniquely determined by

h{x) = {°i Tf ^ V~ X -^ (I 2 )

where z = z(h) is a unique solution of the equation

T{z) = zh (13)

and Xq = Xo(h) = £o/z(h). Further, one has f’ h {xo + 0) — Uq. The function
xo(h) is continuous and xo(0) = 1. The minimal resistance equals

K(h) = \ (p{v{z)) + . (14 )

in particular, TZ(0) = p(0)/2.

If, additionally, the function p satisfies the asymptotic relation p{u) =
cu~ a (1 + o(l)) as u — > +oo, c> 0, a > i/ien

^(/i) = ca (^±1) eo/i” a_1 (l + o(l)), h^+oo, (15)

and i

ft^) = £ (^±1 ) fc-«(l + (l)), /i^+oo. (16)

2 V« + 2,

Let us apply the theorem to the three cases under consideration.

1. First consider the non-convex case. The problem ( fTOl ) we are interested
in is a particular case of (fTTi ) with p(u) = p QC (u) := 1 — u/y/1 + u 2 (the
subscript “nc” stands for “non-convex”). The function p QC itself, however, is
convex, hence uq = andp nc = p nc . Further, one has —l/p’ nc (u) = (l+u 2 ) 3//2 ,
therefore v nc (z) = \/ z 2 / 3 — 1 , Q c = 1, and


T nc (z) = -(2z 2/3 – l)z l/3 Vz 2/3 – 1 – – ln(^ 1/3 + v^ 2/3 – 1). (17)

The formulas (fT71) . (fl”3l . and (fT2l) with x = l/z, determine the solution of
(fTOl) . Notice that, as opposed to the Newton case, the solution is given by
the explicit formulas. However, they contain the parameter z to be defined
implicitly from ( fl~3l) .

Further, according to theorem 2, f’ h (xo + 0) = = f’ h (xo — 0), xq = Xq c ,
hence the solution fh is differentiable everywhere in (0, 1). Besides, one has


x ™(h) = — h~ 3 (l + o(l)) as h^+00. (18)

The minimal resistance is calculated according to (fl4l) ; after some algebra
one gets

njth) = – + 3 + 2 ^ /3 ~ 8Z4/3 V 7 ^/ 3 – 1 + At Mz 1/3 + V 7 ^ 3 -!)-
v ; 2 16z 5 / 3 16z 2 V 7

One also gets from theorem [2] that 7£ nc (0) = 0.5 and


Knc(h) = — /T 2 (l + o(l)) as /i-^+oo. (19)

2. The original Newton problem (case (i) in our classification) is also a
particular case of (fTTl) . with = Pn(u) := 2/ (1 + w 2 ). One has w = 1 and

p K (u) = l?-” , “^”f 1 and after some calculation one gets

r V ; [ 2/(l + M 2 ) if M>1 , &

that £(^ = 1 an d the function Tjv(z), z > 1, in a parametric representation,
is T^v = ~ (3w 4 /4 + w 2 – lnw – 7/4), z = (1 + w 2 ) 2 /(4w), u > 1. From
here one obtains the well-known Newton solution: if < x < Xq then
fh( x ) — 0, and if Xo < x < 1 then fh is defined parametrically: fh =
f (3w 4 /4 + u 2 – In u – 7/4), a; = f where x = 4^/(1 + u 2 ) 2 and u*

is determined from the equation (3m 4 /4 + u 2 — lnu* — 7/4) «*/(l + m*) 2 = fo.
The function is not differentiable at x : one has f’ h (x + 0) = 1 and
f h (x – 0) = 0.

One also has Hn(Q) = 1,


^Jv(^) = t^^ 2 (1 + o(1)) as h^+oc. (20)



a;*(/i) = — /T 3 (l + o(l)) as /i^+oo. (21)


3. The minimal problem in the single impact case with h > M* « 0.54

[ p* jf 11 — o

can also be reduced to (fTTil. withp(ix) = Psi(ix) := < 2/(1 + m 2 ) if m >

where p* = 8(ln(8/5) + arctan(l/2) — 7r/4) « 1.186. This fact can be easily
deduced from [8]; for the reader’s convenience we put the details of deriva-
tion in the next sectionjf) From the above formula one can calculate that
u « 1.808 and » 2.52.

The asymptotic formulas here take the form

a%(h) = & • x£(h)(l + o{l)) as h^+oo (22)



Ksi(h) = — /r 2 (l + o(l)) as h^+oo. (23)

Finally, using the results of [8], one can show that 72. si (0) = n/2—2 arctan(l/2)
0.6435. This will also be made in the next section.

Now we are in a position to compare the solutions in the three cases.
One obviously has TZ QC (h) < lZ S i(h) < lZjsr(h). From the above formulas
one sees that TZ QC (0) = 0.5, K N (Q) = 1, and 7^(0) w 0.6435. Besides,
one has lim h ^ +0O (7^ nc (/i)/7^ iV (/i)) = 1/4 and Yim. h _, +00 (K si (h) /K N (h)) = I.
Thus, for “short” bodies, the minimal resistance in the nonconvex case is
two times smaller than in the Newton case, and 22% smaller, as compared
to the single impact case. For “tall” bodies, the minimal resistance in the
nonconvex case is four times smaller as compared th the Newton case, while
the minimal resistance in the Newton case and in the single impact case are
(asymptotically) the same.

In the three cases of interest, the convex hull of the three-dimensional
optimal body of revolution has a flat disk of radius xo(h) at the front part
of its boundary. One always has xq(0) = 1. For “tall” bodies, one has
lim^ +00 (xg c (/i)/<(/i)) = 1/4 and lim^ +00 (x^) /<(/>)) = $ w 2.52;
that is, the disk radius in the non-convex case and in the single impact case
is, respectively, 4 times smaller and 2.52 times larger, as compared to the
Newton case.

Besides, in the nonconvex case, the front part of the surface of the body’s
convex hull is smooth. On the contrary, in the Newton case, the front part
of the body’s surface has singularity at the boundary of the front disk.

5 We would like to stress that the results presented here about the single impact case
can be found in J8j or can be easily deduced from the main results of [8].


3 Proofs of the results

3.1 Proof of lemma d

It suffices to show that

v z G (x) > v z comG (x) for any x G [0, 1]. (24)

Consider two scenarios of motion for a particle that initially moves ver-
tically downwards, x(t) = x and z{t) = —t. First, the particle hits convG
at a point r G <9(conv G) according to the modified reflection law and then
moves with the velocity v CO nvG( x )- Second, it hits G (possibly several times)
according to the law of elastic reflection, and then moves with the velocity
v G (x). Denote by n the outer unit normal to <9(convG) at r ; on fig. [6] there
are shown two possible cases: r G dG and r ^ dG.

It is easy to see that

(v G {x),n) >0, (25)

where (• , •) means the scalar product. Indeed, denote by r(t) = (x(t),z(t))
the particle position at time t. At some instant t\ the particle intersects
<9(convG) and then moves outside convG. The function (r(t),n) is linear
and satisfies (r(t),n) > (r(ti),n) for t > ti, therefore its derivative (vG(x),n)
is positive.

>From (l25l) and the relations (v C ouvg( x ), n ) — 0, v^ onvG (x) < and
n z > 0, n x > one gets (1241) .

3.2 Proof of lemma [2]

Take a family of piecewise affine even functions f £ : [—1, 1] — * [0, h] such that
f’ £ uniformly converges to f’ G as e — >• + . Require also that the functions f £
are concave and monotone decreasing as x > 0, and / £ (0) = h, / £ (1) = /g(1)-


Consider the family of convex sets G £ G G% mv bounded from above by the
graph of f £ and from below, by the segment — 1 < x < 1, z = 0. Taking into
account (jHj), one gets lim e ^ + R(G £ ) = R(G).

Below we shall determine a family of sets G £t $ e Qh such that lim,5_ > o+ R(G £y s)
R(G £ ) and next, using the diagonal method, select a sequence e n — > 0, 5 n — >
such that Hindoo R{G £nt $ n ) = lim^oo R(G £n ) = R(G). This will finish the

Fix e > and denote by —1 = x_ m < X- m+ \ <…<xo = 0<…<
x m = 1 the jump values of the piecewise constant function f’ £ . (One obviously
has x_j = — Xj.) For each z = 1, . . . , m we shall define a non self-intersecting
curve l t,e,s that connects the points (a^-i, f e (xi-i)) and (x^ f e (%i)) and is
contained in the quadrangle Xi-i < x < Xi, f £ (xi) < z < f £ (xi-i) + (f £ (xi-i +
0) + 8) ■ (x — Xi-i). The curve l~ z ‘ e ‘ S is by definition symmetric to l l,£ ‘ S with
respect to the axis Oz. Let now l £ ‘ 5 := U_ m < i < m /*’ e ” 5 and let G £i $ be the
set bounded by the curve / e,<5 , by the two vertical segments < z < / e (l),
x = ±1, and by the horizontal segment — 1 < x < 1, z = 0.

For an interval I C [0, 1], define

£i(G s ) := J(l + v z GE (x))xdx (26)


flj(G e ,*) := J(l + v z G Jx))xdx. (27)

Denote Zv = one obviously has R(G £ ) = YlT=i Rii(G e ) an d R(G £: s) =

^2iLiRh{Ge,s)- Thus, it remains to determine the curve l^ 5 and prove that

lim i R h (G £tS ) = R h (G £ ). (28)

o — >0~ t ~

This will complete the proof of the lemma.
Note that for x G L, i — 1, . . . , m holds

}2 nixi^+o)

G ‘ y/l + + 0))’


Fix e and i and mark the points P = (x;_i, / e (xj_i)), P’ = (xj, f £ (xi)),
Q = f £ (xi)), and 5 = (x;, f E (xi-i) + (f £ (xi-i + 0) + 5) ■ (x { – x;_i));

see fig. [3 Mark also the point Q<5 = + 5, / £ (^i)), which is located on the
segment QP’ at the distance 5 from Q, and the points Pg = (x^i+8, f £ (x^i+
5)) and Sg = (xj-i + 5, / e (xj_i) + (f’ £ (xi_i + 0) + 5) • 5), which have the same
abscissa as and belong to the segments PP’ and PS, respectively. Denote
by I the line that contains Pg and is parallel to PS. Denote by Ug the arc of


Q Qs

Figure 7: Constructing the curve l l ‘ £ ‘ 5 : a detailed view.

the parabola with vertex Q$ and focus at Ps (therefore its axis is the vertical
line QsPs)- This arc is bounded by the point Qs from the left, and by the
point P$ of intersection of the parabola with /, from the right. Denote by
xf the abscissa of P$ and denote by P’ 5 the point that lies in the line PP’
and has the same abscissa xf. Denote by n$ the arc of the parabola with
the same focus Ps, the axis /, and the vertex situated on / to the left from
Ps. The arc ns is bounded by the vertex from the left, and by the point
S’ s of intersection of the parabola with the line QsPs, from the right. There
is an arbitrariness in the choice of the parabola; let us choose it in such a
way that the arc ns is situated below the line PS. Finally, denote by Js the
perpendicular dropped from the left endpoint of ns to QP’, and denote by
Q’ s the base of this perpendicular.

If xf > Xi, the curve l l ‘ £ ‘ s is the union (listed in the consecutive order) of
the segments PS$ and SsS’ s , the arc ns, the segments Js and Q’sQs, and the
part of 11,5 located to the left of the line P’S.

If xf < x i7 the definition of l l,£ ‘ 5 is more complicated. Define the homo-
thety with the center at P’ that sends P to P’ 5 , and define the curve l l,£,s by
the following conditions: (i) the intersection of 1 % ‘ £,S with the strip region
Xi-i < x < xf is the union of PSs, SsS’ s , ns, Js, Q’sQs, II5, and the interval
PsP’s’, (ii) under the homothety, the curve l l ‘ £ ‘ s moves into itself. The curve
ji,e,s j s un jq Ue iy defined by these conditions; it does not have self-intersections



Figure 8: The curve l % ‘ £ ‘ 5 ,


and connects the points P and P’ . However, it is not piecewise smooth, since
it has infinitely many singular points near P’ . In order to improve the sit-
uation, define the piecewise smooth curve l h£ ‘ s in the following way: in the
strip Xi-i < x < Xi — 5, it coincides with l l ‘ £ ‘ S , the intersection of l l,e,s with
the strip X{ — 5 < x < X{ is the horizontal interval X{ — 5 < x < Xi, z = f s (xi),
and the intersection of l l ‘ £,s with the vertical line x = Xi — 5 is a point or a
segment (or maybe the union of a point and a segment) chosen in such a way
that the resulting curve l l,£,s is continuous.

The particles of the flow falling on the arc 11,5 make a reflection from it,
pass through the focus P$, then make another reflection from the arc its,
and finally move freely, the velocity being parallel to I. Choose 5 < |/e(0 + )|
and 5 < min 1 < i < m _ 1 (/ f !(a;j_i + 0) — f’ £ (xi + 0)), then the particles after the
second reflection will never intersect the other curves l j ‘ £ > 5 , j y£ i. Thus, for
the corresponding values of x, the vertical component of the velocity of the
reflected particle is

If x\ > Xi, the formula ([301) is valid for x G [x^i + 5, Xj\. If x\ < x i? it is valid
for the values x £ + 5, xf). Note, however, that (1301 is also valid for
values of x that belong to the iterated images of x £ + 5, xj] under the

f& i _ 1 + 0) + S


homothety, but do not belong to [xi — 6, Xj\ . Summarizing, (l30l) is true for
x G x^, except for a set of values of measure 0(5). Thus, taking into

account ((26]), {23D, (l29l). and pOl). the convergence is proved. Q.E.D.

3.3 Proof of theorem [2]

Let us first state (without proof) the following lemma.

Lemma 3. Let A > and let the function f G Tk satisfy the condition

I\. f(l) = h, and for almost all x G [0, 1] the value u = f'(x) is a
solution of the problem

xp(u) + Xu —>■ min, u G R+. (31)

Then the function f is a solution of the problem fTI\) and any other solution
satisfies the condition I a with the same value of X.

This simple lemma is a direct consequence of the Pontryagin maximum
principle. Its proof can be found, for example, in [13] or in [T4] .

Now we shall find the function fh satisfying the condition Ia for some
positive A. Let x G [0, 1] be the value for which Ia is fulfilled. Then the value
u = f’ h (x) is also a minimizer for the function xp{u) + Am, and p{u) = p{u).
This implies that (if the function fh really exists then)

nfh]= C P{f h {x))xdx. (32)

Besides, if u > and p is differentiable at u then one has 4-(xp(u) + Xu) = 0,

A v\u)

If u > and p is not differentiable at u, then it has left and right derivatives
at this point and

lxl , .

— T — -“=777—^- (34)

p'(u-0) ~ X ~ p'(u + 0)
If, finally, u = then one has

5 £ ~W) = io – (35)

Put z — 1/A and x = £, /z and rewrite (l33l and (l3H) in terms of the
generalized inverse function: v(zx — 0) <u < v(zx); thus the equality

u = v(zx), (36)


is valid for almost all values x > x . Taking into account (l35l) . substituting
u = f’ h (x), and integrating both parts of (|36l) with respect to x, one comes
to (fT2l . In particular, f’ h (xo + 0) = -u(£o + 0) = uq. Using that //i(l) = h,
one gets (1T31) .

The function T(z)/z is continuous and monotone increasing; it is defined
on [£o, +00) and takes the values from to +00. Therefore the equation
(TT3l) uniquely defines z as a continuous monotone increasing function of h;
in particular, z(0) = £0 and xq(0) = (,o/z(0) = 1. The relations f fT2l and
(1T31) define the function fh solving the minimization problem (fTTT) . From the
construction one can see that this function is uniquely defined.

Recall that 71(h) = lZ[fh\- Integrating by parts the right hand side of
(l32l) . one gets

m) = ^f 1 ^ – jf -r^ f ‘ h{x))df ‘ h{x) –

Taking into account that f’ h (l) = v(z) and xp'(f’ h (x)) = —A = —1/z, one

n(h) = p^MI + L J\ dfhix)t

and integrating by parts once again, one gets (fT4l ). Substituting in ( fT4l ) h =
and using that z(0) = £ , ^(£0) — M o, Y(£ ) — 0, one obtains 7Z(0) = (p(u ) +
M o/£o)/2, and using that p(0) — (.q 1 ^ = p(u ), one obtains 7^.(0) = p(0)/2.

Taking into account the asymptotic of p (which is the same as the asymp-
totics of p: p{u) = cu~ a {l + o(l)), u — * +00), and the asymptotic of
p’\ p'{u) = —cau~ a ~ 1 (l + o(l)), u — > +00, one comes to the formulas

= (Ca)^ ^ (1 + o(l)), f -> +00,

> «. /a + 1 \ , , 1 c+2

1(2) = (caj^+i (1 + o(l)), z — > +00,

\a + 2/

and i

^ = ^/a^2\ ^+i (1 + o(1)); ^ +00 .
ca \a+l J

Substituting them into ( TT4T ) and using the relation xo = £o/z, after a simple
algebra one obtains (fT6l) and (fT5l ). The theorem is proved.

Summarizing, the three-dimensional bodies of revolution minimizing the
resistance are constructed as follows. First, we find the function f£ c mini-
mizing the functional (fTUl) and define the convex set — 1 < x < 1, < z <
h — fh C (\x\). Next, the upper part of its boundary (which is the graph of the


Figure 9: Profiles of optimal solutions in the single impact (a), Newton (b),
and nonconvex (c) cases, for h = 0.8. In the nonconvex case, the profile is
actually a zigzag curve with very small zigzags, as shown on the next figure.

function z = h — /^ c (|x|)) is approximated by a broken line and then sub-
stituted with a curve with rather complicated behavior, according to lemma
[2l The set bounded from above by this curve is “almost convex”: it can be
obtained from a convex set by making small hollows on its boundary. By
rotating it around the axis Oz, one obtains the body of revolution B having
nearly minimal resistance K Z (B).

The vertical central cross sections of optimal bodies in the Newton, single
impact, and nonconvex cases, for h = 0.8, are presented on figure 9.

3.4 Derivation of the asymptotic relations in the single
impact case

For h small (namely, h < M* rs 0.54), a solution in the single impact case
can be described as follows. There are marked several values — 1 < X- 2n+ i <
%-2n+2 < . . . < X2n-2 < %in-\ < 1, n > 2 related to the singular points of
the solution. As h — > + , n — n{h) goes to infinity. One has = — Xk and
x 2 i = (#2i-i + ^2i+i)/2; thus x Q = 0. Besides, one has max fc (i fc — Xk~i) =
X\ = Ah/3. The vertical central cross section of the solution G = G^ Clj 2
is bounded from above by the graph of a continuous non-negative piecewise
smooth even function / = and from below, by the segment — 1 < x < 1,
z = 0. This function has singularities at the points Xf., and the values of
the function at the points Xn-\ coincide: f{x2i-\) = h. On each interval
[x 2 i-i, x 2 i], the graph of / is the arc of parabola with vertical axis and with
the focus at (x2i+i,h). Similarly, on [x 2 i, x 2 i+i\ the graph of / is the arc of
parabola with vertical axis and with the focus at (a^i-ij h). The first parabola
contains the focus of the second one, and vice versa. From this description one
can see that on [a^i-i, x 2 i\, the function equals f(x) = 2?~-^~ ^ +Vu and
on [x 2i , x 2i+1 ), f(x) = 2(^1 +Vu where ^ = h-(x 2i+1 -x 2i _ 1 )/2. On


Figure 10: Detailed view of the zigzag curve.

the intervals [—1, x_ 2 n+i] and [x 2n -i, 1] the graph of the function represents
the so-called “Euler part” of the solution (see |8j).

Note that the solution is not unique. The values x\ and x 2n -i are uniquely
determined, but there is arbitrariness in choice of the intermediate values
X3, . . . , X2n-3 and also in the number n of the independent parameters.

After some calculation, one obtains the value of 1 + Vq for the figure G:


if x e [x 2i -i, x 2i \, 1 + Vq(x) = —

1 +


if X G [x 2 i, X 2 i+l\, 1 + Vq(x)

1 +

X2i+l~ %2i-l

Let us now calculate the integral f x ^ + *(l + Vq(x)) xdx, 1 < i < n — 1.
Since the function 1 +Vq(x), x G [x2i-i, x^i+i] is symmetric with respect to
X — X 2 i^ the integral equals 2x 2i f* 2l+1 (l + Vq(x)) dx. Changing the variable

t = (x — x 2 i-i) / (x 2 i+i — x 2 i-i) and taking into account that 2x 2 i = x 2 i-i +
x 2i+ i, one comes to the integral 2x 2i (x 2i+ i — x 2i _i) Jy 2 2/ (1 + t 2 ) dt = (x\ i+l —
Ai-i)^/ 2 ~ 2arctan(l/2)). Therefore

(1 +v z G {x))xdx = (x2„_! – xI)(tt/2 — 2arctan(l/2)).

J X\


Taking into account that Xi — 4h/3 — > and x 2n (h)-i — 1 as h — > + , one
finally gets

R(Gf) = [ (1 + v z G 4x)) x dx = 7r/2-2arctan(l/2) + o(l), /i -> + ,

that is, 7e 8 i(0) = tt/2 – 2 arctan(l/2) « 0.6435.

If /t > M* 3 the function / = has three singular points: Xi = Xi(h), 0,
and — xi. On the interval [—x±, xi], the graph of / is the union of two
parabolic circs, cts described above with i = 0. On the intervals [—1, —xi]
and [xi, 1], the graph is the “Euler part” of the solution; on both intervals,
/ is a concave monotone function, with /(±1) = and f(±x%) = h. The
part of resistance of G — Gf related to [0, x\] can be calculated:


(l + VQ(x))xdx = xip*,

where p* = 8(ln(8/5) + arctan(l/2) – vr/4) w 1.186. That is, the convex
hull of G represents the solution of the problem ([Til with p{u) = p a i(u) =
j p* if u =

\ 2/(1 + u 2 ) if u > .


[1] I. Newton, Philosophiae naturalis principia mathematica, 1686.

[2] G. Buttazzo, B. Kawohl, On Newton’s problem of minimal resistance.
Math. Intell. 15, No.4, 7-12 (1993).

[3] G. Buttazzo, V. Ferone, B. Kawohl, Minimum problems over sets
of concave functions and related questions. Math. Nachr. 173, 71-89

[4] F. Brock, V. Ferone, B. Kawohl, A symmetry problem in the calculus
of variations. Calc. Var. 4, 593-599 (1996).

[5] G. Buttazzo, P. Guasoni, Shape optimization problems over classes of
convex domains. J. Convex Anal. 4, No. 2, 343-351 (1997).

[6] T. Lachand-Robert, M.A. Peletier, Newton’s problem of the body of
minimal resistance in the class of convex developable functions. Math.
Nachr. 226, 153-176 (2001).


[7] T. Lachand-Robert, M. A. Peletier. An example of non-convex mini-
mization and an application to Newton’s problem of the body of least
resistance. Ann. Inst. H. Poincare, Anal. Non Lin. 18, 179-198 (2001).

[8] M. Comte, T. Lachand-Robert, Newton’s problem of the body of min-
imal resistance under a single-impact assumption. Calc. Var. 12, 173-
211 (2001).

[9] M. Comte, T. Lachand-Robert. Existence of minimizers for Newton’s
problem of the body of minimal resistance under a single-impact as-
sumption. J. Anal. Math. 83, 313-335 (2001).

[10] T. Lachand-Robert and E. Oudet. Minimizing within convex bodies
using a convex hull method. SIAM J. Optim. 16, 368-379 (2006).

[11] A. Yu. Plakhov. Newton’s problem of a body of minimal aerodynamic
resistance. Dokl. Akad. Nauk 390, N°3, 314-317 (2003).

[12] A. Yu. Plakhov. Newton’s problem of the body of minimal resistance
with a bounded number of collisions. Russ. Math. Surv. 58 N°l, 191-
192 (2003).

[13] V. M. Tikhomirov. Newton’s aerodynamical problem. Kvant, no. 5, 11-
18 (1982).

[14] Alexander Plakhov and Delfim Torres. Newton’s aerodynamic problem
in media of chaotically moving particles. Sbornik: Math. 196, 885-933


The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics

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