331

Vol. 18, 1967

Boundary Layer Growth on a Magnetised Circular Cylinder By I(. E. BARRETT, Dpt. of Appl. Mathematics, The University, Liverpool, Great Britain 1. I n t r o d u c t i o n The problem of the growth of the boundary layer on a circular cylinder started impulsively in a non-conducting fluid in a direction perpendicular to its generators was first considered b y BLASlUS (1908), and his work was later extended by GOLDSTEIN and ROSE,HEAD (1936). At t h e start of the motion the velocity field consists of a vortex sheet attached to the body surface and the irrotational flow over the cylinder. As time proceeds the vortex sheet diffuses and is convected to form a boundary layer of width proportional to I/vt, where v is the coefficient of kinematic viscosity and t is the time. Both the authors mentioned, assumed that the Reynolds number R = U o a/v, where U 0 is the free stream velocity and a is the radius of the cylinder, was large and were thus justified in using the unsteady boundary layer equations. As there is an adverse pressure gradient on the rear half of the cylinder the forward stream eventually leaves the surface at a time tr, say, the boundary layer thickening rapidly; the boundary layer approximation breaks down at times substantially greater than tr. The point at which reverse flow occurs will be assumed here to be identical with the point at which the skin friction vanishes, and in such a case the flow will be said to have separated. The extension of this work to the magnetohydrodynamic flows is especially interesting because of the unusual and controversial nature of the steady state flows when large magnetic fields are present. Two factors complicate the problem of the impulsive motion of a body in a conducting fluid. Firstly, there is an additional diffusion parameter, the magnetic diffusivity 2, so that a second boundary of width proportional to l/2-t will be formed. Secondly, an Alfvfin wave is generated at the surface by the impulsive start. If/2 is the magnitude of the normal magnetic field at the surface initially, then the Alfv~n wave will have propagated to a distance of 0(~ L t) b y time ~. Boundary layer equations m a y be utilised only if both the Reynolds number R, and the magnetic Reynolds number R~ = U o a/~,, are large and the Alfv~n wave is confined within the boundary layer region, i.e. 0 ( ~ L t )


i.e. fl, U o t / a m i n ( R , R , ~ ) < O ( 1 ) ,

(1)

where fl, = #/~/22/U~ is the local value of the inverse square of the Alfv4n number. Condition (1) is always satisfied for small enough times unless diffusion is completely absent, and is also satisfied if L is zero over the entire cylinder surface. Thus, we choose to consider the problem of the line dipole magnetised cylinder initially at rest in a uniform magnetic field. The strengths and directions of the dipole and uniform

332

K. E, BARRET'I~

ZAMP

fields can then be chosen to enable the normal field L to satisfyfi, rain (R, Rm) < 0(1) over the entire surface, at times such t h a t U o t/a is 0(1).

2. Equations and Boundary Conditions for Small N o r m a l Fields Consider a perfectly conducting circular cylinder, containing a line magnetic dipole along its axis, at rest in an infinite region of homogeneous, viscous conducting fluid of constant properties to which a uniform magnetic field H 0 is applied. Suppose t h a t at time t ~ 0 the fluid at infinity is started impulsively in the positive x direction with velocity U = U 0 i and t h a t this is maintained at all subsequent times. The equations governing the motion are the Navier-Stokes equations modified in the usual w a y b y the Lorentz force term. These are v, + (v 9 V) v -= ~o-I grad p + v V2v + #

~--1 Curl

h /k h ,

(3)

div v = div h = 0 , j=

Curlh=a(E+#v

(2)

A h),

Curl E = - # h t ,

(4) (5)

where v, p, h and E are the velocity, pressure, magnetic and electric fields and ~, v, # and a are the density, viscosity, magnetic permeability and conductivity

respectively. All four equations are satisfied in the fluid and the last three, with v = 0, in the cylinder. The third and fourth can be combined to eliminate the electric field to give h t - Curl (v A h) = ) . V 2 h , (6) where 2 = (a #)-1 is the magnetic diffusivity. At large distances v and h tend to their uniform values and at the body, v = 0 and the normal magnetic field and the tangential electric field are continuous. As the cylinder is a perfect conductor (5) implies t h a t the magnetic field inside the cylinder is unchanging and (4) implies t h a t the tangential component of the current is zero at the surface. The initial velocity field consists of the potential flow over the cylinder with a vortex sheet attached to the surface and the magnetic field is unchanged b y the impulsive start. In the earliest stages of the flow development diffusion (if present) is the dominant process. Suppose that, when referred to cylindrical polar coordinates (r, 0, z) with origin at the centre of the cylinder, the velocity and magnetic fields have the form

v=(w,~,o),

h=(Z,h, 0).

In addition we will suppose t h a t the dipole field is chosen so t h a t the condition fls rain (R, R,~) < 0(1) , is satisfied. T h a t is, the Alfv6n wave is confined within the b o u n d a r y layer region for times t, such t h a t U o t/a is 0(1).

Vol. 18, 1967

333

Boundary Layer Growth on a Magnetised Circular Cylinder

To describe the boundary layer region we introduce th, e transformation u = Uo u',

w = Uo -f-1/~ w ',

h = H oh',

1 - -R-~/~ H ol',

(7) r=a+-R-lJ2az

',

O=x',

t=a/Uo(,

p=~U~p',

where )~ is the minimum of R and R , . We next write Equations (2), (3) and (6) in terms of these variables, assuming no variation with respect to z, and take the limit h) + oo. This process leads to ut+uu~+wu~=

UUx+

(8)

R-~-Ru, z+fl(hh~+lh~-HH~),

h~ = (u I - w ~)~ + R a ~ R a = ,

(9)

(10)

~~~ .

Equations (8)-(10) are to be solved subject to the boundary conditions that on z=O,

u~w=O,

h~=0,

l=L(x),

for

t~> 0

and as z-+oo,

u--->U(x),

h--~H(x),

u = U(x) ,

h = H(x) ,

for

t~> 0 .

At t= O ,

for

z>0.

In these expressions U=2Sinx,

H=Sin(x+al)+MoHo

1Sin(x+c%),

and L = {-- Cos (x + ~ ) + M 0 H o I Cos (x + ~ ) } R ~ . ~ and ~2 give the directions of the imposed uniform and dipole magnetic fields and M 0 is a measure of the strength of the dipole field. The normal field L(x) is actually of 0(1) and not 0(R ~/~) because of the choice of the dipole field. The solutions of Equations (8)-(10) depend on the ratio of the two Reynolds numbers R and R,,, on the magnitude of the normal magnetic field at the cylinder surface L, and on the freestream Alfv6n number/~-~I~.

3. Zero Normal Field with General Diffusivities Suppose that R m R -1 = s 2, R = R ~ R m and L = 0. The disturbances to the potential flow are then confined within the growing viscous boundary layer. The Blasius variable ~ = z/2 ]/t is an appropriate one to use to describe the boundary layer region at small times. In terms of ~, Equations (8)-(10) become u~+

2r]uv-4tut=4t

UUx+

2~i~wu~--

-~

(

UUx

1

hhx+2t~/~ Z h ~ - ~ / t x

h~/~/+ e~ (2 ~] h v -- 4 t ht) = 2 e 2 t ~12 (w h -- u l)v ,

1 1 u~+~/2-w~ -h~+~l~=O.

)},

(n)

(12)

(13)

334

K. J~7,.BARRETT

ZAMP

The non-conducting problem possesses a solution in power series in t. Here power series in t ~/~ are required except for the special case in which the normal field vanishes everywhere on the surface. To avoid tedious repition, the general case will be treated and it will be assumed t h a t the expansions take the forms oo

oo

u = ~ u ( ~ } ( x , ~) t %

w = 2 r

~) t %

n=0

(14)

n~O

~o

oo

h = Z h ( n ) ( x , ~ ) t n/2 ,

[ - - L(x) -- 2 tll2Zl(n)(x , ~))t ll~ .

(1.5)

The first few approximations yield, anticipating the results u (~) = h (~ = 0 F0(U(0) ) = 0 ,

Go(~ (0)) : 0 ,

G l ( h (1)) = - - 2 2 L qd,~0) ,

F2(u(2)) = 4 (u (~ u~ ) - w (~ u~~ - U U x -

]

X fl L h~l),

G2(h(~)) = 4 e 2 (u (~ l(o)- w(o)h(o))~ ,

]

}(16)

--~ (h(~

h(1) h(xl)@ h(2) h(x2)-Z(~

2 Lh~))} '

]

where F~ and G,, are the differential operators defined b y F~(f) = f ~ + 2 ~ f v -- 2 n f ,

(17)

and g,(g) = g,~ ,1 + e~ (2 r] g,1 - 2 n g) . Also

(19) 0

0

The solutions of Equations (16) are most conveniently expressed as sums of products of functions of x and of ~. W h e n L = 0, i.e. when % = ~ and M 0 = H o

,~(o) = vf~o), ]~(2) ~-

~ o~ = H gO),

G2 (U H x ~21)

@

Ux H

l#~ = o ,

~22)) ,

f(42)'~, u (4) = U {(U U x ) , f (41) 4- ~-7"2 --*Jn

~(~)= C; G f ~ ~) ,

u(3) = h(3) = O ,

(21)

Vol. 18, 1967

335

Boundary Layer Growth on a Magnetised Circular Cylinder

where

Fo(f(~~

~(f~2:))=

= Go(g~~ = 0 ,

4 ( f ~ o ) , f(o, f ~ _ 1),

G2(g~ '))

= - 4 (g~~176

F4(f~ 4:)) = 4 (f~o)f~21)__frlorl)f(21))+,

F4(f~42)): 4

G 2\b~ a(21)'l / ~ 4(g(~176

~(2~) =

-4

'

(f~ojf,(21)__f(o)/~l))

:,

(22)

(
F4(f~~')) = - - 4

(2 g!Ojg?2) _

g(O)g(~22) +

g(O)g~2:)),

The b o u n d a r y conditions are f('*)(O) =f~(")(O) = O,

g(n)(O) = g~)(O) = O,

all n ,

(23)

and

f~o), g~O)-+ l ,

/(r]i) , g(2,i)

i=1,2,3

,

1(4i) i=1,2,3,4,5-+0 ~#

as ~ --> oo. The solution of Equations (22) proceeds iteratively, solutions being fairly easily expressed in terms of the c o m p l e m e n t a r y functions of the equation s, G,(g) = O. One of the c o m p l e m e n t a r y functions of G,~(g) = 0 is p,,~=, a polynomial of degree n, and the other is q,,,, a combination of an exponential and an error function. These solutions are normalised here so t h a t

(P,,~)v=P.-:,,,

P0~,=l,

n>

1,

(q~=)~=q.-:~,,

q0,,=erfeq,

aIln.

Two independent solutions of F.(f) = 0 are/5.1 and q ~ ' these will be denoted b y and q~. Solutions of (22) satisfying (23) are 1 , ~:72

f(o) = ql

f(21) = 4

g(O)=

;b:

p.

]

(q~ - - q0 q2) d?] q- ~

§

1

P l - - -6 -ygll2 go

§

§ 22 /

g(21)

g(~2) _

1 -s ~

1

e2

fT_-fi

qa

qa

6~-1i~ -

+ T--7

%~'

6

--

~

2 x :/~

6 ~1,~ e 3

: q'- d

6 xli~ ea-) - - (1 - - e) p s , , ,

P~

'

(24)

336

K. t;. B ~ r

fg43)

16e 2 {

2e a (3-- e2)

e3 + (1-- e2) 2 q2~' 8 (1ez-q~ 16s ~ {

e2 (1+ 2e) (1-- e) 2

(24)

1-- 3~ a (1 + 2 e ) ( 1 +e)2 ( ~ 4 (1-- ~2) q2 + ~ ~ q-eg8

ibo) 16

(1 + 2 r z (1 -- e) z (q4 --/~4)} ,

2e a

ee ( 1 - e) e

+ 2(1-~)

q2~,

4(1-~=) q 2 - ( 1 - e )

+ (1-- e) ( 1 + e-- e ~ - 3e a ~ (1 + ~)2

f~45>

ZAMP

( f:72)16 ~2 { (I + e~) 83

2e ~)

_

q~

P~ -- -8-q~ 16

} (q4 - P4) ,

~a (I - e) 2

e2 2 (~ - ~ )

P~ 2

_

qo

(1 -- e) (1 + ~2)

~6

(~ + ~)

} (q~ -

p~)

"

The functions fro) and f(42/have been explicitly calculated by H. WUNI)T (1955) in connection with the flow over a yawed cylinder. At large values of ~7 1

(243

u ~ u + ~ # Hx (gx Zt - U H~) t~ + O(t~),

and

h ~H+

(
UZL)t + 0(t~).

At large distances from the surface magnetic and velocity fields are generated if the uniform magnetic and the uniform velocity field are non-parallel. The velocity and magnetic fields have the forms u = U f(~~ + U Uxf~2~l t + O(t2) + O(fl t 2 ) + . . . ,

h = H + O(e 2 t) + .'' ,

(25)

in the boundary layer region. The first approximations describe the initial velocity and magnetic fields and show how the vortex sheet diffuses. The magnetic field is perturbed by a term of O(t). This perturbing term clearly shows that the growing viscous boundary layer contains a second boundary layer of thickness proportional to ] / ~ . The perturbation to the magnetic field affects the velocity series via a term of 0(t~). This term again shows t h a t a secondary boundary layer is formed. The series for the skin friction has the form u. = (~t) -1/2 U [1 + (1 + 3-~7) U~t + {0.05975 (U U~)~- 0.27962 U:} t 2

(26) + 0(ta)] +/3(4 ~ t) -1/2 [{f,(e) U H~ -- f~(e) U, H H , + fa(e) U H 2} t 2 + 0(fl t~)], where fl(e)

4 e2 (2 + 4 e + e 2) 3 (1 + e)'

S 3 '

f~(e)

8 e (2 + e) 3 (1 + e)2

8 3

Vol. 18, 1967

337

B o u n d a r y Layer Growth on a ~'Iagnetised Circuiar Cylinder

and

f~(e)

28 -

3

(1 +

8) 4

(4 + 3 , + : ) .

The three functions f l , f2, fa are shown in Figure 1. The influence of the conductivity on the skin friction is small for e > 2. When fl is large, the diffusivities are equal and 30 25

2.O I5 10

g g

5 Figure 1 The functions /1, /2, /~"

the fields at infinity are aligned, it has been shown by the present author (1966) that near a stagnation point the skin friction has the form

% ~-~ (:~ t)-l/2 2x (1-- @ tS t~ + ...) .

(27)

The second and third approximations to t r given by (27) are about 6~o larger than the first approximation. Equation (26) can be expected to give a reasonable estimate for tr over the entire surface of the cylinder. We now consider the two cases of aligned and crossed fields for infinite conductivity. (a)

Aligned Fields

Solving (26) for tr yields

tj 1 = - 0 . 7 1 2 2 U~ 1 + ~[tt0"7271 - ~-

fl(fl__ f.)j]

U~ - [0.05975+ ~1

flf3] US}1/2

(28) 9

The curves for infinite conductivity for fi = 0, 1, 10, oc are shown in Figure 2. When /5 -- 0 the Goldstein and Rosenhead results are regained. Their calculations confirmed t h a t separation first took place at the rear stagnation point, tr(z~) having the value 0.31. For times greater than tr(z~) t h e y found t h a t the separation points moved symmetrically round from the rear of the cylinder. For small fl, the curves t~(x) are similar to those in the non-conducting problem, separation again developing from the rear stagnation points but at steadily advancing times with increase of ft. When fl is ZAMP 18]22

338

K. E. BARRETT

./

ZAMP

0.5

~"-'---70

0o

30 ~

60 ~

lgg

90 ~

150~

780~

Figure 2 T i m e s of s e p a r a t i o n at v a r y i n g s t a t i o n s for ~ = 0, 1, 10 o o ill a p e f e c t l y c o n d u c t i n g fluid w h e n the fields are aligned a n d the n o r m a l field is zero.

large enough it a p p e a r s t h a t s e p a r a t i o n m a y also develop from t h e front s t a g n a t i o n point. The s e p a r a t i o n t i m e a t s t a t i o n x exceeds t h a t at a - x, the difference b e t w e e n the two s e p a r a t i o n times decreasing w i t h increase of/3. The curve for infinite/5 shows t h a t s e p a r a t i o n occurs s y m m e t r i c a l l y fore a n d aft i m m e d i a t e l y a f t e r t h e impulsive start. I n the limit fi -> oc, /5 t~(0) --> 1/3. This m a y be c o m p a r e d w i t h t h e value of /5 t~(0) = 0.381 for flows for which e = 1. (b)

Crossed

Fields

I n this case 1

t;-:=--0.T122+{[0.7271--~/5(f.

+/~)]

~

-

(0.05975§

1/3f0U"}'/' (29)

~

9

The curves t.(x) which are still symmetric about the 0 - ~ diameter are shown in Figure 3 for a few values of/3. For small/3 the initial separation point is at the rear iv

I

___.7 oo

soo

600

i

~4

goo

..... §

12o0

_4

7500

78oo

Figure 3 Times of separation at varying stations for ~ = 0, 1, 2, 10, oo in a perfectly conducting fluid when the fields are crossed and the normal field is zero. s t a g n a t i o n point, the t i m e at which s e p a r a t i o n occurs being r e t a r d e d with increase of ft. As/5 is increased t h e initial s e p a r a t i o n points are found no longer at t h e rear s t a g n a t i o n p o i n t b u t at points s y m m e t r i c a l l y placed on the rear of the cylinder.

Vol. 18, 1967

339

Boundary Layer Growth on a Magnetised Circular Cylinder

As fi is increased further, the initial separation times decrease and the initial separation points move round to the - z c / 2 - ~/2 diameter, separation eventually occurring instantaneously and symmetrically at 4-7U2 when fi is infinite. In the limit fl + ec,/7 t~ ( • ~/2) + 0.375. The results for aligned fields for a perfectly conducting fluid agrees well with Goldsworthy's work in that/5 is the governing parameter, and t h a t a forward separation bubble is predicted. I t also suggests t h a t a rearward separation bubble will be formed. Goldsworthy excluded this possibility in his steady flow analysis, b y insisting that no closed streamlines could originate from the b o d y surface. 4. S m a l l

Normal

Field with Equal Diffusivities

For simplicity it will be assumed t h a t the diffusivities are equal. When the normal field at the surface is 0(R -lj~) we are still able to use the b o u n d a r y layer Equations (11)-(13). The expansions for the velocity and magnetic field components proceeds in powers of #/2 rather than t when the normal field at the surface is non-zero. Equations (14) and (15) give the forms of the solutions at small times and the earlier approximations satisfy Equations (16). The solutions of (16) for L 4= 0 where they differ from the solutions for a zero normal field are ~(1) :

U L

g~l) ,

u('2) . . . .

+ /5 g L2f~ e2) ,

u (3) = fi {(U L),: H f ~ ~1) + C L H~f(~ ~) + U,r L H f ~ 3 ) } , ~(3) :

V {( v L)x g(31) ~_ y x L g~32) ~_/5 n 3 g~33)}

~ ....

+ Z u L {(u c)~/~~ + u~ Lf~ ~') +/5 L~f~'~}.

The equations satisfied b y some of the previously undefined functions are Fl(g~l)) = _ 2 f ( ~ ,

~(f~22)) = - 2

ovv~'O),

F~(f~ s')) = t (3i) ,

i = 1, 2, 3,

where t (31) =

--4

o7/~ o7~(1),

t (32) = - 4 g~0) g!~) + 4 g(O) ~~(1) - 2 g7,(~1) ,

~ ( 2 3)) : 2i~ ~) , ~ ( s , (4~)) : - 2

t(.3a) = - 2 e(~) o~717 ,

g~?.

The f ' s and g's satisfy the b o u n d a r y conditions (23). Solutions are relatively easy to find b y using an iterative procedure, and are g(1)= ~1q 0 ,

f(22)_ 81 q - i

1 , 4~zi,2

1 r/31) ~7 = _ 3 (q3 - Ps) + T q-1

3 1 1 f(u2) = - - 9 (qa -- f13) + -2- (ql -- Pl) + ~- q-1 + ~ q - u , r 7<~) = 3 ( ~ - p~) -

1

(q~ - #~) + T q - ~ '

~)

-

1

4s ~-~ '

1

f(/~) = 5 8 ~ ~-~

"

The skin friction series which contains terms of the order of 1, t, fi L 2 t, /5 L t 3/2, t e, fl L 2 t 2 and f12 L 4 t 2 now involves two further combinations of the p a r a m e t e r s and L n a m e l y fi L and/7 L 2. Table 1 shows the four main regimes of interest.

340

K. I~. BARRETT

ZA~IP

Table 1 O r d e r s of D o m i n a n t T e r m s

Basic Type

1.

fi L ~ < 1, /~ ~ 1

1, t, f l L t 8~2, t 2

non-magnetic

2.

fl L 2 ~ 1, fl >~ 1

1, t, fl L t a/2, flt 2

zero n o r m a l field a n d s t r o n g u n i f o r m field

3.

fl L 2 >> 1, fl L a ~ 1

1, /5 L 2 t, fl L t a/~, fl t 2

m o d e r a t e n o r m a l field a n d s t r o n g u n i f o r m field

4.

/~ L 2 >~ 1, ~ L a >> 1

1, fi L 2 t, fl L t a2, f12 L a ~2

s t r o n g n o r m a l field

As a further simplification it will be assumed that both the magnetic fields are aligned with the uniform velocity field i.e. that e~ = ~2 = 0. C a s g "[

When fl L 2 and fl are both small all the magnetic terms are small and the separation process is basically that of the non-conducting flow. Case 2

When fl L 2 is small and fl is large the skin friction is approximately given by


1+(1+

34d) U , t + ~

-

H2o!

1 ( 1 + M~ ~ (17

96

H o]

U~-

8 U ~) f l t J .

When L is zero this reduces to the strong uniform field solution. If the uniform field is strong enough separation occurs at both the rear and front stagnation points. The effect of the term 0(t~j2) depends on the strength of the dipole field. If the dipole field is weaker than the uniform applied field, then the separation process is assisted in a symmetrical manner at both stagnation points whereas if the dipole field is the stronger of the applied magnetic fields then the separation process is retarded. The physical explanation for this behaviour is that as M o / H o is increased through unity the induced magnetic field to first order changes sign and consequently so do the current and Lorentz force, the magnetic contribution to the pressure gradient then being advantageous rather than adverse. C~SC 3

The skin friction is approximately given by (G)o= U {1 - @ 6

(1--~M~ H0 ]

zd/2 (1 -- M ~ (3 U 2 - 7 U J/3 2 t~12 3-2H~o) 1 (1 + 21//~ 96 Ho]

U:-8U

2) fit z}

when/5 L 2 >5 1 and/5 L ~ < 1. The separation process is still dominated by the strong uniform field in this case. The term giving the dipole effect is dominated by the term of 0(/5 L ~ t) and this term enhances the chances of separation in a symmetrical manner.

Vol. 18, 1967

Boundary Layer Growth on a Magnetised Circular Cylinder

361

Case d Finally, when/5 L 2 >> 1 and/5 L a >> 1 the skin friction series becomes

{

(u,) 0 = U 1 -

1 (

16 \1-

M o~2 o ~1/~ (1--M~~ ( U 2 -7 H o] / s U 2 ~ t + 3 - f f H g ] ,(3 1 + 5i~

~" ta/2 Ux)/5

(1---M~ Ho ]

2

) U~ t 2 .

This equation only gives zeros for the skin friction which are such that fl L z t > 0(1), so cannot give any reliable information about the flow development. The first, second and fourth terms suggest t h a t the skin friction depends on the function exp ( - 1/4 fl L 2 t) and this is confirmed b y some recent analysis of SEARS (1966).

5. C o n c l u s i o n s For problems in which the magnetic field at the surface is zero it is clear t h a t one can anticipate t h a t wakes will develop in the Alfv~n direction for sufficiently large applied fields strengths. The situation when the normal field at the surface is large, as for example in the case of a non-magnetised cylinder, has not been resolved. The only thing t h a t can be said is t h a t the local Alfv~n wave formed at the surface dominates the overall Alfv6n disturbance arising from the uniform field. REFERENCES BARRETT, K. E. (1967), to be published ill Q. J. M. A. M. BLASlUS, H. (1908), Grenzschichten in Fl~ssigkeiten mit kleiner Reibung, Z. Math. Phys. 56, 1-37. GOLDSTEII%S. alld ROSENI~EAD,L. (1936), Boundary Layer Growth, Proc. Camb. Phil. Soc. 32, 392-401. GOLDSWORTHu F. A. (1961), Magnetohydrodynamic Flows o/a Per/ectly Conducting Viscous Fluid, J.F.M. 17, 519-528. SEARS, W. R. (1966), The Boundary Layer in Crossed Fields m.h.d., J.F.M. 25, Pt. 2, 229-240. WUNDT, I-I. (1955), Wachstum der laminaren Grenzschichten an schrdg angestrdmten Zylindern bei An/ahrt aus dee Ruhe, Ing. Arch. 23, 212-230. Rdsumd

Dans le probl~me considerS, un cylindre parfaitement conducteur et magnetis~ est mis en mouvement par impulsion dana un fluide visqueux incompressible et non-conducteur. On suppose de m~me qu'il y a u n champ uniforme et magn6tique impos6 sur tout le champ d'6coulement. On demontre que la force du champ uniforme et magn6tique et l'amplitude du champ normal initial ~ la surface solide jouent un r61e important dans la croissance de la couche limite sur le cylindre. (Received: October 4, 1966.)