O n e d i m e n s i o n a l m o t i o n o f a v i s c o u s fluid 

C . D E S I M O N E ( * ) - E . P U K I N I ( * * ) - E . S A L U S T I ( * * * ) 

Received on April 6 t h , 1975 

Summary. — T h e effect of t h e f r i c t i o n h a s been s t u d i e d on t h e one 
d i m e n s i o n a l m o t i o n of a viscous fluid. T h i s f r i c t i o n is usually s c h e m a t i z e d in 
v a r i o u s s e m i e m p i r i c a l f o r m u l a e . In t-liis work t h e d i f f e r e n t s c l i e m a t i z a t i o n s 
of t h e f r i c t i o n were n o t s t u d i e d s e p a r a t e l y b u t it was shown t h a t a solution 
e x i s t s for t h e fluid m o t i o n . T h e r e s u l t s give i n f o r m a t i o n on t h e d a m p i n g of 
t h e fluid m o t i o n in t h e case of t h e seiches. 

R iassunto . — Gli e f f e t t i d e l l ' a t t r i t o nel moto di un fluido non possono 
essere t r a s c u r a t i nello s t u d i o di molti f e n o m e n i geofisici. Le origini di tali 
a t t r i t i sono riconducibili a v a r i e f f e t t i fisici che n o r m a l m e n t e vengono sche-
m a t i z z a t i in diverse f o r m u l e s e m i e m p i r i c h e . Sulla base di u n a s c h e m a t i z z a -
zione del m o t o di un fluido lungo u n a sola d i m e n s i o n e si è d e t e r m i n a t a la 
soluzione d e l l ' e q u a z i o n e del m o t o , c o n s i d e r a n d o le varie f o r m e di a t t r i t o 
a g e n t i c o n t e m p o r a n e a m e n t e . I r i s u l t a t i d a n n o i n f o r m a z i o n e sullo s m o r z a -
m e n t o del moto del fluido e sul peso r e l a t i v o dei v a r i a t t r i t i . 

T h e e f f e c t of t h e f r i c t i o n i n t h e m o t i o n of a fluid, e . g . t h e s e a , 
c a n n o t b e n e g l e c t e d i n m a n y g e o p h y s i c a l p h e n o m e n a . I t s o r i g i n c a n 
b e r e l a t e d t o v a r i o u s p h y s i c a l e f f e c t s , so o n e u s u a l l y p r e f e r s t o s c h e m a -
t i z e i t i n d i f f e r e n t s e m i e m p i r i c a l f o r m u l a e . I n a p r e v i o u s w o r k (6) 
s o m e of t h e p o s s i b i l i t i e s of s c h e m a t i z a t i o n f o r t h e o n e d i m e n s i o n a l 
m o t i o n of a fluid h a v e b e e n s t u d i e d . P a r t i c u l a r i t y of t h i s w o r k h a s 
b e e n t h a t t h e s e e f f e c t s w e r e n o t s t u d i e d s e p a r a t e l y b u t i t w a s s h o w n 

(*) I s t i t u t o di Fisica « G. Marconi », U n i v e r s i t à degli S t u d i - R o m a . 
(**) C . N . R . - I . F . A . - R o m a . 
(***) I s t i t u t o di Fisica « G. M a r c o n i » , Università degli S t u d i - R o m a -

I . N . F . N . - Sezione di R o m a . 



118 C. D E SIMONE - R . J ' U R I N I - E. SALUSTI 

t h a t a solution exists if one considers also a d i f f e r e n t t y p e of scliema-
t i z a t i o n . 

I n f o r m u l a e t h e ell'ect was r e l a t e d t o t h e non-linear d i f f e r e n t i a l 
e q u a t i o n : 

~i>2tt u — <j i>2xx [u • li (x)] = v t>-xx u — £ Z u\u\a + f (t, x) [1 ] 
a = o a 

w h e r e t h e s y m b o l u d e n o t e s t h e v e l o c i t y of t h e fluid a t x = const, 
section of t h e c h a n n e l , t h e s u r f a c e of w h i c h is h(x) t h e coefficient v 
is c h a r a c t e r i s t i c of t h e v \ u t y p e f r i c t i o n while y a is t h e o n e r e l a t e d 
t o y u u \ u \

a f r i c t i o n , / is t h e s y m b o l of a n e x t e r n a l force, in o u r case it 
could b e t h e w i n d . I n [1] t h e b o u n d a r y c o n d i t i o n a r e t h o s e of an o p e n 
a n d semi-closed b a s i n . W e h a v e also considered, on t h e basis of com-
m o n sense, t h a t t h e r o u g h n e s s of t h e b o t t o m m a y b e considered as a 
origin of a f r i c t i o n t h e w h i c h is n o t possible t o b e seen t h r o u g h t h e 
u s u a l n u m e r i c a l s t u d i e s t h a t utilize c o m p u t e r s . So we h a v e p r e f e r r e d 
t o t r e a t a flat b o t t o m case w h i c h allows a n a n a l y t i c a l t r e a t m e n t of 
t h e f o r m u l a e . I n t h i s w a y we h a v e s u p p o s e d t h a t t h e s u r f a c e of t h e 
section x=x0 h a s t h e v a l u e : 

h (,T0) = h (x0) + d siny.r0 [2] 

w i t h t w o small v a l u e s of <5 a n d y. O n e can i n t u i t i v e l y say t h a t t h i s 
r o u g h n e s s increases t h e t u r b u l e n c e a n d t h e n it r e n o r m a l i z e s t h e a b o v e 
m e n t i o n e d coefficient. 

T h e p u r p o s e of t h i s w o r k is t o p u t i n t o a c o m p a r i s o n t h e s e effects. 
W e use a n a l y t i c a l m e t h o d s which in some cases a r e r i c h e r with infor-
m a t i o n t h a n t h e p u r e l y ones. T h e r e f o r e we l i m i t ourselves t o a flat 
s c h e m a t i z a t i o n h0 of t h e A d r i a t i c Sea. T h e m e t h o d u s e d , b e c a u s e of 
t h e smallness of t h e v, d coefficients, can b e t h e first order p e r t u r -
b a t i o n of t h e f r e e m o t i o n . Obviously t h i s m e t h o d gives some i n f o r m a t i o n 
f o r n o t a long period of t i m e . W e will discuss t h e r e s u l t s in t h e 
conclusion. 

1 . - T H E P R O B L E M A N D I T S E L E M E N T A R Y S O L U T I O N S 

I n o r d e r to s i m u l a t e a t o u r b e s t t h e A d r i a t i c Sea, d e s p i t e o u r flat 
h y p o t h e s i s li(x) = h0, we use t h e D e f a n t ' s b o u n d a r y conditions. I t 
implies t h a t f o r t h e first a n d t h i r d seiches t h e c h a n n e l is closed 



O N E D I M E N S I O N A L M O T I O N O F A V I S C O U S F L U I D 1 19 

a t Venice a n d is o p e n e d a t O t r a n t o . F o r t h e second seiche, a t t h e con-
t r a r y , t h e c h a n n e l lias t o b e considered closed b o t h a t Venice a n d a t 
O t r a n t o . W e t h e n consider t w o m a i n cases: t h e effect of t h e f r i c t i o n 
on t h e f r e e m o t i o n of t h e w a t e r a n d t h e effect of a s t r o n g w i n d on a 
q u i e t sea. As f o r t h e w i n d , we h a v e a s s u m e d , in t h i s case, also a n 
" a d h o c " s h a p e f o r s e m p l i f y i n g c a l c u l a t i o n . W e s u p p o s e t h a t t h e wind 
s t a r t s s u d d e n l y a t t = 0 a n d t h e f o r c e is m o r e r e l e v a n t a t O t r a n t o 
t h a n in t h e n o r t h e r n p a r t of t h e b a s i n : 

( 0 ' < 0 , r 
1 {x> I) = 1 0 sin HL ( > 0 > - = i 

H a v i n g c o n s i d e r e d t h e s e s c h e m a t i z a t i o n , we s t a r t b y discussing 
t h e first case. 

1 . 1 . - M O T I O N A N D F R I C T I O N O F T U B S E I C H E S 

W e r e p e a t h e r e t h e e q u a t i o n 

1 a 
S2« u — g ~i2xx[h (x)-u] = v S« h xx u •—S X ^t u\u\ [1] 

u a "a 

If we i m p o s e t h a t t h e b a s i n is closed a t Venice a n d opened a t 
O t r a n t o , t h e s u r é l é v a t i o n is zero, we o b t a i n : 

U (0, t) = 3* 11 (X, t)x-L = u (L, t) = 0 

f o r t h e o d d o r d e r seiches. I n t h e second seiche case we h a v e : 

u (0, t) = u (L, t) = 0 

W e t h e n p r o c e e d b y s u p p o s i n g , as described in t h e first p a r a g r a p h , 
t h a t S, v a n d y_a a r e small p a r a m e t e r s , of o r d e r e. W e t h e n develop 
o u r e q u a t i o n in p o w e r of t h i s e p a r a m e t e r . F o r t h e solution, we h a v e : 

U = Uo + £ Ml + £2M2 + • • • 

F o r e = 0 we h a v e f o r t h e o d d o r d e r seiches: 

S2» Uo — gho Wxx Uo = 0; Mo(0, t) = DxUo(L, t) 

w h i c h solution is also f o r t h e second seiche case: 

Uo = A sin x cos \n (J~ n = 1 , 2 , 3 [2] 



1 2 0 C. D E S I M O N E - R . J ' U R I N I - E . SALUSTI 

W e now s c h e m a t i z e t h e f r i c t i o n s as a n effect t h a t s t a r t s a t < = 0 
in order t o m a k e clearer t h e difference b e t w e e n t h e f r i c t i o n s a n d t h e 
f r e e m o t i o n . F o r u = ua + eui, we h a v e t o t a k e i n t o a c c o u n t t h e pro-
perties of 11 „: 

D2ttUi — (jlioV-xx ui = F(u0)\ ui(x, 0) = ~bxiii(x, 0) = 0 

where 

F (Me) = g D2xx (u0 (5 sin y x) + v S< ua —• 2 « Z { u° ' I u° l" } 
o " " 

I n t h e following we p u t a = 1. T h e case a = 0, which is also 
a m o n g t h e m o s t i n t e r e s t i n g ones, can b e easily d e d u c e d f r o m calcula-
t i o n . We deal now with t h e well k n o w n i n h o m o g e n e o u s w a v e p r o b l e m . 
W e will follow t h e C o u r a n t - H i l b e r t ' s classical s c h e m e (4). W e develop 
a t t h e first 

F (uo) = Ax sin fix • /*(<) 

Then we m a k e easier our p r o b l e m b y r e m a r k i n g t h a t we can say 

u = £,Tk(t) sin/9 FT» 

where t h e u n k n o w n T k (t) a r e d e t e r m i n e d b y 

} a2,, Tk(t) + g ho /V Tk(t) = fk(t) 

j TA-(0) = a t Tk(0) = o 

W e t h e n k n o w t h a t 
t 

sin | g h018k (t — s) fk(s) d s Tk(t) = -
ßk | g ho 

a n d it implies t h a t 

u = £fc sinßkX • Tk(t) 

1 . 2 . - E F F E C T O F T H E F R I C T I O N W H E N T H E W I N D M O V E S T H E W A T E R 

W e s t u d y t h e case of a semiopen c h a n n e l in r e s t a t t 0, when 
s u d d e n l y t h e wind p r o d u c e s an e x t e r n a l f o r c e : 

(0 t < 0 
f (x.t) = { r . x ' v ' ' ) 0 sin — t > 0 

also in t h i s case we will s t a r t bv leaving aside t h e f r i c t i o n . 



O N E D I M E N S I O N A L M O T I O N O F A V I S C O U S F L U I D 
1 19 

Tlie motion e q u a t i o n is given b y 

32ÍÍWO — gho t>2xx u0 = $ s i n — 8 (t) 
A 

t h e solution of which is 

0 t < 0 
Uo ( X , t ) = { , . X . t ^ n v ' ' 1 1 $ sin — sin — í > 0 

I A t 

Because of t h e r e m a r k a b l e similarity with t h e previous case, we 
will use also in this case t h e above described m e t h o d to solve t h e equa-
tion a t first order in e. 

T H E A N A L Y T I C A L R E S U L T S 

F o r t h e first seiche we h a v e 

F(uo) = j g ô y + y j 

— y 
i \ 2 

A 
cos 

c o s 

t 
cos — 

t 

Y + J Ì * + 
vA x , t 
— sin — Sin h 

A 2 r A r 

A2 { 2x ) . 21 cos tir ZoA . x t 
7,1 — { cos —r 1 > sin — —— -j sin — cos — 

r ( A ) r |cos t¡r\ r A t 

One m a y r e m a r k t h a t t h e first t e r m is related to t h e roughness 
of t h e b o t t o m . I t is proportional to <5 and it contains t h e y coefficient. 
The second p a r t is r e l a t e d t o t h e l a m i n a r friction in t h e a = 0 a n d 
a — 1 cases a n d provides t h e last two t e r m s . W e r e m a r k here t h a t in 
this case t h e a = 0 p a r t of t h e friction is proportional to t h e l a m i n a r 
friction, we will t h e n n o t discuss it a n y more. W e now describe F(u0) 
as S/t-l/t sin (fikX + (fk) • fk(t) f r o m which it follows: 

Ai = 4" 9 à ( y + T ßi = y + 
71 

; «pi = 
, t 
fi = cos 

t 

A 2 = — — gò\y — ß* = y <p* = T ; A =cos T 



1 2 2 C. D E S I M O N E - R . J ' U R I N I - E . S A L U S T I 

vA 
A 2 r 

p 3 = 9?3 = 0 f3 = sin 

A. = — Xi 
A* 

A s = Xi 
A2 

A - X 

ft = 0 

n 
<P* = 77 

n 
<Ps = T 

/4 - sin 
2í eos t/x 
x |cos t¡x\ 

. 2 í eos t/r 
h = 8 1 1 1 7 l ^ b i l M 

yl0 = Zo 9?« = 0 f» = sin 

Then 

M = A sin — sin — + Ss-JU sin (fax + <p*) T¡,- (í) 
X T 

where 

1\ 

Ti 

gho 
i \2 1 ( t gho y + X 

c o s — 
( T 

gho 
( i \ 2 1 ( t 

— gho [ y - X 
< c o s — 
( V 

cos 

V ( y + y j í j 

V gho [ y — j j < 

T t X- . t 
J 3 = t eos — + —- sin — " T •> r 

This last result is p r o p o r t i o n a l to t h e effect of Xo. The f u n c t i o n 
7',, and T5 are more complicated to describe in order to e l e m e n t a r y 
functions. W e m u s t i n t r o d u c e t h e Sk t i m e i n t e r v a l s : 

S0 0 < i < Si 
71 t 3 
— C — < — 71 

So 
3 t 5 

71 ^ — ^ — 71 
5 t 7 

S2 — JI < — ^ — TI etc. 
2 x 2 

Then in t h e 7 f t h i n t e r v a l we h a v e 

T 
T 4 = 

, , 1 . 2 1 t 21 
l)f c i — sin cos LE TI 1 2 x x x 

2< . 2 « 
- + sin 2 k n 

X X 



O N E D I M E N S I O N A L M O T I O N O F A V I S C O U S F L U I D 
1 19 

F o r t h e second seiche, we can easily calculate t h e result b y p u t t i n g 
A r 

A -> — a n d r -*• — in t h e Tk(t) expressions. 

I n t h e ease of a wind, in analogy with t h e f r e e motion, t h e result is: 

F (Mo) =6 [y + y j cos (y + y j x sin ~ + 

, &x ( 1 \2 I 1 \ . t 0v . x t 
- d - r [ y - T ) C O S v ~ i ) X S M V W S M X 0 0 8 ~x + 

. 21 sin tlx 2x . 21 sin tlx 
sin — r— — — cos —i— sin x x |sin </R| T 2 A x |sin tjx\ 

f r o m which we o b t a i n : 

. _ T / 1 \ 2 . 1 t 
Ai=gò& — \y + — j ; ßv = y + y ; gpi = — ; fx = s i n — 

= g d O J (y — J j ; ßt = y — y ; <p2 = ; /» = sin 

0 1 Í 
vl3 = r— r ; ß3 = -T- ; 933 = 0 ; f3 = cos — 

A- A T 

n r . 3t . . 2t sill tlx 
A i = - 0

2
X l x ; & = 0 ; * = - ;.,4 = sin - ^ J -

2 , . 2t sin t/r 
' ^ = T = 2 ; / 5 = S m 7 IsinW 

a n d t h e Tj.-(i) f u n c t i o n s are 

2'i (i) = Jgho (r + y ) + 7 

sin V<7 Äo [y + y ) t + 

sin — + 
T 

2 *Jg ho [y — j 

gho ( r + j J -
i 

T2 

Vgho (y + T ) ~ 

sin 2 Vi/Äo 
1 \ 1 



1 2 4 C. D E S I M O N E - R . J ' U R I N I - E . S A L U S T I 

T 2 0 ) = T1 (t) J J = > 

T3 (I) = — t sin — 

2', ( 0 = ; 
i 1 . 2< 

— sin b 
r 2 r t + T d — ( — ! ) " ) 

71 (— 1)* 

r2 I 1 . 2t t 2t 
T . W = t ¥ s i n 7 c o s - 7c + - ( l — ( — 1 ) * w ( — 1)* 

We n o t e here a difference between t h e effect of t h e roughness of 
t h e b o t t o m (which gives results limited when t increases) a n d t h e t w o 
other cases, where t h e velocity increases in norm as t oo. I n our 
opinion this effect is d u e to t h e m a i n effect of t h e roughness: it gives 
m a n y small m o v e m e n t b u t it d o e s n ' t decrease t h e kinetic energy. So 
we are encouraged to t r y and see t h e effect of a l a m i n a r friction vAu 
on this p e r t u r b a t i o n of t h e velocity. This friction should decrease t h e 
kinetic energy b u t n o t t h e a m p l i t u d e of t h e effect, because t h e wave-
length of t h e p e r t u r b a t i o n is 1/y which is v e r y large c o m p a r a t e d with A. 
W e s t u d y t h e r e f o r e t h e p a r t of U\ which is p r o p o r t i o n a l to <5, Ui to which 
we apply t h e rA operator. W e o b t a i n : 

m = A g 

cos — 
t 

r + j I cos \y + y I ® 

cos [y + y - ) Vi7 t 
+ 

i \2 
cos c o s s \l() ho 

g-ho [ y - j J 
l 

T * 

t h e n c e 

Ui 
cc 

•• ft cos y x (Jh (t) + c sin ~ sin y x <P2 (t) 



O N E D I M E N S I O N A L MOTION O F A VISCOUS F L U I D 1 2 5 

, A <7 Ô V 
= . - y3 

Vir/to ß Vif /to /t 
. , r~r- . sin 2 V? /to t 

sm3 Vi/ «o y t , , —, h 

/ 
H — cos 2 Vir /to y < + 

ghy3 x ' 

A g ôvy3 \ t 

t2 

*Jg /t o y 

•4 (Vi/ /to Y ) 3 R 

sin Vi/ /to í ? 

02 = — " /—-1- i z ~ cos Jq h0 y t — — : r sin Jg ha yt i c \l g ho (2^0/toy 2ghay2 A ) 

W e r e m a r k t h a t we h a v e obtained velocities which decrease with 
time. Their numerical i m p o r t a n c e will be discussed in t h e following 
p a r t . W e would like to add t h a t one could intuitively assume t h a t this 
linear first p a r t of a negative exponential, a t least for t h e l a m i n a r 
friction t e r m a n d p e r h a p s for t h e others as well. 

3 . - N U M E R I C A L E V A L U A T I O N A N D F I N A L D I S C U S S I O N 

I n order t o e v a l u a t e numerically t h e relative weight of these 
results we h a v e assumed 

L = 106 m v = 1.4 10"2 m 2 s~l; y = 10~2 n r 1 

ho = 3 • 102 m A = 10-! m s"1 

g = 10 m s- 2 Xi = 6 • 10"6 n r 1 

Zo = 10-5 8-i «5 = i m 

W e t h e n h a v e 
Ad t 1 

M l = -2h7V + ; i \2 
X 

— cos ^y + J 
1\ ( t 

X I cos — + 

cos Vir /to [y + y 1 t 

T3 

A ó 
2ho 

1 \2 
y — 

i 
'x2 

cos y + 

1 \ ( t ,—— I 1 \ ) v A T ( t t 
X { COS — COS Vi/ /to ly + y I t - COS y + 

7. 
. t ) . X A21 , ( 1 . 2í t 21 

— sin — sin 7i — ; — (— l)fc - sin cos 
T ) I 4 ( 2 T T r 

, , 2 X A2 X { 4 t ( 2 t 
— hn } cos - y Xi — j — — + (— 1)* j — + 

+ 

• 2 t 
+ sin 2 Ten 

X 
Xo - , 

t t . X 
— cos — sin — 
X X A 





o n e d i m e n s i o n a l m o t i o n o f a v i s c o u s f l u i d 
1 19 

the smaller scale motions which are strongly related to the bottom 
irregularities. 

Pig. 2 - T h e p o s i t i o n a t Venice of t h e f r e e s u r f a c e f d u r i n g 
t = T = 3 h 2 0 m . I 0 is t h e position c o r r e s p o n d i n g t o t h e velocity 

v0, | 1 ( t o vt and f t o v. 

The preceding statement is stressed by the fact t h a t at the second 
order, the bottom roughness gives an internal friction the value of 
which can be larger than the corresponding first order quantity. The 
part of the friction which is proportional to v, the laminar part, gives 
a very small contribution to the general form of the friction, because 
the coupling constant is very small. I t has to remarked, however, t h a t 
its analytical form is very closed to the part proportional to 7.0. This 
analytical similarity could give origin a bit of confusion in the interplay 
of these quantities, in any case we consider here the values usually 
assumed on an experimental field. We can also say t h a t the term 
proportional to X0, usually called linear bottom friction, describes also 
the effect of the friction on the boundary layer. On these same ground, 
moreover, we can say t h a t in the term proportional to 1\ the higher 
velocities give the more remarkable effect. If, however, one has not a 
flat bottom, one could expect t h a t this kind of friction will give diffe-
rent effect for different depth h0. We expect this phenomenon to be 
important for a study of the real Adriatic Sea which is now in progress. 
We can also say t h a t all the frictions by us dealt have a linear effect, 
because we have analyzed the first order of the perturbation expansion 
of the real friction. If we assume on physical field t h a t these effects 

0 . 5 

O 



1 2 8 c . d e s i m o n e - r . j ' u r i n i - e . s a l u s t i 

are exponential, this hypothesis has some experimental evidence and 
we could also say t h a t these terms are proportional to 

„, , „ „ , v At Ad 
ZiA^r, XoAr, — , — 

and at the second order, to 

A g 6 v y3 

g h0 

R E F E R E N C E S 

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