P rojectO ne
V elocityFields inR ectangularand P olarCoordinates
M any° uid° ows andcurrents inoceanography(orgeophysical° uiddynamics)have a predominantly circularcharacterto them. H urricanes, tornadoes
and streams,suchas theG ulfStream,areexamples of° ows whosestructureis
dominated byavortexoragyre.W hen the° owis two-dimensional,itis often
mathematically more economicaltorepresentthese ° ows in polarcoordinates
ratherthaninrectangularcoordinates.T hepurposeofthisprojectistodevelop
themathematicalrepresentations ofvelocityinthesetwocoordinatesystems.
L et v(x;y)be a velocity ¯eld. W e represent this velocity in rectangular
coordinates as
v= v1 i+ v2 j
where i and j are unitvectors in the horizontaland verticaldirections.W ith
(x(t);y(t))denotingthepath ofa° uid particle,wehavethesystem ofcoupled
di®erentialequationsthatrelatethepositionofaparticleattimettoitsvelocity:
dx
dy
= v1 (x;y;t);
= v2 (x;y;t):
dt
dt
(1 )
W hentheoriginalpositionoftheparticleisknown(suchasx(0 )= 1 ;y(0 )= ¡2,
say), one solves the system ofdi®erentialequations in (1 )to getthe particle
path.
W erepresentthevelocityofaparticlein polarcoordinates by
v= vrer + vµeµ
(2)
where er and eµ are unitvectors in the directions ofincreasing r and µ, see
Figure1 .T hegoalofthis projectis to¯nd theanalogueof(1 )interms ofr(t)
and µ(t),thepolarcoordinates of(x(t);y(t)).
1 .R ecallthepolar{ rectangularrelations
x= rcosµ;
y = rsinµ:
(3)
U setheserelations toshowthat
er = cosµi + sinµj;
eµ = ¡sinµi + cosµj:
(4)
j= sinµer + cosµeµ:
(5)
U setheserelations toshow
i = cosµer ¡sinµeµ;
M oreover,showthat
der
= eµ;
dµ
1
deµ
= ¡er:
dµ
R emark1 :N otethatthesubscripts randµ iner andeµ donotdenote
partialdi®erentiations!
R emark2:Itis importanttonote thattheunitvectors i and j donot
@i
@i
dependonxandy,thepositionatwhichtheyapply.H ence, @x
= @y
=0 ,
with similarrelations holdingforj.O n the otherhand,the unitvectors
er and eµ depend onµ oftheposition atwhich theyapply.
2.L et(x(t);y(t))and (r(t);µ(t))denote theparametrization ofthepath of
a particle, the formerin rectangularcoordinates and the latterin polar
coordinates.Itfollows form (3)that
x(t)= r(t)cosµ(t);
y(t)= r(t)sinµ(t):
(6)
R eturningto(1 ),wehavethefollowingrelation between vand xand y:
v=
dx
dy
i+
j:
dt
dt
U sethis relation and (6)toshowthat
v=
dr
dµ
er + r eµ:
dt
dt
(7)
Compare(7)and (2)toconcludethat
dr
= vr;
dt
dµ
1
= vµ:
dt r
(8)
Systems (1 )and (8)are two representations of the same ° uid
° ow.B ecausevr andvµ arefunctions ofrandtheta,system (8)de¯nes a
system ofdi®erentialequations inr(t)andµ(t)that,whencombinedwith
(6),leads tothesamepath (x(t);y(t))thatonewould getfrom (1 ).
3. (a)ConsidertheM erry-go-roundvelocity¯eldv= yi¡xj.Showthat
v= ¡reµ.(H int:U se(5).
(b)L etP = (¡1 ;0 )betheposition occupied bya° uid particleattime
t= 0 .Find the path ofthis particleundervtwice,onceby solving
(1 )and nextby solving(8).P lotthe twoparticle paths.Colorthe
pathinrectangularcoordinates redandtheoneinpolarcoordinates
blue.Combinethetwographs toseethattheyareidentical.
4. (a)Considerthe O seen vortex v = p 2y 2 i ¡ p 2x 2 j. Show that
x +y
x +y
v= ¡eµ.
(b)P lotthepathoftheparticlelocated at(1 ;0 )attime0 fort2 (0 ;3),
¯rstusingthe rectangularrepresentation and nextthe polarrepresentation.
2
5. (a)Considerthe L ine vortex v =
¡1reµ.
y
x
x2 + y2 i ¡ x2 + y2 j.
Show thatv =
(b)P lotthepathoftheparticlelocated at(1 ;0 )attime0 fort2 (0 ;3),
¯rstusingthe rectangularrepresentation and nextthe polarrepresentation.
6. (a)U singtheM erry-go-round velocity¯eld,plotthepaths oftheparticles located at(¡1 ;0 ),(¡2;0 ),(¡3;0 ),and (¡4;0 )attimezerofor
t 2 (0 ;3). Colorthe particle paths red and combine them on the
samescreen.
(b)R epeattheaboveproblem fortheO seen and L inevortices,coloring
theparticlepaths green and blue,respectively.
7.W hatisthequalitativedi®erencebetweenthethreevorticeswehavestudied? In particular, whathappens toastringofdyethatis positioned at
timezeroin each° owalongtheinterval(¡4;¡1 )as times evolves?
3