# P rojectO ne V elocityFields inR ectangularand P olarCoordinates

```P rojectO ne
V elocityFields inR ectangularand P olarCoordinates
M any&deg; uid&deg; ows andcurrents inoceanography(orgeophysical&deg; uiddynamics)have a predominantly circularcharacterto them. H urricanes, tornadoes
and streams,suchas theG ulfStream,areexamples of&deg; ows whosestructureis
dominated byavortexoragyre.W hen the&deg; owis two-dimensional,itis often
mathematically more economicaltorepresentthese &deg; ows in polarcoordinates
ratherthaninrectangularcoordinates.T hepurposeofthisprojectistodevelop
themathematicalrepresentations ofvelocityinthesetwocoordinatesystems.
L et v(x;y)be a velocity &macr;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&deg; uid particle,wehavethesystem ofcoupled
di&reg;erentialequationsthatrelatethepositionofaparticleattimettoitsvelocity:
dx
dy
= v1 (x;y;t);
= v2 (x;y;t):
dt
dt
(1 )
W hentheoriginalpositionoftheparticleisknown(suchasx(0 )= 1 ;y(0 )= &iexcl;2,
say), one solves the system ofdi&reg;erentialequations in (1 )to getthe particle
path.
W erepresentthevelocityofaparticlein polarcoordinates by
v= vrer + v&micro;e&micro;
(2)
where er and e&micro; are unitvectors in the directions ofincreasing r and &micro;, see
Figure1 .T hegoalofthis projectis to&macr;nd theanalogueof(1 )interms ofr(t)
and &micro;(t),thepolarcoordinates of(x(t);y(t)).
1 .R ecallthepolar{ rectangularrelations
x= rcos&micro;;
y = rsin&micro;:
(3)
U setheserelations toshowthat
er = cos&micro;i + sin&micro;j;
e&micro; = &iexcl;sin&micro;i + cos&micro;j:
(4)
j= sin&micro;er + cos&micro;e&micro;:
(5)
U setheserelations toshow
i = cos&micro;er &iexcl;sin&micro;e&micro;;
M oreover,showthat
der
= e&micro;;
d&micro;
1
de&micro;
= &iexcl;er:
d&micro;
R emark1 :N otethatthesubscripts rand&micro; iner ande&micro; donotdenote
partialdi&reg;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 e&micro; depend on&micro; oftheposition atwhich theyapply.
2.L et(x(t);y(t))and (r(t);&micro;(t))denote theparametrization ofthepath of
a particle, the formerin rectangularcoordinates and the latterin polar
coordinates.Itfollows form (3)that
x(t)= r(t)cos&micro;(t);
y(t)= r(t)sin&micro;(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
d&micro;
er + r e&micro;:
dt
dt
(7)
Compare(7)and (2)toconcludethat
dr
= vr;
dt
d&micro;
1
= v&micro;:
dt r
(8)
Systems (1 )and (8)are two representations of the same &deg; uid
&deg; ow.B ecausevr andv&micro; arefunctions ofrandtheta,system (8)de&macr;nes a
system ofdi&reg;erentialequations inr(t)and&micro;(t)that,whencombinedwith
(6),leads tothesamepath (x(t);y(t))thatonewould getfrom (1 ).
3. (a)ConsidertheM erry-go-roundvelocity&macr;eldv= yi&iexcl;xj.Showthat
v= &iexcl;re&micro;.(H int:U se(5).
(b)L etP = (&iexcl;1 ;0 )betheposition occupied bya&deg; 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 &iexcl; p 2x 2 j. Show that
x +y
x +y
v= &iexcl;e&micro;.
(b)P lotthepathoftheparticlelocated at(1 ;0 )attime0 fort2 (0 ;3),
&macr;rstusingthe rectangularrepresentation and nextthe polarrepresentation.
2
5. (a)Considerthe L ine vortex v =
&iexcl;1re&micro;.
y
x
x2 + y2 i &iexcl; x2 + y2 j.
Show thatv =
(b)P lotthepathoftheparticlelocated at(1 ;0 )attime0 fort2 (0 ;3),
&macr;rstusingthe rectangularrepresentation and nextthe polarrepresentation.
6. (a)U singtheM erry-go-round velocity&macr;eld,plotthepaths oftheparticles located at(&iexcl;1 ;0 ),(&iexcl;2;0 ),(&iexcl;3;0 ),and (&iexcl;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&reg;erencebetweenthethreevorticeswehavestudied? In particular, whathappens toastringofdyethatis positioned at
timezeroin each&deg; owalongtheinterval(&iexcl;4;&iexcl;1 )as times evolves?
3
```