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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Water flowing around a cy
linder" }}{PARA 256 "" 0 "" {TEXT -1 233 "Velocity of water  flowing a
round a circular cylinder with axis of cylinder perpendicular to the x
y plane, radius of cylinder is 1 unit.  Far away from teh origin, wate
r is flowing parallel to the x axis with speed 1 unit per second." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 139 
"Since the velocity is defined only in  region x^2+y^2 > 1,  note the \+
use of the Heaviside function to cut the field. The velocity field is \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "restart:\nwith(plots):\nwith(l
inalg):\nassume(x,real): assume(y,real):\nvi := 1 + (y^2-x^2)/(x^2+y^2
)^2;\nvj := -2*x*y/(x^2+y^2)^2;\ncut := Heaviside(x^2+y^2-1):\nvelocit
y := [vi*cut, vj*cut]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 64 "We dra
w the cross section of the cylinder and the velocity field" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 170 "cylinder := plot( [cos(t), sin(t), t=0..2*
Pi], color=blue ):\nvelplot := fieldplot( velocity, x=-3..3, y=-3..3, \+
color=red, arrows=SLIM):    display( \{velplot, cylinder\} );" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 257 220 "Some observations about the velo
city\n(a) velocity = [vi, vj]  is [1,0]  for points away from the cyli
nder i.e. x^2 + y^2 big\n(b) on the boundary of the obstacle,  the wat
er velocity is tangential to the obstacle because " }}{PARA 0 "" 0 "" 
{TEXT 258 67 "      dotproduct( vel, n) = 0 where n is the normal to t
he cylinder" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "n := grad( x^2 + y^
2 -1, [x,y] );\nval := dotprod( [vi,vj], n ):\nassum(x,real): assume(y
,real):\nsimplify(val);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Grav
itational Field" }}{PARA 258 "" 0 "" {TEXT -1 149 "Gravitational force
 field due to a unit mass at the orgin.  Since magnitude of this field
 is very large near (0,0,0) - it is cut off near the origin." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "len := (x^2+y^2+z^2)^(3/2):
\nFi := -x/len;  Fj := -y/len;  Fk := -z/len;\ncut := Heaviside(x^2+y^
2-1/10):\nforce := [Fi*cut, Fj*cut, Fk*cut]:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 119 "with(plots):\nfieldplot3d( force,  x=-1..1, y=-
1..1, z =-1..1, arrows=THICK, color=red, style=patchnogrid, axes=boxed
 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1" 0 }
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