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# hyperbolic

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# hyperbolic   { HYPERBOLIC.PDE

This problem shows the capabilities of FlexPDE in hyperbolic systems.

We analyze a single turn of a helical tube with a programmed flow velocity.

A contaminant is introduced into the center of the flow on the input surface.

Contaminant outflow is determined from the flow equations.

The contaminant concentration should flow uniformly around the helix.

}

title 'Helical Flow: a hyperbolic system.'

select

ngrid=30

regrid=off { Fixed grid works better in hyperbolic systems }

vandenberg   { most effective method  for hyberbolic systems }

 variables  u   definitions  Rin = 1  Rout = 2  R0 = 1.5    dR = 0.3   { width of the input contaminant profile }  gap = 10   { angular gap between input and output faces }  gapr = gap*pi/180 { gap in radians }  cg = cos(gapr)  sg = sin(gapr)  pin = point(Rin*cg,-Rin*sg)  pout = point(Rout*cg,-Rout*sg)    r = magnitude(x,y)  v = 1  vx = -v*y/r  vy = v*x/r  q = 0       { No Source }  sink = 0     { No Sink }   initial values  u = 0   equations  u : div(vx*u, vy*u) + sink*u + q = 0 boundaries

region 1

start (Rout,0)

value(u) = 0       { We know there should be no contaminant on walls  }

arc(center=0,0) angle=360-gap   { positive angle on outside }

nobc(u) { "No BC" on exit plane allows internal solution to dictate outflow }

line to pin

value(u)=0

arc(center=0,0) angle=gap-360   { negative angle on inside }

value(u)=exp(-((x-R0)/dR)^4)   { programmed inflow is supergaussian }

line to (1.2,0) to (1.4,0) to (1.6,0) to (1.8,0) to close { resolve shape }

monitors

contour(u)

plots

contour(u) painted

surface(u)

elevation(u) from (Rin,0.01) to (Rout,0.01)

elevation(u) from (0,Rin) to (0,Rout)

elevation(u) from (-Rin,0.01) to (-Rout,0.01)

elevation(u) from (0,-Rin) to (0,-Rout)

elevation(u) from pout to pin

end