An idealized model of a river flowing around a rock considers a circular rock of radius 1 in the river, which flows across the xy-plane in the positive x-direction. A simple model begins with the potential function .z=x + (x/(x^2+y^2))
a.) Compute the velocity vector field, i.e. . v=grad z
b.) Show that the flow of v is tangent to the circle x^2 + y^2 =1 . This means that no water crosses the circle. The water on the outside must therefore all flow around the circle.
c.) Show that div v=0 . What is the meaning of this in the context of the problem?
2007-12-20 11:57:40 · 1 answers · asked by Andy 2 in Science & Mathematics ➔ Mathematics
dz/dx =1 +1/(x^2+y^2)^2 *(x^2+y^2-2x^2)
dz/dy= -2xy/(x^2+y^2)^2
so grad z=[(x^2+y^2)^2+(y^2-x^2)]/(x^2+y^2)^2*i-2xy/(x^2+y^2)^2j
a tangent vector to the circle is u= y i-x*j with x^2+y^2=1
z=(1+y^2-x^2) i -2xy j and as x^2= 1-y^2
z=2y[y i- xj] so it is tangent
3)Make the calculations
In a flow problem if div v=0 that means the the fluid is not
compressible
2007-12-20 12:36:55 · answer #1 · answered by santmann2002 7 · 0⤊ 0⤋