Write down the complex potential for a source of strength m located at z=ih and a source of strength m located at z= -ih. Show that the real axis is a streamline in the resulting flow field, and so deduce that the complex potential for the two sources is also the complex potential for a flat plate located along y= 0 with a source of strength m located a distance h above it. Obtain the pressure on the upper surface of the plate mentioned above from Bernoulli’s equation. Integrate the pressure difference over the entire surface of the plate, and so show that the force acting on the plate due to the presence of the source is (rho)*m^2/(4*(pi)*h).Take the pressure along the lower surface of the plate to be equal to the stagnation pressure in the fluid. Solution z = x + i y z=x+iy Complex potential general form: F ( z ) = + i F(z)=+i Complex potential for a source of strength m : F ( z ) = m/ 2 log ( z z 0 ) F(z)=m2log(zz0) Bernoulli\'s equation for ideal irrotational flow: t + p + 1 2 G = F ( t ) t+p+12G=F(t) where F ( t ) F(t) is the unsteady Bernoulli constant y=0 is a streamline in the flow field. Separate real and imaginary parts of the complex potential so that I can isolate and show that = 0 at y=0. Tease out from the complex potential + i = m 2 log ( z 2 h 2 ) +i=m2log(z2h2), take the exponential of each side to get rid of the log function, and my i term is tied up in an exponential e + i = e m 2 ( z 2 h 2 ) e+i=em2(z2h2). Since this flow is ideal, irrotational, and steady that Bernoulii\'s equation would simplify down to p + 1 2 G = 0.

1. Write down the complex potential for a source of strength m located at
z=ih and a source of strength m located at z= -ih. Show that the real
axis is a streamline in the resulting flow field, and so deduce that the
complex potential for the two sources is also the complex potential for a
flat plate located along y= 0 with a source of strength m located a distance
h above it.
Obtain the pressure on the upper surface of the plate mentioned
above from Bernoulli’s equation. Integrate the pressure difference over
the entire surface of the plate, and so show that the force acting on the
plate due to the presence of the source is (rho)*m^2/(4*(pi)*h).Take the pressure
along the lower surface of the plate to be equal to the stagnation pressure
in the fluid.
Solution
z = x + i y z=x+iy
Complex potential general form: F ( z ) = + i F(z)=+i
Complex potential for a source of strength m : F ( z ) = m/ 2 log ( z z 0 ) F(z)=m2log(zz0)
Bernoulli's equation for ideal irrotational flow: t + p + 1 2 G = F ( t ) t+p+12G=F(t) where
F ( t ) F(t) is the unsteady Bernoulli constant
y=0 is a streamline in the flow field. Separate real and imaginary parts of the complex potential
so that I can isolate and show that = 0 at y=0. Tease out from the complex potential + i = m 2
log ( z 2 h 2 ) +i=m2log(z2h2), take the exponential of each side to get rid of the log function,
and my i term is tied up in an exponential e + i = e m 2 ( z 2 h 2 ) e+i=em2(z2h2).
Since this flow is ideal, irrotational, and steady that Bernoulii's equation would simplify down to
p + 1 2 G = 0