Pool

$4\mathrm{.}3$ Write down the complex potential for a source of strength $m$ located at $z=ih$ and a source of strength $m$ located at $z=-i.$ 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 the line $y=0$ 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 $\frac{m^2}{4h}.$ Take the pressure along the lower surface of the plate to be equal to the stagnation pressure in the fluid.

$4\mathrm{.}3$ Write down the complex potential for a source of strength $m$ located at $z=ih$ and a source of strength $m$ located at $z=-i.$ 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 the line $y=0$ 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 $\frac{m^2}{4h}.$ Take the pressure along the lower surface of the plate to be equal to the stagnation pressure in the fluid.