Question

A line source of strength Q is situated at the point (a,0); a fixed wall coincides with the y axis. (a) Use the method of images to find where the fluid has largest speed along the wall. (b) Show that the resulting force (integral of pressure) on that part of the wall lying between y=+-b is 2P₀b-1/2pQ²/π²(1/a tan⁻¹(b/a)-b/a²+b²) where P₀ is the pressure of the infinity

Answer 1

To find the point on the wall with the largest speed, use the method of images. Calculate the resulting force on the wall between y = ±b by integrating the pressure over that region.

Step-by-step explanation:

To find the point on the wall with the largest speed, we can use the method of images. We can imagine placing a mirrored image of the source on the other side of the wall. This creates another line source with the same strength Q. The fluid flow around these two sources will combine to create a flow pattern.

The maximum speed will be along the line that connects the two sources. This line will be a vertical line passing through the midway point between the two sources. The fluid will flow along this line and reach its maximum speed at that point on the wall.

For part (b), we need to calculate the resulting force on the wall between y = ±b. This can be done by calculating the pressure at each point on the wall and integrating it over the region between y = ±b. The resulting force will be equal to the integral of pressure over this region.