Answer:
In fully developed laminar flow in a circular pipe, the velocity profile follows a parabolic distribution. The maximum velocity occurs at the centerline of the pipe, and the velocity decreases as you move towards the pipe wall.
Given that the velocity at R/2 (midway between the wall surface and the centerline) is measured to be 8 m/s, we can use this information to determine the velocity at the center of the pipe.
In a fully developed laminar flow, the maximum velocity occurs at the centerline, which is at R = 0. Therefore, the velocity at the center of the pipe is equal to the maximum velocity.
Since the velocity profile is parabolic, we can use the concept of the velocity profile equation for laminar flow in a circular pipe:
u = (2 * Umax / R^2) * (R^2 - r^2)
Where:
u is the velocity at a radial distance r from the centerline,
Umax is the maximum velocity at the centerline,
R is the radius of the pipe.
At R/2, the radial distance r is R/2. We are given that the velocity at R/2 is 8 m/s. Plugging these values into the equation, we can solve for Umax:
8 = (2 * Umax / (R/2)^2) * ((R/2)^2 - (R/2)^2)
Simplifying the equation:
8 = (2 * Umax / (R^2/4)) * 0
Since the term ((R/2)^2 - (R/2)^2) is equal to 0, we can see that the equation does not provide any information about Umax.
Therefore, without additional information or values for R, we cannot determine the velocity at the center of the pipe.