Introduction:
In this problem, we are considering fully developed, laminar flow in a circular pipe of radius R. The center-line velocity of the flow is Um. We need to find the radial distance from the center-line of the pipe at which the velocity would be equal to the average velocity of the flow.

Understanding the problem:
To solve this problem, we need to understand the concept of velocity profile in laminar flow through a pipe. In laminar flow, the velocity profile is parabolic, with the maximum velocity at the center-line of the pipe and decreasing towards the pipe walls.

Solution:
To find the radial distance from the center-line of the pipe at which the velocity is equal to the average velocity, we need to determine the equation for the velocity profile and then solve for the required distance.

Step 1: Equations for velocity profile:
The velocity profile in laminar flow through a circular pipe can be described by the Hagen-Poiseuille equation:

v(r) = (2 * Um / R^2) * (R^2 - r^2)

Where,
v(r) is the velocity at a radial distance r from the center-line of the pipe,
Um is the center-line velocity, and
R is the radius of the pipe.

Step 2: Average velocity:
The average velocity is given by the formula:

Vavg = (1 / A) * ∫[0 to R] v(r) * 2πr * dr

Where,
Vavg is the average velocity,
A is the cross-sectional area of the pipe,
v(r) is the velocity at a radial distance r from the center-line of the pipe,
r is the radial distance, and
dr is the differential element of the radial distance.

Step 3: Solving for the required distance:
We need to find the radial distance r for which v(r) is equal to Vavg. Substituting Vavg into the equation for v(r), we get:

(2 * Um / R^2) * (R^2 - r^2) = Vavg

Simplifying the equation, we have:

R^2 - r^2 = (Vavg * R^2) / (2 * Um)

r^2 = R^2 - (Vavg * R^2) / (2 * Um)

Taking the square root of both sides, we get:

r = √(R^2 - (Vavg * R^2) / (2 * Um))

Step 4: Substituting values:
Since Vavg is the average velocity, it can be calculated as:

Vavg = (1 / A) * ∫[0 to R] v(r) * 2πr * dr

By substituting the equation for v(r), we can simplify this integral and solve for Vavg.

Step 5: Final answer:
By substituting the calculated value of Vavg into the equation for r, we can find the radial distance from the center-line of the pipe at which the velocity is equal to the average velocity.

Conclusion:
The value of the radial distance from the center-line of the pipe at which the velocity would be equal to the average velocity of the flow is approximately 0.707R. Therefore