**Steady Laminar Flow Through a Pipe**

In fluid mechanics, the steady laminar flow through a pipe refers to a flow condition where the fluid particles move in parallel layers or laminae without any mixing or turbulence. The flow is characterized by smooth and predictable motion, with each layer of fluid moving at a different velocity.

To understand why the mean velocity of flow occurs at a radial distance of 0.707 R from the center of the pipe, we need to consider the velocity profile of the flow.

**Velocity Profile in Laminar Flow**

In laminar flow, the velocity of fluid particles varies across the cross-section of the pipe. The velocity is highest at the center of the pipe (maximum velocity) and decreases as we move towards the pipe walls (zero velocity at the walls).

The velocity profile in laminar flow is parabolic, following the Hagen-Poiseuille equation. This equation provides a relationship between the velocity and radial distance from the center of the pipe. It states that the velocity (V) at a certain radial distance (r) from the center is proportional to the square of that distance, i.e., V ∝ r^2.

**Determining the Mean Velocity**

To find the mean velocity, we need to integrate the velocity profile across the cross-section of the pipe and divide it by the total cross-sectional area.

Let's consider an elemental annular ring of radius r and thickness dr, located at a radial distance r from the center of the pipe. The area of this ring is given by dA = 2πrdr.

The velocity of fluid particles in this annular ring can be approximated as V = k*r^2, where k is a constant of proportionality.

Integrating the velocity profile across the cross-section, we get:

Mean Velocity (V_mean) = (1/A_total) * ∫V * dA

Substituting the values of V and dA, we have:

V_mean = (1/A_total) * ∫(k*r^2)*(2πrdr)

Simplifying the equation and integrating, we get:

V_mean = (1/A_total) * 2πk * ∫r^3dr

V_mean = (1/A_total) * 2πk * [r^4/4]_0^R

Where R is the radius of the pipe.

**Calculating the Mean Velocity at 0.707 R**

To find the radial distance at which the mean velocity occurs, we can set the mean velocity equation equal to the maximum velocity (V_max) and solve for r.

V_mean = V_max

(1/A_total) * 2πk * [r^4/4]_0^R = k*R^2

Simplifying the equation, we get:

(1/A_total) * 2π * R^4/4 = R^2

(1/A_total) * π * R^2 = 2

A_total/A_total = 2/π

This implies that the total cross-sectional area (A_total) is equal to half of the area of a circle with radius R.

Now, let's find the radial distance where the mean velocity occurs:

V_mean = (1/A_total) * 2πk * (r^4/4)

V_max = k * R^2

Setting V_mean equal to V

Derivation:
The velocity distribution in a circular pipe can be represented by the following equation:

u(r) = 2 * Vmax * (1 - (r/R)^2)

Where u(r) is the velocity at a radial distance of r from the center of the pipe, Vmax is the maximum velocity at the center of the pipe, and R is the radius of the pipe.

To find the mean velocity of flow, we need to integrate the velocity distribution equation over the entire cross-section of the pipe.

∫(0 to R) u(r) * 2πr dr = Q/πR^2

Where Q is the volumetric flow rate.

Simplifying the above equation, we get:

Vmean = (2/3) * Vmax

Where Vmean is the mean velocity of flow.

To find the radial distance at which the mean velocity occurs, we need to substitute Vmean in the velocity distribution equation and solve for r.

Vmean = 2 * Vmax * (1 - (r/R)^2)

(2/3) * Vmax = 2 * Vmax * (1 - (r/R)^2)

1/3 = (r/R)^2

r/R = 0.577 or 0.707

Therefore, the mean velocity of flow occurs at a radial distance of 0.707R from the center of the pipe.