Answer:

Introduction:
In fluid flow, the average velocity is the average value of the velocity at different points within the flow field. On the other hand, the maximum velocity is the highest velocity that occurs within the flow field. The ratio of average velocity to maximum velocity can provide insights into the distribution of velocities within the flow. In the case of steady laminar flow in circular pipes, the ratio is equal to 1/2.

Explanation:

Steady Laminar Flow in Circular Pipes:
Steady laminar flow refers to a type of flow where the fluid particles move in parallel layers, with no mixing or turbulence. Circular pipes are a common conduit for fluid flow, and the flow characteristics can be analyzed using the Hagen-Poiseuille equation.

Hagen-Poiseuille Equation:
The Hagen-Poiseuille equation describes the flow of a viscous fluid in a circular pipe. It relates the flow rate (Q) to the pressure gradient (∆P), pipe radius (r), fluid viscosity (μ), and pipe length (L). The equation is given as:

Q = (π * r^4 * ∆P) / (8 * μ * L)

where:
Q = flow rate
π = pi (approximately 3.14159)
r = pipe radius
∆P = pressure gradient
μ = fluid viscosity
L = pipe length

Relation between Maximum Velocity and Average Velocity:
In steady laminar flow, the velocity profile within a circular pipe is parabolic. This means that the velocity is zero at the pipe walls and reaches a maximum value at the center of the pipe. The average velocity is calculated by dividing the flow rate by the cross-sectional area of the pipe.

The maximum velocity occurs at the center of the pipe, where the velocity profile is at its peak. The average velocity, on the other hand, takes into account the velocities at all points within the flow field. Since the velocity profile is parabolic, the average velocity is lower than the maximum velocity.

Ratio of Average Velocity to Maximum Velocity:
To find the ratio of average velocity to maximum velocity, we need to calculate the average velocity and maximum velocity separately.

Average Velocity:
The average velocity can be calculated using the formula:
V_avg = Q / A
where:
V_avg = average velocity
Q = flow rate
A = cross-sectional area of the pipe

Maximum Velocity:
The maximum velocity occurs at the center of the pipe, where the velocity profile is parabolic. The maximum velocity (V_max) can be calculated using the formula:
V_max = 2 * V_avg

Calculation:
Let's assume that the flow rate (Q) and pipe radius (r) are known.

Average Velocity:
The average velocity can be calculated using the formula:
V_avg = Q / A
where:
A = π * r^2 (cross-sectional area of the pipe)

Maximum Velocity:
The maximum velocity (V_max) can be calculated using the formula:
V_max = 2 * V_avg

Ratio of Average Velocity to Maximum Velocity:
The ratio of average velocity to maximum velocity can be calculated as:
V_avg / V_max = (Q / A) / (2 * (Q / A))
= 1 / 2

Therefore, the ratio of average velocity to maximum