A cylindrical pipe 30 cm in diameter carries water at a volumetric flow rate of 10-3 m3/s.

The pattern of the flow in the pipe ...

The pattern of the flow in the pipe can be determined based on the Reynolds number (Re). The Reynolds number is a dimensionless quantity used to predict the flow regime in a fluid flow, based on its velocity, density, viscosity, and characteristic length.

To calculate the Reynolds number, we can use the formula:

Re = (velocity x diameter) / viscosity

where:
- velocity is the average flow velocity of the water in the pipe,
- diameter is the diameter of the pipe, and
- viscosity is the dynamic viscosity of water.

Since the water flow rate is given as 10^-3 m^3/s, and the diameter of the pipe is given as 30 cm (0.3 m), let's calculate the average flow velocity:

velocity = flow rate / cross-sectional area

The cross-sectional area of the pipe can be calculated using the formula:

cross-sectional area = π * (diameter/2)^2

Let's plug in the values and calculate the average flow velocity:

cross-sectional area = π * (0.3/2)^2 = π * 0.075^2 = 0.0177 m^2

velocity = (10^-3 m^3/s) / (0.0177 m^2) = 0.056 m/s

Now, let's calculate the Reynolds number by considering the dynamic viscosity of water, which is approximately 0.001 Pa.s:

Re = (0.056 m/s * 0.3 m) / 0.001 Pa.s = 16.8 / 0.001 = 16800

The flow regime can be determined based on the Reynolds number:
- For Re < 2300, the flow is laminar.
- For 2300 < Re < 4000, the flow is transitional.
- For Re > 4000, the flow is turbulent.

In this case, with a Reynolds number of 16800, the flow in the pipe is considered turbulent.

To determine the pattern of flow in the cylindrical pipe, you need to consider the Reynolds number, which is a dimensionless quantity used to predict the flow regime.

The Reynolds number (Re) is defined as the ratio of inertial forces to viscous forces and is calculated using the following formula:

Re = (ρVD) / μ

Where:
- ρ is the density of the fluid (in kg/m³). In this case, it is the density of water, which is approximately 1000 kg/m³.
- V is the velocity of the flow (in m/s).
- D is the diameter of the pipe (in m).
- μ is the dynamic viscosity of the fluid (in Pa·s). For water at room temperature, it is about 0.001 Pa·s.

Given that the diameter of the pipe is 30 cm (or 0.3 m) and the volumetric flow rate is 10^-3 m³/s, we can calculate the velocity using the following formula:

V = Q / A

Where:
- Q is the volumetric flow rate (in m³/s).
- A is the cross-sectional area of the pipe (in m²).

Given that the diameter is 30 cm, the radius (r) is 0.15 m. Therefore, the cross-sectional area (A) is calculated as:

A = πr²

Knowing these values, we can now calculate the velocity and the Reynolds number to determine the flow pattern.

1. Calculate the cross-sectional area:
A = π(0.15)² = 0.0707 m²

2. Calculate the velocity:
V = (10^-3) / 0.0707 = 14.14 m/s

3. Calculate the Reynolds number:
Re = (1000 * 14.14 * 0.3) / 0.001 = 4,242,000

The Reynolds number obtained is very high, indicating that the flow in the pipe is turbulent. In turbulent flow, the fluid moves in irregular patterns, characterized by eddies and vortices.