# Streamline & Turbulent Flow

In a streamline (or laminar) flow, the fluid flows in parallel layers with no turbulence or mixing between the layers. In streamline flow, the velocity and direction of the fluid at any point remain constant over time.

In a turbulent flow, there are irregular fluctuations in velocity, pressure, and direction. This type of flow is typically associated with high flow rates or low viscosity fluids and is characterized by a high level of energy and mixing.

The Reynolds number is the ratio of inertial forces to viscous forces within the fluid. It is given by the equation
\begin{align}
Re = \frac{\rho v d}{\mu}
\end{align}
where $\rho$ is the fluid density, $v$ is the fluid velocity, $d$ is a characteristic length scale (such as the diameter of a pipe), and $\mu$ is the fluid viscosity.

The Reynolds number is a dimensionless parameter used in fluid mechanics to predict the type of flow (either laminar or turbulent). When the Reynolds number is below a critical value, the flow is laminar. When the Reynolds number exceeds this critical value, the flow becomes turbulent. The critical Reynolds number for a given flow system depends on the geometry of the system, the fluid properties, and the flow rate.

## Problems from IIT JEE

**Problem (JEE Mains 2022):**
If $\rho$ is the density and $\eta$ is coefficient of viscosity of fluid which flows with a speed $v$ in the pipe of diameter $d$, the correct formula for Reynolds number $R_e$ is

- $R_e=\frac{\eta d}{\rho v}$
- $R_e=\frac{\rho v}{\eta d}$
- $R_e=\frac{\rho v d}{\eta}$
- $R_e=\frac{\eta}{\rho v d}$

**Problem (JEE Mains 2019):**
Water from a pipe is coming at a rate of 100 liters per minute. If the radius of the pipe is 5 cm, the Reynolds number for the flow is of the order of (density of water = 1000 kg/m^{3}, coefficient of viscosity of water = 1 mPa/s)

- $10^3$
- $10^4$
- $10^2$
- $10^6$

**Problem (JEE Mains 2021):**
What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18 litre/min to 0.48 litre/min? The radius of the tap and viscosity of water are 0.5 cm and $10^{-3}$ Pa-s, respectively. (density of water is 1000 kg/m^{3})

- steady flow to unsteady flow
- unsteady to steady flow
- remains steady flow
- remains turbulent flow

## Related

- Viscosity and Poiseuille's Law
- Stokes' Law and Terminal Velocity
- Equation of Continuity
- Bernoulli's Theorem and its Applications