What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18

Q: Why do we use 1000 and 2000 as the critical Reynolds numbers here?

While many textbooks use 2000 and 4000 as the critical values for pipe flow, the specific convention used in this problem's context defines $R_e 2000$ as turbulent. Always adapt to the specific thresholds provided or implied by the exam's standard.

Q: Why did we have to convert liters per minute to cubic meters per second?

The Reynolds number is a dimensionless quantity. To ensure all units cancel out perfectly, every parameter must be in standard SI units. Liters and minutes are not standard SI units for volume and time.

Q: Does the density of water change when the flow rate increases?

No, water is considered an incompressible fluid under normal conditions. Its density remains constant at $10^3 \text{ kg/m}^3$ regardless of the flow rate.

Q: What exactly happens physically when the flow becomes unsteady?

When the flow transitions from steady (laminar) to unsteady, the smooth, parallel layers of water start to mix. Small fluctuations and lateral velocity components begin to appear, disrupting the glassy appearance of the stream.

Q: Why did we use $D = 2r$ instead of just $r$ in the Reynolds formula?

The characteristic length scale for flow through a circular pipe is its diameter $D$, not its radius. Using the radius would result in a Reynolds number that is off by a factor of 2, leading to incorrect conclusions.

Question

What will be the nature of flow of water from a circular tap, when its flow rate increased from $0.18 	ext{ L/min}$ to $0.48 	ext{ L/min}$ ? The radius of the tap and viscosity of water are $0.5 	ext{ cm}$ and $10^{-3} 	ext{ Pa-s}$, respectively. (Density of water = $10^3 	ext{ kg/m}^3$)

Options:

(A) Unsteady to steady flow

(B) Remains steady flow

(C) Remains turbulent flow

(D) Steady flow to unsteady flow

Sigma Percentile · Accepted Answer

### The Transition of Flow: From Steady to Unsteady

Imagine a circular tap pouring out water. As you slowly open the tap, the water stream starts off smooth and glassy. But as you open it further, the flow becomes chaotic and irregular. This beautiful physical phenomenon is governed by a single dimensionless quantity: the **Reynolds Number**.

In this problem, we are given a tap with a radius of $0.5 	ext{ cm}$, and water with a viscosity of $10^{-3} 	ext{ Pa-s}$ and a density of $10^3 	ext{ kg/m}^3$. The flow rate is increased from $0.18 	ext{ L/min}$ to $0.48 	ext{ L/min}$. Our goal is to determine how the nature of the flow changes.

### The Master Equation: Reynolds Number

To determine whether a flow is steady (laminar), unsteady (transitional), or turbulent, we calculate the Reynolds number ($R_e$). It is defined as the ratio of inertial forces to viscous forces:

$$R_e = \frac{ho v D}{\eta}$$

Here, $ho$ is the density, $v$ is the velocity, $D$ is the diameter of the pipe, and $\eta$ is the dynamic viscosity.

Since we are given the volume flow rate $Q$, we can express the velocity as $v = \frac{Q}{A} = \frac{Q}{\pi r^2}$. The diameter is simply $D = 2r$. Substituting these into our formula gives us a highly useful, simplified expression:

$$R_e = \frac{ho \left(\frac{Q}{\pi r^2}ight) (2r)}{\eta} = \frac{2ho Q}{\pi r \eta}$$

### Crunching the Numbers

Before we plug in the numbers, we must ensure all units are in the standard SI system. The flow rates are given in liters per minute, which we must convert to cubic meters per second by multiplying by $10^{-3}$ and dividing by $60$.

**Initial State:**
For the initial flow rate $Q_1 = 0.18 	ext{ L/min}$:

$$R_{e1} = \frac{2 	imes 10^3 	imes \left(\frac{0.18 	imes 10^{-3}}{60}ight)}{\pi 	imes (0.5 	imes 10^{-2}) 	imes 10^{-3}}$$

Simplifying the powers of ten, we get:

$$R_{e1} = \frac{1200}{\pi} \approx 382$$

**Final State:**
For the increased flow rate $Q_2 = 0.48 	ext{ L/min}$:

$$R_{e2} = \frac{2 	imes 10^3 	imes \left(\frac{0.48 	imes 10^{-3}}{60}ight)}{\pi 	imes (0.5 	imes 10^{-2}) 	imes 10^{-3}}$$

$$R_{e2} = \frac{3200}{\pi} \approx 1019$$

### The Verdict

Now, we compare our calculated Reynolds numbers against the standard critical values for pipe flow. According to the problem's context:
- If $R_e < 1000$, the flow is **steady**.
- If $1000 < R_e < 2000$, the flow is **unsteady**.
- If $R_e > 2000$, the flow is **turbulent**.

Our initial Reynolds number is $382$, which is well below $1000$, indicating that the flow starts as **steady**. However, our final Reynolds number is $1019$, which just crosses the $1000$ threshold, pushing the flow into the **unsteady** regime.

Therefore, as the tap is opened further, the nature of the water flow changes from **steady flow to unsteady flow**.

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Visualized Solution

$Analyzing the Tap Flow$

$Reynolds Number$

$Formula in terms of Flow Rate$

$Initial Reynolds Number (R_{e 1})$

$Final Reynolds Number (R_{e 2})$

$Nature of Flow$

$The Way Forward$

The Sigma Insight: Flow of Fluid

The Transition of Flow

The Master Equation

Crunching the Numbers

The Verdict

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Tap Flow

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Reynolds Number

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Formula in terms of Flow Rate

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Initial Reynolds Number (Re1​)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Reynolds Number (Re2​)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Nature of Flow

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward