Three points `A,B`,and `C` on a steady flow of a non-viscon and inconpressible fluid are observed. The pressure, velocity and height of the point `A, B` and C are (2,3,1),(1,2,2) and (4,1,2) respectively.Density of the fluid is `1kgm^(-3)` and all other parameters are given in SI unit. Then which of the following is correct?(`g=10ms^(-2))`

Options

A. Point `A` and `B` lie on the same stream line
B. Point `B` and `C` lie on the same stream line
C. Point `C` and `A` lie on the same stream line
D. None of the above

Explanation

To solve the problem, we will apply Bernoulli's theorem, which states that for a steady flow of an incompressible and non-viscous fluid, the following equation holds:

\[ P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant} \]

Where:
- \( P \) = pressure
- \( \rho \) = density of the fluid
- \( g \) = acceleration due to gravity
- \( h \) = height
- \( v \) = velocity

Given data:
- For point A: \( P_A = 2 \, \text{Pa}, v_A = 3 \, \text{m/s}, h_A = 1 \, \text{m} \)
- For point B: \( P_B = 1 \, \text{Pa}, v_B = 2 \, \text{m/s}, h_B = 2 \, \text{m} \)
- For point C: \( P_C = 4 \, \text{Pa}, v_C = 1 \, \text{m/s}, h_C = 2 \, \text{m} \)
- Density \( \rho = 1 \, \text{kg/m}^3 \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

### Step 1: Calculate Bernoulli's equation for point A

Using the values for point A:
\[ P_A + \rho g h_A + \frac{1}{2} \rho v_A^2 = 2 + (1)(10)(1) + \frac{1}{2}(1)(3^2) \]
Calculating:
\[ = 2 + 10 + \frac{1}{2}(1)(9) \]
\[ = 2 + 10 + 4.5 = 16.5 \]

### Step 2: Calculate Bernoulli's equation for point B

Using the values for point B:
\[ P_B + \rho g h_B + \frac{1}{2} \rho v_B^2 = 1 + (1)(10)(2) + \frac{1}{2}(1)(2^2) \]
Calculating:
\[ = 1 + 20 + \frac{1}{2}(1)(4) \]
\[ = 1 + 20 + 2 = 23 \]

### Step 3: Calculate Bernoulli's equation for point C

Using the values for point C:
\[ P_C + \rho g h_C + \frac{1}{2} \rho v_C^2 = 4 + (1)(10)(2) + \frac{1}{2}(1)(1^2) \]
Calculating:
\[ = 4 + 20 + \frac{1}{2}(1)(1) \]
\[ = 4 + 20 + 0.5 = 24.5 \]

### Step 4: Compare the results

From our calculations:
- For point A: \( 16.5 \)
- For point B: \( 23 \)
- For point C: \( 24.5 \)

Since the values of Bernoulli's equation at points A, B, and C are different, it indicates that these points do not lie on the same streamline.

### Conclusion

Thus, the correct answer is that none of the points A, B, and C lie on the same streamline, which corresponds to option 4: "None of the above."

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