Question

Match the Gradually Varied Flow (GVF) profiles on a mild slope (\(M_1, M_2, M_3\)) with the nature of their water surface slope (\(dy/dx\)) relative to the flow direction.

Profile & Nature of Slope (\(dy/dx\)) 
P. \(M_1\) Profile & 1. Positive (Rising Curve) 
Q. \(M_2\) Profile & 2. Negative (Drawdown Curve) 
R. \(M_3\) Profile & 3. Positive (Rising Curve) 

Select the correct classification: 

• A. \(M_1\) is Rising, \(M_2\) is Rising, \(M_3\) is Falling
• B. \(M_1\) is Rising, \(M_2\) is Falling, \(M_3\) is Rising
• C. \(M_1\) is Falling, \(M_2\) is Rising, \(M_3\) is Falling
• D. \(M_1\) is Rising, \(M_2\) is Falling, \(M_3\) is Falling

Hint:

For any GVF profile, remember this rule: The water surface always tends towards the normal depth line (\(y_n\)). If the flow is subcritical (\(Fr<1\), as in M1 and M2), this happens from upstream. If the flow is supercritical (\(Fr>1\), as in M3), this happens from downstream control.

Answer 1

The Correct Option is B

Step 1: Identifying the GVF Equation:
The slope of the water surface is given by: \[ \frac{dy}{dx} = S_0 \left[ \frac{1 - (y_n/y)^{10/3}}{1 - (y_c/y)^3} \right] \text{ (or more simply) } \frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} \] On a mild slope, the normal depth (\(y_n\)) is greater than the critical depth (\(y_c\)).
Step 2: Profile Zone Analysis:
M1 Profile (Zone 1): Depth \(y>y_n>y_c\). - Since \(y>y_n\), \(S_f<S_0\) (numerator is positive). - Since \(y>y_c\), flow is subcritical, \(Fr<1\) (denominator is positive). - Result: \(dy/dx>0\). The curve is Rising. M2 Profile (Zone 2): Depth \(y_n>y>y_c\). - Since \(y<y_n\), \(S_f>S_0\) (numerator is negative). - Since \(y>y_c\), flow is subcritical, \(Fr<1\) (denominator is positive). - Result: \(dy/dx<0\). The curve is Falling (Drawdown). M3 Profile (Zone 3): Depth \(y_n>y_c>y\). - Since \(y<y_n\), \(S_f>S_0\) (numerator is negative). - Since \(y<y_c\), flow is supercritical, \(Fr>1\) (denominator is negative). - Result: Negative / Negative = Positive. \(dy/dx>0\). The curve is Rising.
Step 3: Conclusion:
The profiles follow the pattern: M1 (Rising), M2 (Falling), and M3 (Rising). This matches Option (B).