Question

Gradually Varied Flow (GVF) profiles in open channels given in Column 1 are to be matched with the water surface slopes in Column 2 in the table below.

Column 1 (GVF Profile)	Column 2 (Water Surface Slope)
(P) M1	(I) Positive
(Q) M2	(II) Negative
(R) M3	(III) Zero

Which of the following options is/are NOT correct?

• A. (P) - (I); (Q) - (II); (R) - (I)
• B. (P) - (I); (Q) - (II); (R) - (III)
• C. (P) - (II); (Q) - (I); (R) - (I)
• D. (P) - (I); (Q) - (III); (R) - (II)

Hint:

Use the GVF sign rule \(dy/dx=(S_0-S_f)/(1-Fr^2)\): M1 and M3 give a positive slope, M2 gives a negative slope; check that against each option.

Answer 1

The Correct Option is B, C, D

Step 1: Write the governing GVF equation.
For gradually varied flow in a prismatic channel, the slope of the water surface relative to the channel bed is given by:
\[ \frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} \]
where \(S_0\) is the bed slope, \(S_f\) is the friction slope (energy loss slope corresponding to the actual depth \(y\)), and \(Fr\) is the Froude number at depth \(y\). The sign of \(dy/dx\) tells us whether the water surface profile rises (positive), falls (negative), or stays parallel to the bed (zero) as we move in the direction of flow.

Step 2: Fix the depth ordering for a mild channel.
All three profiles M1, M2, M3 occur on a mild slope, where the normal depth \(y_n\) is greater than the critical depth \(y_c\) (\(y_n > y_c\)). By definition: M1 zone: \(y > y_n > y_c\) (depth above both normal and critical depth). M2 zone: \(y_c < y < y_n\) (depth between critical and normal depth). M3 zone: \(y < y_c < y_n\) (depth below critical depth).

Step 3: Evaluate the sign of \(dy/dx\) for M1.
In the M1 zone, \(y > y_n\), so the actual depth is larger than normal depth, meaning the flow is slower than normal, so the friction slope \(S_f\) (which decreases as depth increases) is smaller than the bed slope \(S_0\); hence \(S_0 - S_f > 0\). Also \(y > y_c\) means the flow is subcritical, so \(Fr < 1\) and \(1 - Fr^2 > 0\). A positive numerator over a positive denominator gives \(dy/dx > 0\): M1 has a positive water surface slope. This matches (P) - (I).

Step 4: Evaluate the sign of \(dy/dx\) for M2.
In the M2 zone, \(y < y_n\), so the friction slope \(S_f\) (higher at shallower depth, for the same discharge) is larger than the bed slope \(S_0\); hence \(S_0 - S_f < 0\). Since \(y > y_c\), the flow is still subcritical, \(Fr < 1\), so \(1 - Fr^2 > 0\). A negative numerator over a positive denominator gives \(dy/dx < 0\): M2 has a negative water surface slope. This matches (Q) - (II).

Step 5: Evaluate the sign of \(dy/dx\) for M3.
In the M3 zone, \(y < y_n\) as well, so as in M2, \(S_f > S_0\) and \(S_0 - S_f < 0\). But here \(y < y_c\), so the flow is supercritical, \(Fr > 1\), making \(1 - Fr^2 < 0\). A negative numerator over a negative denominator gives \(dy/dx > 0\): M3 also has a positive water surface slope, not zero. This matches (R) - (I), not (R) - (III) and not (R) - (II).

Step 6: Check each option against the correct mapping.
The correct matching is (P) - (I), (Q) - (II), (R) - (I), which is exactly option (A). So option (A) is correct. Option (B), (P) - (I); (Q) - (II); (R) - (III), wrongly assigns M3 a zero slope; it is NOT correct. Option (C), (P) - (II); (Q) - (I); (R) - (I), swaps the signs for M1 and M2; it is NOT correct. Option (D), (P) - (I); (Q) - (III); (R) - (II), wrongly assigns M2 a zero slope and M3 a negative slope; it is NOT correct.

Step 7: Final Answer.
Options (B), (C), and (D) are the ones that are NOT correct.
\[ \boxed{\text{Options (B), (C), (D) are not correct}} \]