Fluid Mechanics

The formula for critical depth in an open channel with a triangular cross-section is given by: Where: - = discharge - = acceleration due to gravity This formula is derived from the specific energy considerations of open channel flow.
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The sequent depth ratio for a hydraulic jump is related to the Froude number of the supercritical flow by the equation: Given: Using the sequent ratio equation and solving for , we get:
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The equation for critical flow is given by: This equation represents the condition for the critical flow in an open channel where: - = discharge - = cross-sectional area - = acceleration due to gravity When this equation holds true, the flow is considered to be critical.
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Step 1: Understanding the Dimensionless Numbers and Forces.
- (A) Euler's number: Euler's number is often used in fluid mechanics to describe the behavior of fluid flows under certain conditions, but it is not directly associated with any particular force type. - (B) Froude's number: The Froude number is a dimensionless number used to describe the influence of gravity forces in fluid flow. - (C) Mach number: The Mach number compares the speed of an object to the speed of sound in the surrounding medium. It is often associated with compressibility effects. - (D) Weber number: The Weber number relates to the ratio of inertial forces to surface tension forces in fluid dynamics.
Step 2: Matching the Forces.
- (A) Euler's number: Matches with (I) Pressure force. - (B) Froude's number: Matches with (II) Gravity force. - (C) Mach number: Matches with (IV) Compressibility force. - (D) Weber number: Matches with (III) Surface Tension.
Step 3: Conclusion.
The correct matching is (1) (A) - (I), (B) - (II), (C) - (IV), (D) - (III).

Step 1: Formula for Kinematic Viscosity.
The kinematic viscosity is given by the formula: Where: - is the kinematic viscosity (in ) - is the dynamic viscosity (in or poise) - is the density (in )
Step 2: Converting units.
- The given dynamic viscosity poise. - , so: - The specific gravity , and density .
Step 3: Calculating Kinematic Viscosity.
Using the formula for kinematic viscosity:
Step 4: Conclusion.
Therefore, the correct kinematic viscosity is , and the correct answer is (1).

Step 1: Understanding the Problem.
The beam is subjected to a uniformly distributed load of 120 kN. The length of the cantilever beam is 3 m. To find the support reactions, we need to calculate the vertical reaction and the moment at the fixed en(D)
Step 2: Calculating the Vertical Reaction.
The total vertical load on the beam is given as 120 kN. Since the beam is in static equilibrium, the vertical reaction at the fixed support must equal the total load:
Step 3: Calculating the Moment Reaction.
The moment reaction at the fixed end can be found by considering the moment equilibrium about the fixed en(D) For a uniformly distributed load, the moment at the fixed end is calculated as: The load is uniformly distributed over the beam, so the centroid of the load is at the midpoint of the beam, which is at 1.5 m. Thus, the moment at the fixed end is:
Step 4: Conclusion.
Thus, the support reactions at the fixed end are 120 kN vertical and 180 kN-m moment. Therefore, the correct answer is option (2).

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Step 1: Understanding Shear Stress Calculation.
The formula for shear stress ( ) at the neutral axis in a beam is given by: Where: - = Shear force = 50 kN = 50,000 N - = Area of cross-section The triangular cross-section area is: Converting area to m , we get:
Step 2: Calculating Shear Stress.
Now, we calculate the shear stress:
Step 3: Conclusion.
Therefore, the shear stress at the neutral axis is 3.2 N/mm , making option (2) the correct answer.

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Step 1: Understanding the Failure Theories.
The maximum principal stress theory is considered the most conservative failure theory for ductile materials. This theory assumes that failure occurs when the maximum principal stress reaches the material's ultimate tensile strength.
Step 2: Conclusion.
Thus, the most conservative failure theory for ductile materials is the maximum principal stress theory, making option (1) the correct answer.

Final Answer:

Step 1: Understanding the fluids and their characteristics.
- (A) Newtonian fluid: A fluid where the relationship between shear stress and velocity gradient is linear, with constant viscosity, and a zero velocity gradient at no flow. - (B) Non-Newtonian fluid: The relationship between shear stress and velocity gradient is non-linear, and they have a definite yield stress. - (C) Ideal Fluid: This fluid is assumed to have a linear relationship between shear stress and velocity gradient, but is an idealized concept. - (D) Ideal Plastic: For this fluid, the relationship is non-linear, with a definite yield stress.
Step 2: Matching the relationships.
- (A) Newtonian fluid: Matches with (I) Zero velocity gradient (because at no flow, the velocity gradient is zero). - (B) Non-Newtonian fluid: Matches with (IV) Non-linear relationship. - (C) Ideal Fluid: Matches with (III) Linear relationship. - (D) Ideal Plastic: Matches with (II) Non-linear with definite yield stress.
Step 3: Conclusion.
Therefore, the correct matching is (1) (A) - (I), (B) - (IV), (C) - (III), (D) - (II).