Q: Hydraulic power transmission in a pipeline: if inlet pressure is 90 kg/cm² and the pressure drop along the line is 10 kg/cm², what is the efficiency of transmission (based on head ratio)?

Introduction / Context: In long pipeline power transmission, useful head at the outlet is reduced by frictional losses. A common performance metric is the efficiency of transmission, defined as the ratio of the available head (or pressure) at delivery to the head supplied at the inlet, ignoring mechanical conversion losses at the receiver. Given Data / Assumptions: Inlet pressure = 90 kg/cm² (proportional to head supplied). Pressure drop along the line = 10 kg/cm². Steady incompressible flow with negligible elevation change and no additional minor losses considered beyond the stated drop. Concept / Approach: Efficiency of transmission η_t is typically taken as η_t = Head at outlet / Head at inlet = (H_in − h_loss) / H_in. With pressures used as head proxies for the same fluid, η_t = (P_in − ΔP) / P_in. Step-by-Step Solution: Compute outlet pressure: P_out = 90 − 10 = 80 kg/cm².Compute efficiency: η_t = P_out / P_in = 80 / 90.Evaluate the fraction: 80/90 = 8/9 ≈ 0.888... .Convert to percent: η_t ≈ 88.8%. Verification / Alternative check: This matches the general optimum-power result where the maximum transmitted power occurs when h_loss = H/3 (η_t = 2/3 ≈ 66.7%). Since here h_loss/H = 10/90 ≈ 0.111 < 1/3, the efficiency is higher than 66.7%, consistent with 88.8%. Why Other Options Are Wrong: (a) 66.6% corresponds to the special case of maximum power, not this scenario. (b) and (c) arise from incorrect ratios. (e) is unrelated to the given numbers. Common Pitfalls: Mixing gauge and absolute pressures (here only differences matter); confusing efficiency of transmission with turbine or pump efficiency. Final Answer: 88.8%

Q: Energy grade concepts: the Total Energy Line (TEL) for steady incompressible flow represents the sum of which heads?

Introduction / Context: In pipeline and open-channel analysis, the Energy Grade Line (EGL or TEL) and Hydraulic Grade Line (HGL) visualize how energy varies along the flow. Correctly identifying what the Total Energy Line includes is essential for head-loss calculations and pump/turbine sizing. Given Data / Assumptions: Steady, incompressible flow of a single-phase fluid. Bernoulli equation applies along a streamline (with pump/turbine and losses as appropriate). Elevation is measured from a chosen datum plane. Concept / Approach: The Total Energy Line (TEL) plots the total head H_total = p/(ρ g) + V^2/(2 g) + z at points along the system. The Hydraulic Grade Line (HGL) is the sum of pressure head and elevation head only, HGL = p/(ρ g) + z. The vertical distance between the TEL and HGL equals the velocity head, V^2/(2 g). Step-by-Step Solution: Start from Bernoulli: p1/(ρ g) + V1^2/(2 g) + z1 + h_p − h_t − h_f = p2/(ρ g) + V2^2/(2 g) + z2.Identify TEL at a point as p/(ρ g) + V^2/(2 g) + z.Identify HGL at a point as p/(ρ g) + z (TEL minus velocity head).Recognize head losses drop the TEL downstream in the direction of flow. Verification / Alternative check: Plot a simple pipeline: TEL slopes downward by h_f between sections; when pipe diameter changes, a step occurs in TEL due to velocity head change, while HGL may step differently depending on pressure change. Why Other Options Are Wrong: (a), (b), and (c) list individual components only; TEL is the sum of all three. (e) contradicts standard head plotting practice used in hydraulics and fluid mechanics. Common Pitfalls: Confusing TEL with HGL; forgetting to include pump head additions or turbine head extractions; misplacing the datum. Final Answer: All of the above summed (pressure + velocity + elevation)

Q: Velocity field classification in two-dimensional flow: for u = a x and v = b y, what is the point called where the local velocity becomes zero?

Introduction / Context: In fluid kinematics, understanding the velocity field helps identify special locations such as stagnation points, saddle points, and vortical centers. These points control streamlines and pressure distributions and are fundamental in potential-flow theory and CFD post-processing. Given Data / Assumptions: Two-dimensional velocity components u = a x and v = b y in Cartesian coordinates. a and b are constants; incompressibility is not assumed unless a + b = 0. Steady flow field; no body forces considered for this kinematic classification. Concept / Approach: A stagnation point is defined as a point in the flow where the local fluid velocity vector is zero. For the given linear field, the velocity components vanish when x = 0 and y = 0 simultaneously, i.e., at the origin. The term “stationary point” is a general mathematical phrase; in fluid mechanics, the precise term for zero velocity is “stagnation point.” Step-by-Step Solution: Set u = a x = 0 ⇒ x = 0.Set v = b y = 0 ⇒ y = 0.Therefore, the unique location with u = v = 0 is at (x, y) = (0, 0).Classify that location: by definition, it is a stagnation point. Verification / Alternative check: Compute speed V = √(u^2 + v^2) = √((a x)^2 + (b y)^2). This is zero only when x = 0 and y = 0, confirming the classification. Why Other Options Are Wrong: (a) “Critical point” is ambiguous without additional context; (b) “neutral point” is not standard velocity-field terminology; (d) “stationary point” is mathematical jargon but lacks the specific fluid meaning; (e) incorrect because a precise standard term exists. Common Pitfalls: Confusing stagnation with separation points; forgetting that pressure is typically maximum at stagnation in inviscid steady flow due to Bernoulli’s principle along streamlines. Final Answer: stagnation point

Q: Dynamic viscosity (μ): select the correct SI unit.

Introduction / Context: Dynamic viscosity quantifies a fluid’s resistance to shear deformation and appears in Newton’s law of viscosity, τ = μ (du/dy). Correct units prevent errors in calculations of laminar flow, boundary layers, and Reynolds numbers. Given Data / Assumptions: Newton’s law: shear stress τ has units of force per area (N/m²). Velocity gradient du/dy has units of s^-1. Standard SI coherent unit system is used. Concept / Approach: From τ = μ (du/dy), rearrange to μ = τ / (du/dy). Substituting dimensions gives μ units = (N/m²) / (s^-1) = N·s/m². Since N/m² is a Pascal (Pa), μ is equivalently expressed as Pa·s. This is distinct from kinematic viscosity ν = μ/ρ with units m²/s. Step-by-Step Solution: Identify τ units: N/m².Identify du/dy units: s^-1.Compute μ units: (N/m²)/(s^-1) = N·s/m² = Pa·s.Relate to base units: N = kg·m/s² ⇒ μ = (kg·m/s²)·s / m² = kg/(m·s). Verification / Alternative check: For water at 20°C, μ ≈ 1.002 × 10^-3 Pa·s. Converting to base SI gives ≈ 1.002 × 10^-3 kg/(m·s), consistent with the derivation. Why Other Options Are Wrong: (a) kg·s/m² is not dimensionally equivalent to kg/(m·s); it inverts time. (c) N·s²/m³ mismatches stress-rate dimensions. (d) m²/s is kinematic viscosity ν, not μ. (e) kg/m³ is density, another property entirely. Common Pitfalls: Confusing Pa·s with m²/s; mixing SI with CGS (poise, centipoise: 1 cP = 1 mPa·s). Final Answer: Newton·second per square metre (N·s/m² = Pa·s)

Q: In flow measurement practice, a stepped notch is essentially a combination of which type(s) of basic notches to extend the measurement range?

Introduction / Context: Laboratory flumes and small channels often require a single gauge to measure wide ranges of discharge. A stepped notch is a practical device that changes effective crest length with head, enabling better sensitivity at low flows while accommodating higher flows without drowning the notch. Given Data / Assumptions: Steady, free-flow conditions over a sharp crest (fully aerated nappe). Steps are arranged vertically so progressively higher steps engage as head increases. Objective is to vary effective rectangular width with stage. Concept / Approach: A stepped notch is constructed from a series of rectangular openings at different elevations, each “step” adding rectangular width as the head rises. This provides discrete increments in effective length b used in Q = C_d * b * √(2 g) * H^(3/2), improving resolution across a range of flows. It is different from a compound notch that deliberately mixes triangular and rectangular openings in one crest to achieve a single continuous head–discharge law. Step-by-Step Solution: Identify the aim: extend range and sensitivity without changing the weir plate.Mechanism: stacked rectangular apertures activate sequentially with increasing H.Therefore the stepped notch is a combination of rectangular notches of different sizes/levels. Verification / Alternative check: Most textbooks describe stepped notches as “multiple rectangular notches placed one above another.” Compound (or combination) notches, on the contrary, mix V and rectangular shapes in the same crest opening. Why Other Options Are Wrong: (b) and (e) misidentify the geometry. (c) describes a compound notch, not a stepped notch. (d) cannot be correct because all listed variants do not describe the same device. Common Pitfalls: Confusing “stepped” with “compound.” The former stacks similar rectangular openings; the latter blends dissimilar notch shapes. Final Answer: Rectangular notches of different sizes (at different steps)

Q: For the most economical (best hydraulic) trapezoidal open-channel section, which geometric condition must hold true? Choose the classical proportionality used in design.

Introduction / Context: The “most economical” or “best hydraulic” section minimizes wetted perimeter for a given area, thereby maximizing hydraulic radius and reducing friction losses. For trapezoidal channels used in irrigation and drainage, a celebrated condition ties the top width and the sloping side length. Given Data / Assumptions: Prismatic trapezoidal channel with side slope z:1 (horizontal:vertical). Goal: minimize wetted perimeter P for a given flow area A. Steady, uniform open-channel flow concept (e.g., Manning/Chezy context). Concept / Approach: Using calculus of variations or standard derivations, the conditions for the best trapezoidal section are: (1) hydraulic radius R = A/P = y/2, and (2) half of the top width equals the sloping side length. This condition directly shapes the cross section and simplifies design checks. Step-by-Step Solution: Let b be bed width, y depth, and sloping side length s = √(y^2 + (z y)^2) = y √(1 + z^2).Top width T = b + 2 z y ⇒ T/2 = b/2 + z y.Optimality yields T/2 = s (i.e., half top width equals one sloping side length). Verification / Alternative check: Applying the condition returns the known result R = y/2 at optimum, consistent with minimal P for a given A. Why Other Options Are Wrong: (a) and (c) are not general optimality conditions. (e) is not a standard best-section identity. Only (b) matches the classical economical trapezoid result. Common Pitfalls: Confusing the optimal trapezoid with the rectangular or semicircular best sections; each has different characteristic ratios. Final Answer: Half of the top width equals one sloping side length

Q: Fire hose nozzles are generally designed with which outlet shape to achieve a high-velocity jet at subsonic conditions for water?

Introduction / Context: Firefighting requires long-reach, coherent water jets to deliver momentum to targets. Nozzle geometry controls pressure-to-velocity conversion and jet quality. The appropriate shape depends on whether the working fluid is compressible and on the speed regime. Given Data / Assumptions: Fluid: water (essentially incompressible at these conditions). Subsonic speeds; no compressible-flow choking phenomena. Objective: maximize jet velocity and coherence. Concept / Approach: For incompressible subsonic flows, a convergent nozzle reduces cross-sectional area to convert pressure energy into kinetic energy (Bernoulli principle). Convergent–divergent designs are used for compressible gases to achieve supersonic speeds across a throat; this is unnecessary and detrimental for liquid water jets. A purely divergent section would decelerate the flow and degrade the jet. Step-by-Step Solution: Start with upstream pressure p_1 and velocity V_1 in the hose.Convergent reduction to area A_2 increases V_2 while reducing static pressure (p_2).Select a smooth bell mouth/contour to minimize separation and energy losses for a stable, coherent jet. Verification / Alternative check: Practical firefighting nozzles and hydraulic texts recommend smooth convergent contours with slight straightening sections to preserve jet coherence and minimize turbulence at the exit. Why Other Options Are Wrong: Divergent (a) causes diffusion (velocity drop). Convergent–divergent (c) is for compressible supersonic gases. Cylindrical (d) offers less acceleration and may produce a weaker jet. A mere “Venturi throat only” (e) is not how water jets are produced at hose tips. Common Pitfalls: Confusing gas nozzle design with liquid jet production; liquids do not require a divergent section to accelerate. Final Answer: Convergent

Q: The line joining the heights to which water would rise in vertical piezometer tubes fitted at different sections of a pressurized conduit is called what?

Introduction / Context: Visualizing energy distribution in pressurized pipe systems uses two companion curves: the Hydraulic Grade Line (HGL) and the Energy Grade Line (EGL). Piezometric tubes (or pressure taps plus elevation) reveal the HGL directly as water levels at points along a pipeline. Given Data / Assumptions: Pressurized, steady, incompressible flow in a closed conduit. Neglect air entrainment; tubes measure static head (p/γ + z). No pump/turbine stations between measurement points. Concept / Approach: The height to which water rises in a piezometer equals the piezometric head (pressure head + elevation head). The locus of these heights is known as the Hydraulic Grade Line (also called piezometric line or pressure grade line). “Hydraulic gradient” strictly refers to the slope of the HGL, not the line itself. Step-by-Step Solution: At a section: H_p = p/γ + z (ignoring velocity head in piezometer height).Connect H_p values across sections to obtain the HGL.Recognize that EGL = HGL + V^2/(2 g). Verification / Alternative check: In a uniform pipe with friction, HGL slopes downward in the flow direction; pumps cause a jump up, while turbines cause a drop, aligning with standard energy accounting. Why Other Options Are Wrong: (a) “Hydraulic gradient” names the slope, not the line. (e) “All of the above” would include (a), which is not a name for the line itself in strict usage; hence the best single term is HGL (d). (b) and (c) are synonyms for HGL, but only one option should be chosen; the standard engineering term is the Hydraulic Grade Line. Common Pitfalls: Confusing HGL with EGL; the difference is the velocity head. Final Answer: Hydraulic grade line

Question 1

Froude number classification in open-channel flow: when is the flow subcritical (tranquil)?

Accepted Answer

Introduction / Context: The Froude number Fr is a key dimensionless parameter in open-channel hydraulics used to classify flow regimes. Correctly identifying subcritical, critical, and supercritical states guides design decisions for channels, spillways, and energy dissipation structures. Given Data / Assumptions: Prismatic channel with steady, uniform flow conceptually. Hydraulic depth D_h = A/T (area over top width) used for wide channels; for rectangular sections, D_h ≈ flow depth y. Froude number defined as Fr = V / √(g * D_h). Concept / Approach: Flow classification is based on the relative importance of inertia to gravity waves. When Fr < 1, surface gravity waves can propagate upstream, and the flow is subcritical (tranquil). At Fr = 1, flow is at the critical state. When Fr > 1, the flow is supercritical (rapid), dominated by inertia; disturbances cannot move upstream. Step-by-Step Solution: Compute Fr = V / √(g * D_h).Compare Fr to unity: Fr < 1 indicates subcritical; Fr = 1 indicates critical; Fr > 1 indicates supercritical.Infer engineering implications: hydraulic jumps occur when supercritical transitions to subcritical, requiring energy dissipation considerations. Verification / Alternative check: For a wide rectangular channel, substitute D_h ≈ y. Example: if V = 0.8 m/s and y = 1.0 m, g = 9.81 m/s^2, then Fr ≈ 0.8 / 3.13 ≈ 0.26 < 1, confirming subcritical flow. Why Other Options Are Wrong: (b) Fr = 1.0 defines the critical condition, not subcritical. (c) Fr > 1.0 is supercritical. (d) and (e) are inconsistent with the standard definition and would misclassify many flows. Common Pitfalls: Using geometric depth instead of hydraulic depth in irregular sections; ignoring that local accelerations can change Fr along a transition. Final Answer: less than 1.0

Question 2

Weirs with end contractions: compared to the channel width, how does the effective weir length (crest width) change?

Accepted Answer

Introduction / Context: When a sharp-crested rectangular weir is placed in a channel and is not extended to the sidewalls, flow contracts at the ends. This phenomenon is known as end contraction and must be accounted for when determining effective weir length in discharge calculations. Given Data / Assumptions: Sharp-crested rectangular weir set in a channel with clearances to the sidewalls (contracted weir). Free, ventilated nappe and standard approach conditions. Head measured at a prescribed distance upstream to minimize velocity head effects. Concept / Approach: End contractions reduce the effective length through which water passes. Empirical formulations correct the nominal crest length b by subtracting a quantity proportional to the head H, such as L_e = b − 0.1nH (with n the number of end contractions, often 2). Hence, the effective length is less than the channel width, and discharge is computed with L_e rather than the full channel width. Step-by-Step Solution: Identify whether the weir is fully suppressed (no end contraction) or contracted.For contracted weirs, apply an effective length smaller than the physical crest length.Use the appropriate discharge equation Q = C_d * (2/3) * L_e * √(2g) * H^(3/2), inserting L_e < channel width. Verification / Alternative check: Compare to a suppressed weir (crest flush with sidewalls): there, L_e equals the channel width and no end-contraction correction is required. The difference confirms that with end contractions, L_e is necessarily smaller. Why Other Options Are Wrong: (a) Holds only for fully suppressed weirs. (c) Half-width is arbitrary and generally incorrect. (d) Incorrect given standard practice. (e) Physically impossible with end contractions. Common Pitfalls: Neglecting to subtract contraction corrections; measuring head too close to the crest; not ventilating the nappe. Final Answer: It is less than the channel width due to end contractions.

Question 3

Hydraulic power transmission in a pipeline: if inlet pressure is 90 kg/cm² and the pressure drop along the line is 10 kg/cm², what is the efficiency of transmission (based on head ratio)?

Accepted Answer

Introduction / Context: In long pipeline power transmission, useful head at the outlet is reduced by frictional losses. A common performance metric is the efficiency of transmission, defined as the ratio of the available head (or pressure) at delivery to the head supplied at the inlet, ignoring mechanical conversion losses at the receiver. Given Data / Assumptions: Inlet pressure = 90 kg/cm² (proportional to head supplied). Pressure drop along the line = 10 kg/cm². Steady incompressible flow with negligible elevation change and no additional minor losses considered beyond the stated drop. Concept / Approach: Efficiency of transmission η_t is typically taken as η_t = Head at outlet / Head at inlet = (H_in − h_loss) / H_in. With pressures used as head proxies for the same fluid, η_t = (P_in − ΔP) / P_in. Step-by-Step Solution: Compute outlet pressure: P_out = 90 − 10 = 80 kg/cm².Compute efficiency: η_t = P_out / P_in = 80 / 90.Evaluate the fraction: 80/90 = 8/9 ≈ 0.888... .Convert to percent: η_t ≈ 88.8%. Verification / Alternative check: This matches the general optimum-power result where the maximum transmitted power occurs when h_loss = H/3 (η_t = 2/3 ≈ 66.7%). Since here h_loss/H = 10/90 ≈ 0.111 < 1/3, the efficiency is higher than 66.7%, consistent with 88.8%. Why Other Options Are Wrong: (a) 66.6% corresponds to the special case of maximum power, not this scenario. (b) and (c) arise from incorrect ratios. (e) is unrelated to the given numbers. Common Pitfalls: Mixing gauge and absolute pressures (here only differences matter); confusing efficiency of transmission with turbine or pump efficiency. Final Answer: 88.8%

Question 4

Hydrostatics concept check: what is the correct definition of the centre of buoyancy for a floating or submerged body?

Accepted Answer

Introduction / Context: The behaviour of floating and submerged bodies—stability, trim, and righting moments—depends on the relative locations of three key points: the centre of gravity (G), the centre of buoyancy (B), and for floats/ships, the metacentre (M). Understanding what B represents is fundamental to naval architecture and hydraulic design. Given Data / Assumptions: Static conditions in a homogeneous, incompressible fluid. Body either fully submerged or floating in equilibrium. Gravity acts uniformly; pressure distribution is hydrostatic. Concept / Approach: Buoyant force equals the weight of the displaced fluid and acts upward through a specific point. That point is the centroid of the displaced fluid volume, known as the centre of buoyancy B. It is not the body’s centroid (unless densities and shapes cause coincidence by chance) nor the centre of pressure, which pertains to distributed forces on surfaces and depends on the second moment of area and depth. Step-by-Step Solution: Identify displaced fluid volume when the body is submerged or partially immersed.Compute the geometric centroid of that displaced volume; this point is B.Place the buoyant force vector upward through B with magnitude ρ g * Volume displaced.Compare B with G to assess stability; for small heel angles, stability depends on M being above G. Verification / Alternative check: Archimedes’ principle provides magnitude of buoyant force; hydrostatic pressure integration over the wetted surface shows resultant passes through the centroid of displaced volume, confirming the definition. Why Other Options Are Wrong: (a) Centre of gravity of the body depends on mass distribution, not displaced liquid. (c) Centre of pressure is a different hydrostatic resultant point on a surface, not the line of action of buoyancy in general. (d) and (e) are unrelated or misleading descriptions. Common Pitfalls: Confusing B with G; assuming B is fixed (it shifts as the submerged geometry changes during heel or sinkage). Final Answer: The centroid of the displaced fluid volume.

Question 5

Energy grade concepts: the Total Energy Line (TEL) for steady incompressible flow represents the sum of which heads?

Accepted Answer

Introduction / Context: In pipeline and open-channel analysis, the Energy Grade Line (EGL or TEL) and Hydraulic Grade Line (HGL) visualize how energy varies along the flow. Correctly identifying what the Total Energy Line includes is essential for head-loss calculations and pump/turbine sizing. Given Data / Assumptions: Steady, incompressible flow of a single-phase fluid. Bernoulli equation applies along a streamline (with pump/turbine and losses as appropriate). Elevation is measured from a chosen datum plane. Concept / Approach: The Total Energy Line (TEL) plots the total head H_total = p/(ρ g) + V^2/(2 g) + z at points along the system. The Hydraulic Grade Line (HGL) is the sum of pressure head and elevation head only, HGL = p/(ρ g) + z. The vertical distance between the TEL and HGL equals the velocity head, V^2/(2 g). Step-by-Step Solution: Start from Bernoulli: p1/(ρ g) + V1^2/(2 g) + z1 + h_p − h_t − h_f = p2/(ρ g) + V2^2/(2 g) + z2.Identify TEL at a point as p/(ρ g) + V^2/(2 g) + z.Identify HGL at a point as p/(ρ g) + z (TEL minus velocity head).Recognize head losses drop the TEL downstream in the direction of flow. Verification / Alternative check: Plot a simple pipeline: TEL slopes downward by h_f between sections; when pipe diameter changes, a step occurs in TEL due to velocity head change, while HGL may step differently depending on pressure change. Why Other Options Are Wrong: (a), (b), and (c) list individual components only; TEL is the sum of all three. (e) contradicts standard head plotting practice used in hydraulics and fluid mechanics. Common Pitfalls: Confusing TEL with HGL; forgetting to include pump head additions or turbine head extractions; misplacing the datum. Final Answer: All of the above summed (pressure + velocity + elevation)

Question 6

Definition threshold for a “large orifice”: choose the correct head-to-diameter criterion used to classify an orifice as large (versus small).

Accepted Answer

Introduction / Context: Orifices are categorized as “small” or “large” depending on the head of liquid relative to the orifice size. This classification determines whether the velocity can be assumed uniform across the opening and which analytical approach to use in discharge calculations. Given Data / Assumptions: Sharp-edged orifice discharging under a head H. Diameter (or characteristic dimension) of orifice = d. Steady incompressible flow, neglecting viscosity in first-order classification. Concept / Approach: For a small orifice, the head is large compared with the orifice size, so the variation of velocity of approach across the opening is negligible and the jet is assumed to have nearly uniform velocity. The conventional rule of thumb is that an orifice is “small” when H ≥ 5d. Conversely, it is considered “large” when the head is not sufficiently greater than the orifice size—often expressed as H < 5d—so velocity and pressure vary appreciably over the opening and more detailed analysis (e.g., subdividing the opening) is required. Step-by-Step Solution: Compare the given head H to the orifice size d.Apply the rule: small if H ≥ 5d; large if H < 5d.Conclude: an orifice is called “large” when H is less than five times its diameter. Verification / Alternative check: Handbooks and standard hydraulics texts adopt the 5d criterion, sometimes with minor variations depending on orifice shape; the engineering implication remains the same. Why Other Options Are Wrong: (b) H > 10d defines a very “small” orifice regime. (c) An equality threshold is not the classification rule by itself. (d) Head dependence is central to the definition. (e) H < d is too restrictive; many large orifices operate with H between d and 5d. Common Pitfalls: Assuming a single discharge coefficient works for all heads; ignoring that non-uniform velocity distribution increases with larger relative orifice size. Final Answer: Head H is less than five times the orifice diameter d (H < 5d).

Question 7

Velocity field classification in two-dimensional flow: for u = a x and v = b y, what is the point called where the local velocity becomes zero?

Accepted Answer

Introduction / Context: In fluid kinematics, understanding the velocity field helps identify special locations such as stagnation points, saddle points, and vortical centers. These points control streamlines and pressure distributions and are fundamental in potential-flow theory and CFD post-processing. Given Data / Assumptions: Two-dimensional velocity components u = a x and v = b y in Cartesian coordinates. a and b are constants; incompressibility is not assumed unless a + b = 0. Steady flow field; no body forces considered for this kinematic classification. Concept / Approach: A stagnation point is defined as a point in the flow where the local fluid velocity vector is zero. For the given linear field, the velocity components vanish when x = 0 and y = 0 simultaneously, i.e., at the origin. The term “stationary point” is a general mathematical phrase; in fluid mechanics, the precise term for zero velocity is “stagnation point.” Step-by-Step Solution: Set u = a x = 0 ⇒ x = 0.Set v = b y = 0 ⇒ y = 0.Therefore, the unique location with u = v = 0 is at (x, y) = (0, 0).Classify that location: by definition, it is a stagnation point. Verification / Alternative check: Compute speed V = √(u^2 + v^2) = √((a x)^2 + (b y)^2). This is zero only when x = 0 and y = 0, confirming the classification. Why Other Options Are Wrong: (a) “Critical point” is ambiguous without additional context; (b) “neutral point” is not standard velocity-field terminology; (d) “stationary point” is mathematical jargon but lacks the specific fluid meaning; (e) incorrect because a precise standard term exists. Common Pitfalls: Confusing stagnation with separation points; forgetting that pressure is typically maximum at stagnation in inviscid steady flow due to Bernoulli’s principle along streamlines. Final Answer: stagnation point

Question 8

Dynamic viscosity (μ): select the correct SI unit.

Accepted Answer

Introduction / Context: Dynamic viscosity quantifies a fluid’s resistance to shear deformation and appears in Newton’s law of viscosity, τ = μ (du/dy). Correct units prevent errors in calculations of laminar flow, boundary layers, and Reynolds numbers. Given Data / Assumptions: Newton’s law: shear stress τ has units of force per area (N/m²). Velocity gradient du/dy has units of s^-1. Standard SI coherent unit system is used. Concept / Approach: From τ = μ (du/dy), rearrange to μ = τ / (du/dy). Substituting dimensions gives μ units = (N/m²) / (s^-1) = N·s/m². Since N/m² is a Pascal (Pa), μ is equivalently expressed as Pa·s. This is distinct from kinematic viscosity ν = μ/ρ with units m²/s. Step-by-Step Solution: Identify τ units: N/m².Identify du/dy units: s^-1.Compute μ units: (N/m²)/(s^-1) = N·s/m² = Pa·s.Relate to base units: N = kg·m/s² ⇒ μ = (kg·m/s²)·s / m² = kg/(m·s). Verification / Alternative check: For water at 20°C, μ ≈ 1.002 × 10^-3 Pa·s. Converting to base SI gives ≈ 1.002 × 10^-3 kg/(m·s), consistent with the derivation. Why Other Options Are Wrong: (a) kg·s/m² is not dimensionally equivalent to kg/(m·s); it inverts time. (c) N·s²/m³ mismatches stress-rate dimensions. (d) m²/s is kinematic viscosity ν, not μ. (e) kg/m³ is density, another property entirely. Common Pitfalls: Confusing Pa·s with m²/s; mixing SI with CGS (poise, centipoise: 1 cP = 1 mPa·s). Final Answer: Newton·second per square metre (N·s/m² = Pa·s)

Question 9

In a steady-flow situation in fluid mechanics, which property at a fixed point in the flow field changes with time (if any)? (Assume a continuum fluid and classical definitions of steady versus unsteady flow.)

Accepted Answer

Introduction / Context: Understanding what “steady flow” means is fundamental in hydraulics and aerodynamics. The term appears in Bernoulli applications, open-channel analysis, and turbomachinery design. Students often confuse steady with uniform, so this question clarifies the time aspect at a fixed spatial location. Given Data / Assumptions: A continuum fluid (liquid or gas) moving in a control volume or along a streamline network. Classical definition: steady versus unsteady is judged at a fixed point in space. No phase change or discontinuities at the observation point. Concept / Approach: By definition, steady flow means that all flow properties at a given point do not vary with time. Typical properties include velocity components, pressure, density, and temperature. They may vary from point to point in space, but at any one location they remain constant in time while the flow is steady. Step-by-Step Solution: State the definition: For steady flow, ∂(property)/∂t at a fixed point = 0.Properties include u, v, w (velocity components), p (pressure), ρ (density), and T (temperature).Therefore, at a fixed point, velocity, pressure, and density do not change with time under steady conditions. Verification / Alternative check: Contrast with “uniform flow,” where properties do not change with position along a direction. A flow can be steady but non-uniform (e.g., a nozzle with changing area: velocity varies along the axis but not with time at any one cross-section). Why Other Options Are Wrong: Velocity (a), pressure (b), and density (c) may vary spatially in steady flow, but at a fixed location they do not vary with time; hence selecting any of them is incorrect. “Temperature only” (e) is also incorrect; temperature is a property like others and would also remain constant at a point in steady flow. Common Pitfalls: Equating “steady” with “constant everywhere,” which is false. Steady talks about time at one point; uniform talks about space at one instant. Final Answer: None of these

Question 10

In flow measurement practice, a stepped notch is essentially a combination of which type(s) of basic notches to extend the measurement range?

Accepted Answer

Introduction / Context: Laboratory flumes and small channels often require a single gauge to measure wide ranges of discharge. A stepped notch is a practical device that changes effective crest length with head, enabling better sensitivity at low flows while accommodating higher flows without drowning the notch. Given Data / Assumptions: Steady, free-flow conditions over a sharp crest (fully aerated nappe). Steps are arranged vertically so progressively higher steps engage as head increases. Objective is to vary effective rectangular width with stage. Concept / Approach: A stepped notch is constructed from a series of rectangular openings at different elevations, each “step” adding rectangular width as the head rises. This provides discrete increments in effective length b used in Q = C_d * b * √(2 g) * H^(3/2), improving resolution across a range of flows. It is different from a compound notch that deliberately mixes triangular and rectangular openings in one crest to achieve a single continuous head–discharge law. Step-by-Step Solution: Identify the aim: extend range and sensitivity without changing the weir plate.Mechanism: stacked rectangular apertures activate sequentially with increasing H.Therefore the stepped notch is a combination of rectangular notches of different sizes/levels. Verification / Alternative check: Most textbooks describe stepped notches as “multiple rectangular notches placed one above another.” Compound (or combination) notches, on the contrary, mix V and rectangular shapes in the same crest opening. Why Other Options Are Wrong: (b) and (e) misidentify the geometry. (c) describes a compound notch, not a stepped notch. (d) cannot be correct because all listed variants do not describe the same device. Common Pitfalls: Confusing “stepped” with “compound.” The former stacks similar rectangular openings; the latter blends dissimilar notch shapes. Final Answer: Rectangular notches of different sizes (at different steps)

Question 11

For the most economical (best hydraulic) trapezoidal open-channel section, which geometric condition must hold true? Choose the classical proportionality used in design.

Accepted Answer

Introduction / Context: The “most economical” or “best hydraulic” section minimizes wetted perimeter for a given area, thereby maximizing hydraulic radius and reducing friction losses. For trapezoidal channels used in irrigation and drainage, a celebrated condition ties the top width and the sloping side length. Given Data / Assumptions: Prismatic trapezoidal channel with side slope z:1 (horizontal:vertical). Goal: minimize wetted perimeter P for a given flow area A. Steady, uniform open-channel flow concept (e.g., Manning/Chezy context). Concept / Approach: Using calculus of variations or standard derivations, the conditions for the best trapezoidal section are: (1) hydraulic radius R = A/P = y/2, and (2) half of the top width equals the sloping side length. This condition directly shapes the cross section and simplifies design checks. Step-by-Step Solution: Let b be bed width, y depth, and sloping side length s = √(y^2 + (z y)^2) = y √(1 + z^2).Top width T = b + 2 z y ⇒ T/2 = b/2 + z y.Optimality yields T/2 = s (i.e., half top width equals one sloping side length). Verification / Alternative check: Applying the condition returns the known result R = y/2 at optimum, consistent with minimal P for a given A. Why Other Options Are Wrong: (a) and (c) are not general optimality conditions. (e) is not a standard best-section identity. Only (b) matches the classical economical trapezoid result. Common Pitfalls: Confusing the optimal trapezoid with the rectangular or semicircular best sections; each has different characteristic ratios. Final Answer: Half of the top width equals one sloping side length

Question 12

Fire hose nozzles are generally designed with which outlet shape to achieve a high-velocity jet at subsonic conditions for water?

Accepted Answer

Introduction / Context: Firefighting requires long-reach, coherent water jets to deliver momentum to targets. Nozzle geometry controls pressure-to-velocity conversion and jet quality. The appropriate shape depends on whether the working fluid is compressible and on the speed regime. Given Data / Assumptions: Fluid: water (essentially incompressible at these conditions). Subsonic speeds; no compressible-flow choking phenomena. Objective: maximize jet velocity and coherence. Concept / Approach: For incompressible subsonic flows, a convergent nozzle reduces cross-sectional area to convert pressure energy into kinetic energy (Bernoulli principle). Convergent–divergent designs are used for compressible gases to achieve supersonic speeds across a throat; this is unnecessary and detrimental for liquid water jets. A purely divergent section would decelerate the flow and degrade the jet. Step-by-Step Solution: Start with upstream pressure p_1 and velocity V_1 in the hose.Convergent reduction to area A_2 increases V_2 while reducing static pressure (p_2).Select a smooth bell mouth/contour to minimize separation and energy losses for a stable, coherent jet. Verification / Alternative check: Practical firefighting nozzles and hydraulic texts recommend smooth convergent contours with slight straightening sections to preserve jet coherence and minimize turbulence at the exit. Why Other Options Are Wrong: Divergent (a) causes diffusion (velocity drop). Convergent–divergent (c) is for compressible supersonic gases. Cylindrical (d) offers less acceleration and may produce a weaker jet. A mere “Venturi throat only” (e) is not how water jets are produced at hose tips. Common Pitfalls: Confusing gas nozzle design with liquid jet production; liquids do not require a divergent section to accelerate. Final Answer: Convergent

Question 13

The line joining the heights to which water would rise in vertical piezometer tubes fitted at different sections of a pressurized conduit is called what?

Accepted Answer

Introduction / Context: Visualizing energy distribution in pressurized pipe systems uses two companion curves: the Hydraulic Grade Line (HGL) and the Energy Grade Line (EGL). Piezometric tubes (or pressure taps plus elevation) reveal the HGL directly as water levels at points along a pipeline. Given Data / Assumptions: Pressurized, steady, incompressible flow in a closed conduit. Neglect air entrainment; tubes measure static head (p/γ + z). No pump/turbine stations between measurement points. Concept / Approach: The height to which water rises in a piezometer equals the piezometric head (pressure head + elevation head). The locus of these heights is known as the Hydraulic Grade Line (also called piezometric line or pressure grade line). “Hydraulic gradient” strictly refers to the slope of the HGL, not the line itself. Step-by-Step Solution: At a section: H_p = p/γ + z (ignoring velocity head in piezometer height).Connect H_p values across sections to obtain the HGL.Recognize that EGL = HGL + V^2/(2 g). Verification / Alternative check: In a uniform pipe with friction, HGL slopes downward in the flow direction; pumps cause a jump up, while turbines cause a drop, aligning with standard energy accounting. Why Other Options Are Wrong: (a) “Hydraulic gradient” names the slope, not the line. (e) “All of the above” would include (a), which is not a name for the line itself in strict usage; hence the best single term is HGL (d). (b) and (c) are synonyms for HGL, but only one option should be chosen; the standard engineering term is the Hydraulic Grade Line. Common Pitfalls: Confusing HGL with EGL; the difference is the velocity head. Final Answer: Hydraulic grade line

Question 14

In flow visualization, the curve whose tangent at every point gives the instantaneous direction of the fluid velocity is called what?

Accepted Answer

Introduction / Context: Several distinct curves are used to visualize flows: streamlines, path lines, streak lines, and timelines. Textbook definitions differ subtly but matter for unsteady flows and for interpreting flow-visualization experiments (smoke lines, dye injection, or PIV tracers). Given Data / Assumptions: Instantaneous velocity field V(x, y, z, t) is known at a given time t. We seek the curve whose tangent aligns with V at each point at that same instant. Concept / Approach: A streamline is defined such that at a fixed instant, its tangent vector is exactly the local velocity direction. In steady flow, streamlines, path lines (particle trajectories), and streak lines coincide. In unsteady flows they differ: a path line is the actual path of one particle over time; a streak line traces all particles that have passed through a particular point; a timeline connects multiple particles released simultaneously. Step-by-Step Solution: Define streamline: dy/dx = v/u (in 2D), so its tangent matches the velocity direction.At any instant, integrate the differential relation to obtain streamline shapes.Use streamlines to infer shear layers, separation, and recirculation zones. Verification / Alternative check: In numerical CFD post-processing, streamlines are seeded to show direction fields; they match analytical definitions by construction. Why Other Options Are Wrong: Path line and streak line depend on particle history, not the instantaneous field. “Potential line” is a line of constant velocity potential, orthogonal to streamlines in potential flow, not aligned with the velocity direction. Common Pitfalls: Assuming all three coincide even in unsteady flows; this is only true for steady conditions. Final Answer: Streamline

Question 15

Among standard open-channel shapes, which cross-section is the most efficient hydraulically (i.e., provides the maximum discharge for a given area/wetted perimeter)?

Accepted Answer

Introduction / Context: “Most efficient” channel sections minimize wetted perimeter for a given cross-sectional area, thereby maximizing the hydraulic radius R = A/P and reducing frictional head loss. This concept guides shape selection in sewers, aqueducts, and irrigation canals. Given Data / Assumptions: Steady, uniform flow approximations with Manning or Chezy resistance laws. Comparison among geometric families for equal area or conveyance goals. No freeboard, stability, or constructability constraints considered—purely hydraulic efficiency. Concept / Approach: Among common shapes, the semicircular channel provides the least wetted perimeter for a given area because a circle encloses maximum area per unit perimeter. A half-circle retains this extremal property for an open channel with a free surface. Consequently, for equal area, semi-circular sections yield the largest hydraulic radius and hence the highest discharge at the same slope and roughness. Step-by-Step Solution: Hydraulic radius R = A/P; larger R reduces friction term.For a semicircle of radius r: A = (π r^2)/2; P = π r.For other shapes (rectangular, triangular), P is larger for the same A ⇒ smaller R. Verification / Alternative check: Isoperimetric inequality shows circles maximize area for a given perimeter; this extends to the open half for channels. Why Other Options Are Wrong: Rectangular and triangular sections are simpler to build but are not hydraulically optimal. “Half hexagon” is a specific trapezoid and does not beat the semicircle in A/P performance. Parabolic channels can be efficient but still do not surpass the semicircle in the idealized comparison. Common Pitfalls: Confusing hydraulic efficiency with practical design constraints (erosion, lining, cost). Real projects often choose trapezoids for constructability despite lower hydraulic efficiency. Final Answer: Semi-circular

Question 16

An open container full of water is moved vertically downward with a uniform linear acceleration a. Compared with the static (at rest) condition, the pressure at the container's bottom will be:

Accepted Answer

Introduction / Context: Accelerating containers create an “apparent gravity” that modifies hydrostatic pressure distributions. This is important for sloshing analysis, accelerometer calibration with liquids, and spacecraft fluid handling (where effective gravity can be reduced or reversed). Given Data / Assumptions: Vertical, uniform downward acceleration a. Open container of water (free surface exposed to atmosphere). No wave motion; quasi-static distribution during uniform acceleration. Concept / Approach: In an accelerating frame moving downward at acceleration a, the apparent gravity is g_eff = g − a (downward). Hydrostatic pressure varies as p = p_atm + ρ g_eff h. When 0 < a < g, g_eff is reduced, decreasing pressure at depth relative to the static case. If a → g, g_eff → 0 and the fluid becomes effectively weightless; pressure becomes nearly uniform at p_atm. Step-by-Step Solution: Define effective weight per unit volume: γ_eff = ρ (g − a).Bottom pressure: p_bottom = p_atm + ρ (g − a) h.Compare with static: p_static = p_atm + ρ g h ⇒ p_bottom < p_static for a > 0. Verification / Alternative check: Limit checks: a = 0 gives static pressure. a = g makes g_eff = 0 so p is nearly atmospheric throughout. These checks confirm the trend of decreasing pressure with increasing downward acceleration. Why Other Options Are Wrong: (a) contradicts the reduction in g_eff. (b) would only be true for a = 0. (d) is impossible unless in a perfect vacuum and even then p = p_atm at the free surface. (e) is incorrect; pressure depends explicitly on acceleration through g_eff. Common Pitfalls: Confusing downward acceleration with deceleration; also forgetting that hydrostatic pressure depends on effective body force, not merely depth. Final Answer: Less than static pressure

Question 17

In the CGS (centimetre–gram–second) system, the standard unit of kinematic viscosity ν is:

Accepted Answer

Introduction / Context: Fluid property tables report viscosity in either dynamic (μ) or kinematic (ν) forms and in various unit systems. Engineers must convert correctly between SI and CGS to avoid large numerical errors in design and simulation. Given Data / Assumptions: Kinematic viscosity ν = μ/ρ. CGS unit system is used. We distinguish dynamic (poise) from kinematic (stoke) viscosity units. Concept / Approach: In CGS, dynamic viscosity has unit poise (P) = g/(cm·s). Kinematic viscosity has unit stoke (St) = cm^2/s. Commonly, centistokes (cSt) are used in lubrication (1 cSt = 0.01 St). Step-by-Step Solution: Recall ν units: length^2/time; in CGS → cm^2/s = stoke.Relation to SI: 1 St = 1 cm^2/s = 1e−4 m^2/s.If μ is given in poise and ρ in g/cm^3, ν = μ/ρ in stokes directly. Verification / Alternative check: Water at 20°C: μ ≈ 1 cP, ρ ≈ 1 g/cm^3 ⇒ ν ≈ 1 cSt = 0.01 St, matching common reference data. Why Other Options Are Wrong: Poise (b) is dynamic viscosity unit. Newton and Pascal-second are SI force and SI dynamic viscosity units, respectively. “None” is unnecessary because stoke is the correct CGS kinematic viscosity unit. Common Pitfalls: Mixing up cSt with St; a factor of 100 error is frequent in oil selection and Reynolds number calculations. Final Answer: Stoke

Question 18

Metacentric height (GM) is defined as the distance between which two points of a floating body for small-angle stability analysis?

Accepted Answer

Introduction / Context: Initial stability of ships and floating structures is quantified by the metacentric height GM. A positive GM indicates a restoring moment for small heel angles. Designers track GM for safety, comfort, and performance under load changes and free-surface effects. Given Data / Assumptions: Small-angle heel so that the concept of a single metacentre M is valid. Homogeneous fluid; hydrostatic buoyancy principles apply. Rigid body, no flooding or free-surface sloshing effects included. Concept / Approach: Metacentric height GM is defined as the algebraic distance between the metacentre M and the centre of gravity G: GM = BM − BG. The sign of GM governs stability: GM > 0 yields a righting couple; GM < 0 yields instability. While BM (metacentric radius) depends on geometry of the waterplane, BG depends on loading and ballast. Step-by-Step Solution: Identify points: G (weight line of action), B (buoyancy centre), M (intersection of buoyant lines for small heel).Define GM = distance from G to M (positive if M is above G).Relate to righting arm: GZ ≈ GM * sin θ for small θ. Verification / Alternative check: Inclining experiment measures GM experimentally by shifting a known mass and observing equilibrium angle; result accords with the GM definition above. Why Other Options Are Wrong: (a) GB is not GM; (c) BM is the metacentric radius, not GM. (d) and (e) do not state the correct definition for GM. Common Pitfalls: Confusing GM with BM; remembering that GM depends on both BM and BG is crucial for design changes. Final Answer: Metacentre (M) and centre of gravity (G)

Question 19

In fluid mechanics, an orifice is defined as which of the following? Select the most accurate description of the opening and flow condition.

Accepted Answer

Introduction / Context: Understanding what constitutes an orifice versus a mouthpiece or open-channel opening is fundamental in hydraulics. Exam questions often test whether learners can distinguish these devices by geometry and flow conditions, which directly affect discharge equations and coefficients. Given Data / Assumptions: The opening is in a tank or vessel wall or base. Flow issues due to a head difference across a thin boundary (thin plate idealization). No requirement for the jet to run full in a long tube (that would be a mouthpiece). Concept / Approach: An orifice is a sharp-edged opening of closed perimeter in a relatively thin wall. The jet forms immediately and contracts to a vena-contracta. The upstream water level for standard orifice flow is above the top edge of the opening, creating a positive head. Mouthpieces, by contrast, have appreciable length (2–3 diameters or more) that alters jet development and coefficients. Step-by-Step Solution: Identify the defining geometry: a sharp-edged opening with closed perimeter → orifice.Check upstream level condition: it must be above the top for pressurized orifice flow.Recognize prolonged sides as a mouthpiece, not an orifice. Verification / Alternative check: Standard texts define an orifice by the thin-plate opening with a sharp edge; introducing a length (tube) changes the device to an internal or external mouthpiece and changes coefficients Cd, Cc, Cv. Why Other Options Are Wrong: Upstream level below top edge: that indicates weir/open-channel behavior, not classic orifice head.Partially full flow: describes open-channel conditions, not a pressurized jet through a plate.Prolonged sides 2–3D: that is a mouthpiece, not a simple orifice. Common Pitfalls: Confusing weirs (free surface control) with orifices (pressurized jet). Assuming any opening is an orifice even when there is a tube length. Final Answer: An opening with a closed perimeter of regular shape through which water flows

Question 20

The differential equation of steady inviscid motion along a streamline, dp/ρ + g dz + v dv = 0, is attributed to which scientist?

Accepted Answer

Introduction / Context: Bernoulli’s equation is foundational in fluid mechanics, typically obtained by integrating the Euler equation of motion along a streamline. Many learners know Bernoulli, but the underlying differential form and its attribution to Euler are crucial historical and conceptual points. Given Data / Assumptions: Steady flow along a streamline. Inviscid (no viscosity) and incompressible fluid. Body force due to gravity only. Concept / Approach: The Euler equation expresses Newton’s second law for a fluid element neglecting viscosity. Along a streamline it reduces to the differential form: dp/ρ + g dz + v dv = 0. Integrating this relation yields Bernoulli’s constant: p/ρ + g z + v^2/2 = constant. Step-by-Step Solution: Start with momentum balance for inviscid flow (Euler’s equation).Project along a streamline to obtain dp/ρ + g dz + v dv = 0.Integrate to recover Bernoulli’s energy equation. Verification / Alternative check: All standard derivations of Bernoulli’s equation reference Euler’s equation as the starting point for inviscid flows, confirming the attribution to Leonhard Euler, not Bernoulli or others listed. Why Other Options Are Wrong: Bernoulli: associated with the integrated result, not the differential equation form.Cauchy–Riemann: complex analysis relations, unrelated to fluid momentum balance.Laplace: potential equation, not the momentum equation used here. Common Pitfalls: Attributing everything about Bernoulli’s theorem to Bernoulli, overlooking Euler’s contribution. Confusing potential flow equations with momentum equations. Final Answer: Leonhard Euler

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