Continuity, Euler and Bernoulli Equations - Differential Analysis - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2007

The velocity of an airstream (ρ = 1.0 kg/m³) is to be measured using a pitot-static tube. The level difference between the two arms of the manometer is 2 cm of water. The velocity is

A

0.02 m/s

B

2.0 m/s

C

10 m/s

D

20 m/s

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2

2008 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2008

The momentum equation (Euler),
\[ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial x}, \]
is valid if and only if the flow is

A

unsteady

B

laminar

C

steady

D

inviscid

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3

2008 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2008

The velocity in a wind tunnel is being measured using a Pitot-static tube connected to a vertical U-tube manometer. The density of air is 1.2 kg/m³ and the deflection of the manometer is 24 mm. The manometric fluid is water. The velocity measured by the Pitot-static tube is:

A

14.1 m/s

B

20.0 m/s

C

22.0 m/s

D

400 m/s

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4

2008 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2008

Water is flowing with volume flow rate Q through a pipe whose diameter reduces to half across a reducer. If the flow is frictionless, compare the manometer reading \( h_1 \), \( h_2 \) and \( h_3 \) corresponding to the three different inclinations of the pipe \( \theta_1 = 30^0 \), \( \theta_2 = 0^0 \) and \( \theta_3 = -30^0 \). Note that only the pipe tilts, while the manometer always stays vertical.

A

\(h_1 > h_2 > h_3 \)

B

\(h_1 < h_2 < h_3 \)

C

\(h_1 = h_2 = h_3 \)

D

\(h_1 = h_3 \) and \(h_1 > h_2 \)

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5

2009 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2009

A nozzle has inlet and outlet diameters of 10 cm and 5 cm, respectively. If it discharges air at a steady rate of 0.1 m³/s into the atmosphere, the gauge pressure (static) at the nozzle inlet will be

A

1.26 kPa

B

1.46 kPa

C

3.52 kPa

D

3.92 kPa

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6

2009 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2009

Under what conditions is the equation Δ•ρV⃗ = 0 valid ?
P : Steady incompressible flow
Q : Unsteady incompressible flow
R : Steady compressible flow
S : Unsteady compressible flow

A

P, Q, R

B

Q, R, S

C

P, R, S

D

P, Q, S

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7

2010 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2010

Expressions for P, Q and R are

A

P = 0; Q = 0; R = 0

B

P = U_∞; Q = 0; R = 0

C

P = 0 ; Q = \(\frac{\pi}{2δ}\) ; R = U_∞

D

P = U_∞; Q = \(\frac{\pi}{2δ}\) ; R = 0

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8

2011 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2011

A pitot-static probe is inserted in an air flow. A manometer connected to this probe having Hg as the manometric fluid shows a difference of 30 mm. Assume a probe factor of 1. Assuming ρ_air = 1.23 kg/m³, ρ_Hg = 13600 kg/m³ and g = 10 m/s², the speed of the air flow is approximately

A

66.5 m/s

B

81.5 m/s

C

76.5 m/s

D

92.5 m/s

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9

2011 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2011

In an inviscid incompressible flow, the velocity field is given by \(\vec{V} = x\hat{i} + y\hat{j}\) m/s and the body force per unit mass is given by \(\vec{g} = -10\hat{k}\) m/s². The pressure at the point (0, 0, 0) is 101 Pa. Assuming that the density of the fluid is 1 kg/m³, the pressure at the point (1, 1, 1) for this flow is

A

100 Pa

B

105 Pa

C

95 Pa

D

90 Pa

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10

2012 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2012

The water level in a gas-pressurized tank with a large cross-sectional area is maintained constant as shown in the figure below. The water level in the tank is 4.2 m above the pipe centerline as indicated in the figure. The gas pressure is 130 kPa. The atmospheric pressure, gravitational acceleration and density of water may be taken as 100 kPa, 10 m/s² and 1000 kg/m³, respectively. Neglecting losses, the maximum velocity (in m/s) of water at any location in the horizontal portion of the delivery pipe for the pressure NOT to drop below atmospheric pressure, is

A

1.3

B

4.2

C

10

D

12

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11

2012 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2012

The velocity of an air stream is 20 m/s. The densities of mercury and air are 13600 kg/m³ and 1.2 kg/m³, respectively. The gravitational acceleration may be taken as 10 m/s². When a Pitot-static tube is placed in the stream, assuming the flow to be incompressible and frictionless, the difference between the stagnation and static pressure in the flow field (in mm Hg) would approximately be

A

1760

B

1.76

C

0.57

D

0.57 × 10⁻⁵

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12

2013 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2013

Bernoulli’s equation is valid for the following type of flow:

A

Compressible, steady, inviscid

B

Incompressible, steady, viscous

C

Compressible, unsteady, viscous

D

Incompressible, steady, inviscid

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13

2014 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2014

The gravity driven flow over a hump of height \( h \) in a canal is shown in the figure. The height of the free surface from the canal bed at upstream of the hump is \( H \). The free surface height reduces to \( H_1 \) above the hump.
Assuming the canal bed to be horizontal, the discharge per unit width is given by

A

\(\sqrt{\frac{2g(H - H_1 - h)}{\frac{1}{H_1^2} - \frac{1}{H^2}}} \)

B

\(\sqrt{\frac{2gh}{\frac{1}{(H_1 + h)^2} - \frac{1}{H^2}}} \)

C

\(\sqrt{\frac{2g(H - H_1)}{\frac{1}{(H_1 + h)^2} - \frac{1}{H^2}}} \)

D

\(\sqrt{\frac{2g(H - H_1)}{\frac{1}{H_1^2} - \frac{1}{H^2}}} \)

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14

2015 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2015

A steady, two-dimensional, inviscid and incompressible flow field is described in rectangular Cartesian coordinates as \(u = ax\) and \(v = -ay\), where \(u\) and \(v\) are the components of the velocity vector in the \(x\) and \(y\) directions, respectively. Gravity acts along the negative \(y\)-direction. The pressure distribution, with the reference pressure taken as zero at the origin, with usual notation, is given by

A

\(-\frac{1}{2}\rho a^2 (x^2 + xy + y^2) - \rho gy\)

B

\(-\frac{1}{2}\rho a^2 (x^2 - xy + y^2) - \rho gy\)

C

\(-\frac{1}{2}\rho a^2 (x^2 + y^2) - \rho gy\)

D

\(-\frac{1}{2}\rho a^2 (x^2 - y^2) - \rho gy\)

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15

2016 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2016

Consider a fully developed, steady, incompressible, 2-D, viscous channel flow with uniform suction and blowing velocity \(v_0\), as shown in the figure given below. The centerline velocity of the channel is 10 m/s along the x-direction. If the value of \(v_0\) at both the walls is 1 m/s, the value of the y-component of velocity inside the flow field is ________ m/s.

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16

2016 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2016

Exhaust from a kitchen goes into the atmosphere through a tapered chimney as shown. The area of cross-section of chimney at location-1 is twice of that at location-2. The flow can be assumed to be inviscid with constant exhaust density of 1 kg/m³ and acceleration due to gravity, \(g = 9.8\) m/s². If the steady, uniform exhaust velocity at location-1 is \(U = 1\) m/s, the pressure drop across the chimney is ________ Pa.

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17

2017 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2017

An inviscid incompressible fluid of density 1000 kg/m³ is flowing in a horizontal pipe of tapered cross-section with a flow rate of 4000 cm³/s. The area of cross-section at two different locations ‘A’ and ‘B’ are 10 cm² and 20 cm², respectively. The velocity of the fluid at the location ‘A’ is 4 m/s and pressure is 5 N/m². The pressure (N/m²) at location ‘B’ would be ______

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18

2018 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2018

The velocity field and the surface normal vector are given by, V̅ = u î + v ĵ + w k̂ and n̅ = n₁ î + n₂ ĵ + n₃ k̂, respectively. If Euler equations are to be solved, the boundary condition that must be satisfied at the wall is,

A

V̅ · n̅ = 0

B

V̅ = 0

C

∇ · V̅ = 0

D

∇ × n̅ = 0

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19

2018 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2018

A pitot-static tube is used to measure air velocity in a duct by neglecting losses. The density of air is \(1.2\) kg/m\(^3\). If the difference between the total and static pressures is \(1\) kPa, the velocity of air at the measuring location, in m/s, is ______.

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20

2019 · Engineering Sciences · Differential Analysis · Continuity, Euler and Bernoulli Equations

Engineering Sciences (XE) 2019

The velocity field in Cartesian coordinates in a two-dimensional steady incompressible flow of a fluid with density ρ is V = x̂ − ŷ. Assuming no body and line forces, the magnitude of pressure gradient ∇p at point (1,1) is

A

√2 ρ

B

ρ

C

ρ/√2

D

ρ/2

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