Continuity Equation MCQ Quiz in मराठी - Objective Question with Answer for Continuity Equation - मोफत PDF डाउनलोड करा
Last updated on Mar 16, 2025
Latest Continuity Equation MCQ Objective Questions
Top Continuity Equation MCQ Objective Questions
Continuity Equation Question 1:
The water is flowing through a pipe having areas 0.05 m2 and 0.01 m2 at sections 1 and 2 respectively. The rate of flow through the pipe is 50 litres per second in section 1. The velocity at sections 1 and 2 are:
Answer (Detailed Solution Below)
Continuity Equation Question 1 Detailed Solution
Concept:
Continuity equation: It is the conservation of mass flow rate.
- ρ1A1V1 = ρ1A1V1
For incompressible fluid density will be constant thus continuity equation will be:
- A1V1 = A2V2
where, A1, A2 = area of section 1 & 2 respectively, V1, V2 = velocity of section 1 & 2 respectively
The flow rate of liquid is equal to Q = AV.
Calculation:
Given:
Area: A1 = 0.05 m2, A2 = 0.01 m2.
Flow rate: Q = 50 l/s = 50 x 10-3 m3/s.
Q = A1V1 = A2V2
\(V_1 = \frac{Q}{A_1} = \frac{50 \times 10^{-3}}{0.05} = 1\ m/s\)
\(V_2 = \frac{Q}{A_2} = \frac{50 \times 10^{-3}}{0.01} = 5\ m/s\)
Continuity Equation Question 2:
Consider the two-dimensional velocity field given by \(\vec V = \left( {5 + {a_1}x + {b_1}y} \right)\hat i + \left( {4{a_2}x + {b_2}y} \right)\hat j,\) where a1, b1, a2 and b2 Are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?
Answer (Detailed Solution Below)
Continuity Equation Question 2 Detailed Solution
Concept:
Every fluid flow must satisfy its corresponding mass conservation (continuity) equation.
For a 2 – dimensional incompressible flow
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial \nu }}{{\partial y}} = 0\)
\(\vec V = u\hat i + \nu \hat j\)
Calculation:
u = 5 + a1x + b1y, ν = 4 + a2x + b2y
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial \nu }}{{\partial y}} = 0 \Rightarrow {a_1} + {b_2} = 0\)
Continuity Equation Question 3:
A pipeline carries oil at a velocity of 2 m/s through a 25 cm pipe. At another section the diameter is 20 cm. find the velocity of oil.
Answer (Detailed Solution Below)
Continuity Equation Question 3 Detailed Solution
Concept:
For steady, incompressible flow, continuity equation is given by,
A1V1 = A2V2
Calculation:
Given:
d1 = 25 cm, V1 = 2 m/s
d2 = 20 cm, V2 = ?
As per the continuity equation,
A1V1 = A2V2
⇒ \(\frac{\pi }{4} × d_1^2 × {V_1} = \frac{\pi }{4} × d_2^2 × {V_2}\)
252 × 2 = 202 × V2
V2 = 3.125 m/s
Continuity Equation Question 4:
Continuity Equation for three dimensional flow is:
Answer (Detailed Solution Below)
Continuity Equation Question 4 Detailed Solution
Explanation:
Continuity Equation is based on the principle of conservation of mass. For a fluid flowing through a pipe at all the cross-sections, the quantity of fluid per second is constant.
The continuity equation is given as \({ρ _1}{A_1}{V_1} = \;{ρ _2}{A_2}{V_2}\)
Generalized equation of continuity.
\(\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} + \frac{{\partial ρ }}{{\partial t}} = 0\)
This equation can be written in vector form as,
Case 1: For steady flow \(\frac{{\partial ρ }}{{\partial t}} = 0\) then the above equation will become,
\(\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} = 0\)
Case 2: For Incompressible flow, ρ is constant, therefore the continuity equation of steady incompressible for three-dimensional flow is,
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)
For two dimensional flow
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)
Continuity Equation Question 5:
For compressible fluid, continuity equation -
Answer (Detailed Solution Below)
Continuity Equation Question 5 Detailed Solution
Explanation:
The continuity Equation is based on the principle of conservation of mass. For a fluid flowing through a pipe at all the cross-sections, the quantity of fluid per second is constant.
The continuity equation is given as \({ρ _1}{A_1}{V_1} = \;{ρ _2}{A_2}{V_2}\) (Compressible fluid)
Density ρ = C for an incompressible fluid.
∴ Continuity equation for an incompressible fluid A1V1 = A2V2
Additional Information
Generalized equation of continuity.
\(\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} + \frac{{\partial ρ }}{{\partial t}} = 0\)
This equation can be written in vector form as,
Case 1: For steady flow \(\frac{{\partial ρ }}{{\partial t}} = 0\) then the above equation will become,
\(\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} = 0\)
Case 2: For Incompressible flow, ρ is constant, therefore the continuity equation of steady incompressible for three-dimensional flow is,
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)
\(\nabla .\vec V = 0\)
For two dimensional flow
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)
Continuity Equation Question 6:
The general equation of continuity for three-dimensional flow an incompressible fluid for steady flow is:
Answer (Detailed Solution Below)
Continuity Equation Question 6 Detailed Solution
Explanation:
Continuity equation in three dimensions:
\(\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {\rho u} \right) + \frac{\partial }{{\partial y}}\left( {\rho v} \right) + \frac{\partial }{{\partial z}}\left( {\rho w} \right) = 0\)
The above equation is valid for:
- Steady and unsteady flow.
- Uniform and non-uniform flow.
- Compressible and incompressible flow.
For Steady flow:
\(\frac{\partial }{{\partial x}}\left( {\rho u} \right) + \frac{\partial }{{\partial y}}\left( {\rho v} \right) + \frac{\partial }{{\partial z}}\left( {\rho w} \right) = 0\;\left( \because{\frac{{\partial \rho }}{{\partial t}} = 0} \right)\)
If the fluid is Incompressible and Steady:
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\;\;\left( \because{\rho = constant} \right)\)
Continuity Equation Question 7:
Find the value of ‘a’ for fluid flow given by V = (axy2 + 2y)î + (3xy + x2y)ĵ.
It is given that flow is steady incompressible flow at (1, 1).Answer (Detailed Solution Below)
Continuity Equation Question 7 Detailed Solution
Concept:
Continuity equation should be satisfied for incompressible, steady flow,
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)
Calculation:
\(\frac{\partial }{{\partial x}}\left( {ax{y^2} + 2y} \right) + \frac{\partial }{{\partial y}}\left( {3xy + {x^2}y} \right) = 0\)
(ay2) + (3x + x2) = 0
At (1, 1)
a + (3 + 1) = 0
Continuity Equation Question 8:
An ideal flow of a liquid obeys
Answer (Detailed Solution Below)
Continuity Equation Question 8 Detailed Solution
Explanation:
Ideal fluid:
A fluid, which is incompressible and is having no viscosity, is known as an ideal fluid. Ideal fluid is only an imaginary fluid as all the fluids, which exist, have some viscosity.
Hence a fluid which has constant density and has zero viscosity is known as ideal fluid
Ideal fluid obeys the following two equations
1) Continuity equation
2) Bernouilli’s equation
Important Points
1) Continuity equation:
The equation based on the principle of conservation of mass is called the continuity equation. Thus for the fluid flowing through the pipe at all the cross-sections, the quantity of fluid per second is constant.
ρ1 × A1 × V1 =ρ2 × A2 × V2
ρ1 and ρ2 is the density of liquid at section 1 and section 2 respectively
A1 and A2 is an area of pipe at section 1 and section 2 respectively
V1 and V2 is velocity of liquid section 1 and section 2 respectively
For an ideal fluid i.e incompressible fluid, ρ1 = ρ2, and the continuity equation reduces to,
A1 × V1 = A2 × V2
2) Bernoulli’s equation:
Bernoulli’s equation in the form of energy per unit weight is given by,
\(\frac{{\rm{P}}}{{\rm{\gamma }}} + \frac{{{{\rm{V}}^2}}}{{2{\rm{g}}}} + {\rm{Z}} = {\rm{H}}\)
Where,
\(\frac{{\rm{P}}}{{\rm{\gamma }}} = {\rm{Static\;head}},\frac{{{{\rm{V}}^2}}}{{2{\rm{g}}}} = {\rm{Dynamic\;head}},{\rm{\;Z}} = {\rm{dattum\;head}},{\rm{\;H}} = {\rm{total\;head}}\)
The assumptions made in the derivation of Bernoulli’s equation is
1) The fluid is ideal, i.e Viscosity is zero
2) The flow is steady
3) The flow is incompressible
4) The flow is irrotational
Continuity Equation Question 9:
Which equation is based on the principle of conservation of mass?
Answer (Detailed Solution Below)
Continuity Equation Question 9 Detailed Solution
Explanation:
Continuity equation |
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Bernoulli’s equation |
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Discharge equation |
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Linear equation |
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Continuity Equation Question 10:
The continuity equation for steady incompressible flow is expressed in vector notations as
Answer (Detailed Solution Below)
Continuity Equation Question 10 Detailed Solution
Explanation:
- The law of conservation of mass states that mass can neither be created nor be destroyed.
- The rate at which mass enters the region = Rate at which mass leaves the region + Rate of accumulation of mass in the region
- The above statement can be expressed analytically in terms of velocity and density field of flow and the resulting expression is known as the equation of continuity or the continuity equation.
\(\begin{array}{l} \frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {\rho u} \right) + \frac{\partial }{{\partial y}}\left( {\rho v} \right) + \frac{\partial }{{\partial z}}\left( {\rho w} \right) = 0\\ \frac{{\partial \rho }}{{\partial t}} +\nabla .\left({} {{\rm{\rho \vec V}}} \right) = 0 \end{array}\)
Important Points
- The continuity equation applies to all fluids, compressible and incompressible flow, Newtonian and non-Newtonian fluids.
- It expresses the law of conservation of mass at each point in a fluid and must, therefore, be satisfied at every point in a flow field.
- So, the continuity equation is connected with the conservation of mass and it can be applied to viscous/non-viscous, the compressibility of the fluid, or the steady/unsteady flow.