# Computational Fluid Dynamics Questions and Answers – Governing Equations – Reynolds Transport Theorem

«
»

This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Governing Equations – Reynolds Transport Theorem”.

1. The Reynolds transport theorem establishes a relationship between __________ and ___________
a) Control mass system, Control volume system
b) Differential equation, Integral equation
c) Non-conservative equation, Conservative equation
d) Substantial derivative, Local derivative

Explanation: Equations formed by considering the control mass system and control volume system are not the same even if the same physical law is used. A relation between these equations is established by Reynolds transport theorem.

2. Let B denote any property of a fluid flow. The statement of Reynolds transport theorem is “The instantaneous total change of B inside the _____________ is equal to the instantaneous total change of B within the ______________ plus the net flow of B into and out of the _____________”
a) Control volume, Control mass, Control volume
b) Control volume, Control volume, Control mass
c) Control mass, Control mass, Control volume
d) Control mass, Control volume, Control volume

Explanation: Statement of Reynolds Transport Theorem: “The instantaneous total change of B inside the control mass is equal to the instantaneous total change of B within the control volume plus the net flow of B into and out of the control volume”.

3. Consider the following terms:
MV → Material Volume (Control Mass)
V → Control Volume
S → Control Surface
B → Flow property
b → Intensive value of B in any small element of the fluid
ρ → Density of the flow
t → Instantaneous time
$$\vec{v}$$ → Velocity of fluid entering or leaving the control volume
$$\vec{n}$$ → Outward normal vector to control surface
Which of these equations is the mathematical representation of Reynolds transport theorem in the above terms?
a) $$(\frac{dB}{dt})_{MV} = \frac{d}{dt}(\int_sb \rho dS) + \int_vb \rho \vec{v}.\vec{n} dV$$
b) $$(\frac{dB}{dt})_{MV} = \frac{d}{dt}(\int_vb \rho dV) + \int_sb \rho \vec{v}.\vec{n} dS$$
c) $$(\frac{dB}{dt})_V = \frac{d}{dt}(\int_{MV}b \rho MV) + \int_sb \rho \vec{v}.\vec{n} dS$$
d) $$(\frac{dB}{dt})_{MV} = \int_vb \rho dV + \frac{d}{dt}(\int_sb \rho \vec{v}.\vec{n} dS)$$

Explanation:
$$(\frac{dB}{dt})_{MV} →$$ Instantaneous total change of B in material volume
$$\frac{d}{dt} (\int_vb \rho dV) →$$ Instantaneous total change of ” B” within control volume
$$\int_sb \rho \vec{v}. \vec{n}dS →$$ Net flow of B into and out of the control volume through control surfaces
Reynolds transport theorem states:
(Instantaneous total change of B in material volume)=(Instantaneous total change of B within control volume + Net flow of B into and out of control volume through control surfaces)
Therefore,
$$(\frac{dB}{dt})_{MV} = \frac{d}{dt}(\int_vb \rho dV) + \int_sb \rho \vec{v}.\vec{n}dS$$.

4. Leibniz rule is applied to which of these terms in deriving Reynolds transport theorem?
a) Volume integral term of control volume
b) Differential term of material volume
c) Surface integral term of control volume
d) Volume integral term of material volume

Explanation: Using Leibniz rule, the differentiation of an integral term can be reduced. Here, differential of integral exists in the Volume integral term of Control Volume which is given by $$\frac{d}{dt}(\int_vb \rho dV).$$

5. Why a surface integral is used to represent flow of B into and out of the control volume?
a) Control volume is moving
b) Flow of fluid is through the control surfaces
c) Fluid only on the control surfaces
d) Control volume is stationary

Explanation: Fluid can enter into or exit from the control volume through the control surface. If this flow velocity is integrated along the control surfaces, we can get the net inflow or outflow of fluid to the control volume.

6. When is Leibniz rule applicable to control volume?
a) When control volume is moving
b) When control volume is deforming
c) When control volume is fixed
d) In all conditions

Explanation: Leibniz rule is applicable to a system only if a system variable is independent. When control volume is fixed, position of the control volume becomes independent. So, Leibniz rule is applicable only to fixed control volumes.

7. Let,
V → Control Volume
B → Flow property
b → Intensive value of B in any small element of the fluid
ρ → Density of the flow
t → Instantaneous time
Which of these terms represent the ‘instantaneous total change of the flow property within the control volume’ after Leibniz rule is applied?
a) $$\frac{d}{dt}(\int_vb \rho dV)$$
b) $$\int_v \frac{\partial}{\partial T}(b \rho)dV$$
c) $$\rho \int_v \frac{\partial b}{\partial T} dV$$
d) $$\rho \int_v \frac{\partial \rho}{\partial b} dV$$

Explanation: According to Leibniz rule, if the variation of f(x, t) is independent of t,
$$\frac{d}{dx} \int f(x,t)dt = \int \frac{\partial}{\partial x}f(x,t)dt$$
Instantaneous total change of the flow property within the control volume is given by,
$$\frac{d}{dt}(\int_vb \rho dV)$$
Applying Leibniz rule,
$$\frac{d}{dt}(\int_vb \rho dV) = \rho \int_v \frac{\partial b}{\partial T}dV$$.

8. Gauss divergence theorem is used to convert a surface integral to volume integral. This is used in Reynolds Transport theorem. What is the purpose of this conversion?
a) Simplifying the term
b) Differentiating the flow property
d) Grouping terms related to control volume

Explanation: One term related to control volume is a volume integral. The other term is a surface integral. To group these two terms together, Gauss Divergence Theorem is used.

9. Gauss divergence is applied to which of these terms?
a) Instantaneous total change of B inside the control mass
b) Instantaneous total change of B within the control volume
c) Net flow of B into and out of the control volume
d) Net flow of B into and out of the control mass

Explanation: The term representing ‘net flow of B into and out of the control volume’ is a surface integral. This surface integral is converted into a volume integral using the Gauss divergence theorem.

10. Let,
V → Control Volume
b → Intensive value of B in any small element of the fluid
ρ → Density of the flow
$$\vec{v}$$ → Velocity of fluid entering or leaving the control volume
After applying Gauss divergence theorem, how does the term representing ‘net flow of B into and out of the control volume’ look like?
a) $$\int_v \nabla.(\rho \vec{v}b)dV$$
b) $$\int_s \nabla.(\rho \vec{v}b)dS$$
c) $$\int_v(\rho \vec{v}b)dV$$
d) $$\int_s(\rho \vec{v}b)dS$$

Explanation: The term representing ‘net flow of B into and out of the control volume’ is
$$\int_s b \rho \vec{v}.\vec{n}dS$$
Applying Gauss divergence theorem,
$$\int_s b \rho \vec{v}.\vec{n}dS = \int_v \nabla.(\rho \vec{v}b)dV$$.

11. The final equation of Reynolds transport theorem can be used to drive ____________ form of the conservation laws in fixed regions.
a) Eucledian
b) Lagrangian
c) Eulerian
d) Cartesian