1. How is the substantial derivative of velocity vector denoted?
a) \(\frac{D\vec{V}}{Dt}\)
b) \(\frac{d\vec{V}}{dt}\)
c) \(\frac{\partial \vec{V}}{\partial t}\)
d) \(\frac{D\vec{V}}{Dx}\)
Explanation: \(\frac{D\vec{V}}{Dt}\) is the substantial derivative. \(\frac{d\vec{V}}{dt}\) is the local derivative. \(\frac{\partial \vec{V}}{\partial t}\) is the partial derivative.
2. Expand the substantial derivative Dρ/Dt.
a) \(\frac{D\rho}{Dt}=\frac{d\rho}{dt}+u \frac{d\rho}{dx}+v\frac{d\rho}{dy}+w\frac{d\rho}{dz}\)
b) \(\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+u \frac{d\rho}{dy}+v\frac{d\rho}{dz}+w\frac{d\rho}{dx}\)
c) \(\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+u \frac{d\rho}{dy}+v\frac{d\rho}{dz}+w\frac{d\rho}{dx}\)
d) \(\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+v\frac{\partial \rho}{\partial y}+w \frac{\partial \rho}{\partial z}\)
Explanation: As the location coordinates (x, y, z) vary with time,
\(\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial \rho}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial \rho}{\partial z}\frac{\partial z}{\partial t}\)
\(\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+v\frac{\partial\rho}{\partial y}+\frac{\partial \rho}{\partial t}\)
3. Substantial derivative applies to ____________?
a) Both stationary and moving models
b) Only moving models
c) Only stationary models
d) Neither stationary nor moving models
Explanation: Substantial derivatives arise as the coordinates move and they vary with time. So, they are applicable only to moving models.
4. The simplified form of substantial derivative can be given by __________
a) \(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+\nabla T\)
b) \(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+\nabla .T\)
c) \(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+\vec{V}.\nabla T\)
d) \(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+\nabla \times T\)
Explanation:
\(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}+w \frac{\partial T}{\partial z}\)
\(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+(u\vec{i}+v\vec{j}+w\vec{k}).(\frac{\partial T}{\partial x} \vec{i}+\frac{\partial T}{\partial y}\vec{j}+\frac{\partial T}{\partial z}\vec{k})\)
\(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+\vec{V}.\nabla T.\)
5. Which of these statements best defines local derivative?
a) Time rate of change
b) Spatial rate of change
c) Time rate of change of a moving point
d) Time rate of change at a fixed point
Explanation: Local derivative is the term \(\frac{\partial}{\partial t}\) of a property. This defines the time rate of change of a property at a particular point with the assumption that the point is fixed.
6. A flow property has substantial derivative. What does this imply?
a) The property is a function of both time and space
b) The property is a function of time only
c) The property is a function of space only
d) The property is independent of time and space
Explanation: If a property has substantial derivative, it is differentiable by both time and space. This means that it is a function (i.e., dependent on) of time and space.
7. Substantial derivative = _____ + _____
a) Partial derivative, convective derivative
b) Local derivative, convective derivative
c) Local derivative, partial derivative
d) Total derivative, convective derivative
Explanation: Substantial derivative is the addition of local derivative (based on fixed model) and convective derivative (based on motion of the model)
8. Which of these terms represent the convective derivative of temperature (T)?
a) \(\vec{V}.\nabla T\)
b) \(\frac{DT}{Dt}\)
c) ∇T
d) \(\frac{\partial T}{\partial t}\)
Explanation: \(\vec{V}.\nabla T\) (dot product of velocity vector gradient of T) is the convective derivative which is the time rate of change due to the movement of the fluid element.
9. Substantial derivative is the same as ________ of differential calculus.
a) Partial derivative
b) Instantaneous derivative
c) Total derivative
d) Local derivative
Explanation: Substantial derivative is the same as total derivative. However, total derivative is completely mathematical.
\(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}+w \frac{\partial T}{\partial z}\)
\(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}+w \frac{\partial T}{\partial z}\).
10. Which of these is not an equivalent to for substantial derivative?
a) Lagrangian derivative
b) Material derivative
c) Total derivative
d) Eulerian derivative
Explanation: Eulerian derivative means the local derivative (\(\frac{\partial}{\partial t}\)). Material and Lagrangian derivatives are the other names for substantial derivative. Total derivative is mathematical equivalent to substantial derivative.