Engineering Questions with Answers - Multiple Choice Questions

# Computational Fluid Dynamics – Governing Equations – Substantial Derivative

1 - Question

How is the substantial derivative of velocity vector denoted?
a) DV⃗ Dt
b) dV⃗ dt
c) ∂V⃗ ∂t
d) DV⃗ Dx
Explanation: DV⃗ Dt is the substantial derivative. dV⃗ dt is the local derivative. ∂V⃗ ∂t is the partial derivative.

2 - Question

Expand the substantial derivative Dρ/Dt.
a) DρDt=dρdt+udρdx+vdρdy+wdρdz
b) DρDt=∂ρ∂t+udρdy+vdρdz+wdρdx
c) DρDt=dρdz+u∂ρ∂y+v∂ρ∂z+w∂ρ∂t
d) DρDt=∂ρ∂t+u∂ρ∂x+v∂ρ∂y+w∂ρ∂z
Explanation: As the location coordinates (x, y, z) vary with time, DρDt=∂ρ∂t+∂ρ∂x∂x∂t+∂ρ∂y∂y∂t+∂ρ∂z∂z∂t DρDt=∂ρ∂t+u∂ρ∂x+v∂ρ∂y+∂ρ∂t

3 - Question

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 - Question

The simplified form of substantial derivative can be given by __________
a) DTDt=∂T∂t+∇T
b) DTDt=∂T∂t+∇.T
c) DTDt=∂T∂t+V⃗ .∇T
d) DTDt=∂T∂t+∇×T
Explanation: DTDt=∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z DTDt=∂T∂t+(ui⃗ +vj⃗ +wk⃗ ).(∂T∂xi⃗ +∂T∂yj⃗ +∂T∂zk⃗ ) DTDt=∂T∂t+V⃗ .∇T.

5 - Question

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 ∂∂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 - Question

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 - Question

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 - Question

Which of these terms represent the convective derivative of temperature (T)?
a) V⃗ .∇T
b) DTDt
c) ∇T
d) ∂T∂t
Explanation: V⃗ .∇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 - Question

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. DTDt=∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z DTDt=∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z.

10 - Question

Which of these is not an equivalent to for substantial derivative?
a) Lagrangian derivative
b) Material derivative
c) Total derivative
d) Eulerian derivative