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# 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

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Answer: aExplanation: 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

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Answer: dExplanation: 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

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Answer: bExplanation: 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

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Answer: cExplanation: 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

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Answer: dExplanation: 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

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Answer: aExplanation: 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

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Answer: bExplanation: 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

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Answer: aExplanation: 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

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Answer: cExplanation: 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

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Answer: dExplanation: Eulerian derivative means the local derivative (∂∂t). Material and Lagrangian derivatives are the other names for substantial derivative. Total derivative is mathematical equivalent to substantial derivative.