Question

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) \(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+\nabla \times T\)

This question was addressed to me by my school teacher while I was bunking the class.

My enquiry is from Governing Equations topic in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is:

(c) DTDt=∂T∂t+V⋅∇TDtDT​=∂t∂T​+V⋅∇T

Explanation:

The substantial derivative (also called the material derivative) describes the rate of change of a field variable (like temperature, TT) as observed by a moving fluid element. It accounts for both the local rate of change at a fixed point in space and the convective change due to the fluid's motion.

The general form of the substantial derivative is:

DTDt=∂T∂t+V⋅∇TDtDT​=∂t∂T​+V⋅∇T

• ∂T∂t∂t∂T​ is the local rate of change (change with respect to time at a fixed location),
• V⋅∇TV⋅∇T is the convective term, describing how the temperature field is carried along by the fluid motion.

Thus, the correct answer is (c).