Question

Physical significance of substantial time derivative

Answer 1

The substantial derivative (or total derivative) is defined as the time rate of change of a fluid property of a fluid element as it travels through a given flow field (velocity field). It is denoted by

D/Dt=∂/∂t + (q·grad)

where q is the velocity field vector and grad denotes gradient.

∂/∂t is the local derivative - the time rate of change at a fixed point.

q·grad is the convective derivative - the time rate of change due to the movement of a fluid element through a flow field whose properties are spatially different (i.e., over here it's 20° Celsius; over there it's 25°).

In somewhat more palatable form,

D/Dt=∂/∂t + u*∂/∂x + v*∂/∂y + w*∂/∂z

where u, v, and w are the x, y, and z components of velocity vector q. 


hope so help u

Answer 2

D/Dt=∂/∂t + (q·grad)

where q is the velocity field vector and grad denotes gradient.

∂/∂t is the local derivative - the time rate of change at a fixed point.

q·grad is the convective derivative - the time rate of change due to the movement of a fluid element through a flow field whose properties are spatially different (i.e., over here it's 20° Celsius; over there it's 25°).

In somewhat more palatable form,

D/Dt=∂/∂t + u*∂/∂x + v*∂/∂y + w*∂/∂z

where u, v, and w are the x, y, and z components of velocity vector q.