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.


For example:

Take density (ρ). Density over a large volume varies with space and time: ρ=f(x,y,z,t).

The derivative of ρ, therefore, is:

dρ=(∂ρ/∂x)dx + (∂ρ/∂y)dy + (∂ρ/∂z)dz + (∂ρ/∂t)dt.

The velocity (q) of a fluid flowing through this volume is a vector made up of three components:

q=(u,v,w).

Therefore:

dx=u*dt
dy=v*dt
dz=w*dt

To find the substantial derivative Dρ/Dt (which is the total change of ρ as the fluid element we're following moves from point 1 to point 2 during time t) - we need dρ/dt.

Dividing the above equation for dρ by dt, and substituting in the above three differential equations for velocity, we get

Dρ/Dt=dρ/dt=u*(∂ρ/∂x) + v*(∂ρ/∂y) + w*(∂ρ/∂z) + (∂ρ/∂t)

(Which, incidentally, equals ∂ρ/∂t + (q·grad)ρ.)

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