Why do we include both local and temporal acceleration in fluid mechanics but only consider temporal acceleration in solid-body mechanics?

Question

I am a beginner in physics, and I was studying fluid mechanics, specifically Newton's second law, when I was surprised to find that the expression for acceleration was composed of both local and temporal acceleration. This puzzled me because, in solid-body mechanics, we did not use this approach and only considered temporal acceleration.

I would like to ask you about the reason we use both local acceleration, which arises due to spatial changes in velocity, and temporal acceleration, which arises due to temporal changes in velocity, when calculating acceleration in fluid mechanics? This is mathematically expressed as:

$$a = \frac{Dv}{Dt} = \frac{∂v}{∂t} + (v ⋅ ∇) v$$

where temporal acceleration is represented by $\frac{∂v}{∂t}$ and local acceleration is represented by $(v ⋅ ∇) v$.

In contrast, why do we only consider temporal acceleration in solid-body mechanics, neglecting any spatial changes in velocity? In solid-body mechanics, acceleration is expressed as:

$$a = \frac{Dv}{Dt} = \frac{∂v}{∂t}$$

where only the change in velocity with respect to time is considered, without accounting for spatial variations.

Answer 1

I've never heard the phrase local acceleration used in this context before, and I think it hurts more than it helps.

They key thing to recognize is that the $v$ you are differentiating is not the velocity of a particle in any direct sense. Rather, it is a function of space and time, such that $v(x,t)$ is the average velocity of the fluid at the point $x$ and time $t$.

Now, imagine a tiny graduate student in a tiny raft which is being carried along by the flowing fluid. If their position at time $t$ is given by $x_r(t)$ ($r$ is for "raft"), then because their velocity is determined by the fluid, their velocity at time $t$ is given by

$$v\big(x_r(t), t\big)$$ and their acceleration by $$\frac{d}{dt} v\big(x_r(t), t\big) = \frac{\partial v}{\partial t}\big(x_r(t),t\big) + x'_r(t) \cdot \nabla v\big(x_r(t),t\big) \rightarrow \frac{\partial v}{\partial t} + (v\cdot \nabla) v$$

Typically we talk about a tiny parcel of fluid flowing along rather than a graduate student, but the same idea applies. The acceleration of a parcel can be broken down into two parts - first, the fluid velocity could be changing at the parcel's location, and second, the fluid velocity might be different in different locations.

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By way of example, one might imagine two limiting cases. In the first, we might have a fluid flowing uniformly through a pipe, with the flow velocity increasing with time. The velocity is the same everywhere in space (so $(v\cdot \nabla)v = 0$), but each second the flow rate is going up. This would correspond to the $\partial v/\partial t$ contribution.

Secondly, you might imagine a pipe which is steadily becoming narrower, causing the fluid velocity to increase the further you go down the pipe (since the flow rate must be constant). At any given point, the fluid velocity is constant in time (so $\partial v/\partial t=0$) but the fluid velocity changes with position, corresponding to the $(v\cdot \nabla)v$ contribution.

In a generic flow, both of these things are happening, so the acceleration of a fluid parcel co-moving with the fluid velocity is the sum of the two contributions.