MHB What is a velocity field and its relationship to a fluid element's motion?

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A velocity field is defined as a function that maps each point in a fluid's domain to its velocity at a specific time. For a given time t, the velocity field is represented as u(·, t), where the dot signifies that the position variable is fixed while time is held constant. This notation emphasizes that the velocity field depends on both position and time, but for clarity, it is often simplified to focus on the position variable. The distinction is important because the velocity u requires two arguments: position and time. Understanding this concept is crucial for analyzing fluid motion and dynamics.
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Hey! :o

Let the fluid occupies the space $D \subset \mathbb{R}^n, n=2 \text{ or } 3$.
$\overrightarrow{x}$ is a point of $D$.
We consider the element of the fluid that is at the position $\overrightarrow{x}$ at the time $t$ , and moves along the trajectory $\Gamma$.

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Let $\overrightarrow{u}(\overrightarrow{x}, t)$ the velocity of this element. For a given time $t$, $\overrightarrow{u}(\cdot , t)$ is a vector field over $D$, and is called velocity field.

Could you explain to me the last part:

"For a given time $t$, $\overrightarrow{u}(\cdot , t)$ is a vector field over $D$, and is called velocity field."

?? (Wondering)
 

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mathmari said:
Hey! :o

Let the fluid occupies the space $D \subset \mathbb{R}^n, n=2 \text{ or } 3$.
$\overrightarrow{x}$ is a point of $D$.
We consider the element of the fluid that is at the position $\overrightarrow{x}$ at the time $t$ , and moves along the trajectory $\Gamma$.



Let $\overrightarrow{u}(\overrightarrow{x}, t)$ the velocity of this element. For a given time $t$, $\overrightarrow{u}(\cdot , t)$ is a vector field over $D$, and is called velocity field.

Could you explain to me the last part:

"For a given time $t$, $\overrightarrow{u}(\cdot , t)$ is a vector field over $D$, and is called velocity field."

?? (Wondering)

Hi! (Blush)

Formally, we would say that the velocity field $\overrightarrow v$ is a function of position to velocity, that is, $\overrightarrow v: D \to \mathbb{R}^n$, given by $\overrightarrow v(\overrightarrow{x}) = \overrightarrow{u}(\overrightarrow{x}, t)$.
The latter can also written as $\overrightarrow v(\cdot) = \overrightarrow{u}(\cdot, t)$, without changing the meaning, where $\cdot$ is an arbitrary symbol. (Nerd)To abbreviate it, we would like to say that $\overrightarrow{u}$ is the velocity field, but that would not be correct, since $\overrightarrow{u}$ takes 2 arguments and we need one argument with a fixed $t$.
So we abbreviate it as $\overrightarrow{u}(\cdot, t)$ with the understanding that $\cdot$ is the placeholder for the implicit argument. (Wasntme)
 
I see... Thanks for the explanation! (flower)
 
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