This **question** belongs to the **physics** domain and aims to explain the **concepts** of **velocity,** velocity **field,** and **flow** field.

Velocity can be **described** as the rate of **transformation** of the object’s position concerning a **frame** of concern and **time.** It sounds complex but **velocity** is essentially **speeding** in a particular **direction.** Velocity is a vector **quantity,** which means it requires both the **magnitude** (speed) and **direction** to describe **velocity.** The SI unit of velocity is **meter** per **second** $ms^{-1}$. **Acceleration** is the change in **magnitude** or the **direction** of the **velocity** of a body.

The **velocity** field indicates an **allocation** of velocity in a **region.** It is **represented** in a **functional** form as $V(x,y,z,t)$ **implying** that velocity is a part of the **time** and **spatial** coordinates. It is **helpful** to recall that we are **examining** fluid flow **beneath** the Continuum Hypothesis that allows us to **express** velocity at a point. **Further,** velocity is a vector **quantity** having **direction** and **magnitude.** This is **demonstrated** by noting the **velocity** field as:

\[ \overrightarrow{V} =\overrightarrow{V}(x,y,z,t) \]

**Velocity** has three **components,** one in each **direction,** that is $u,v$, and $w$ in $x, y$, and $z$ **directions,** respectively. It is typical to write \overrightarrow{V} as:

\[ \overrightarrow{V} = u\overrightarrow{i} + v\overrightarrow{j} + w\overrightarrow{k} \]

It is **precise** that each of $u,v,$ and $w$ can be **functions** of $x,y,z,$ and $t$. Thus:

\[ \overrightarrow{V} = u(x,y,z,t) \overrightarrow{i} + v(x,y,z,t) \overrightarrow{j} + w(x,y,z,t) \overrightarrow{k} \]

The way of **examining** the fluid motion that **emphasis** on explicit locations in the **space** via the fluid **flows** as the time passes is the **Eulerian specification of the flow field.** This can be **pictured** by **seating** on the bank of a river and overseeing the water pass the **patched** location.

The **stagnation** point is a point on the **surface** of a solid body **engaged** in a fluid **streamlet** which directly meets the **stream** and at which the **streamlines** separate.

## Expert Answer

In **two-dimensional** flows, The gradient of the streamline$\dfrac{dy}{dx}$, must be equivalent to the **tangent** of the angle that velocity vector **creates** with the x axis.

Velocity field **component’s** are given as:

\[ u = x+y \]

\[ v= xy^3 +16 \]

\[ w=0\]

Here we have $V=0$, therefore:

\[ u = x+y \]

\[ 0 = x+y \]

\[ x = -y \]

\[ v = xy^3 +16 \]

\[ 0 = xy^3 +16 \]

\[ -16 = xy^3 \]

\[ -16 = (-y)y^3 \]

\[ 16 = y^4 \]

\[ y_{1,2} = \pm 2 \]

## Numerical Answer

**Stagnation** points are $A_1(-2,2)$ and $A_2(2,-2)$.

## Example

The **velocity** field of a flow is **given** by $V= (5z-3)I + (x+4)j + 4yk$, where $x,y,z$ in feet. Determine the **fluid** speed at the origin $(x=y=z=0)$ and on the x axis $(y=z=0)$.

\[u=5z-3\]

\[v=x+4\]

\[w=4y\]

At origin:

\[u=-3\]

\[v=4\]

\[w=0\]

So that:

\[V=\sqrt{u^2 + v^2 + w^2}\]

\[V=\sqrt{(-3)^2 + 4^2 }\]

\[V= 5\]

**Similarly,** on the x-axis:

\[u=-3\]

\[v=x+4 \]

\[w=0\]

\[V=\sqrt{(-3)^2 + (x+4)^2 } \]

\[V=\sqrt{x^2 +8x +25 } \]