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The components of a velocity field are given by u= x+y, v=xy^3 +16, and w=0. Determine the location of any stagnation points (V=0) in the flow field.

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:

\[u=-3\]

\[v=x+4 \]

\[w=0\]

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

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

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