### Flow Classification : Incompressible Flow VS Compressible Flow

In general, **liquid** is called an **incompressible fluid**, and **gas** a **compressible fluid**. Nevertheless, even in the case of a liquid it becomes necessary to take compressibility into account whenever the liquid is highly pressurized, such as oil in a hydraulic machine. Similarly, even in the case of a gas, the compressibility may be disregarded whenever the change in pressure is small. As a criterion for this judgement, the **Mach number M** is used, whose value, however, varies according to the nature of the situation.

### Flow Classification : Irrotational Flow VS Rotational Flow

Fluid particles running through a narrow channel flow, while undergoing deformation and **rotation**, are shown in the following figure.

\[\zeta = \frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}}\]

The previous equation gives what is called the **vorticity** for the z axis. The case where the **vorticity** is **zero**, namely the case where the fluid movement obeys is called **irrotational flow**.

As shown in the following figure, a cylindrical vessel containing liquid spins about the vertical axis at a certain angular velocity. The liquid makes a rotary movement along the flow line, and, at the same time, **the element itself rotates**. This is shown in the upper diagram of the figure, which shows how wood chips float, a well-studied phenomenon. In this case, it is a **rotational flow**, and it is called a **forced vortex flow**. Shown in the same figure is the case of rotating flow which is observed whenever liquid is made to flow through a small hole in the bottom of a vessel. Although the liquid makes a rotary movement, its micro elements always face the same direction without performing rotation. This case is a kind of **irrotational flow** called **free vortex flow**.

Hurricanes, eddying water currents and tornadoes are familiar examples of natural vortices. Although the structure of these vortices is complex, the basic structure has a **forced vortex** at its **center** and a **free vortex** on its **periphery**. Many natural vortices are generally of this type.

If the **flow velocities** are given as follows, show respectively whether the flows are **rotational** or **irrotational** (*k* = constant) :

\[u = - ky,v = kx\]

\[\begin{array}{c}

\frac{{\partial v}}{{\partial x}} = k\\

\frac{{\partial u}}{{\partial y}} = - k\\

\zeta = \frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} = k - ( - k) = 2k \ne 0

\end{array}\]

Therefore, the given flow is **rotational**.

\[u = {x^2} - {y^2},v = - 2xy\]

\[\begin{array}{c}

\frac{{\partial v}}{{\partial x}} = - 2y\\

\frac{{\partial u}}{{\partial y}} = - 2y\\

\zeta = \frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} = - 2y - ( - 2y) = 0

\end{array}\]

Therefore, the given flow is **irrotational**.

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