# Flow Classification (Compressibility – Rotation)

### 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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