Vorticity Equation in Rectangular Coordinates

In a two-dimensional motion, the vorticity at a point P, which is located perpendicular to the plane, is equal to the limit of the ratio of the circulation in an infinitesimal circuit embracing P to the area of the circuit.

A flow possesses vorticity if any of its elements are rotating (spinning). It is a convenient way to investigate the motion of a circular element, treating it as a solid, at the instant of time considered. Let P(x, y) be the center of the circular element and u and v are the velocity components, along x – and y-directions, respectively, as shown in Figure 5.3.

Подпись: Figure 5.3 A fluid element and appropriate coordinates and velocity components.

Let us assume that the fluid element consists of numerous fluid particles of mass Am each, such as one at the point Q(x + Sx, y + Sy). At point Q, the velocity components, along x – and y-directions,

Подпись: respectively, are:
Подпись: du du u H Sx d Sy dx dy
Подпись: and

dv dv

v +—- Sx +—– Sy.

dx dy

The moment of momentum (or angular momentum) of the fluid element about point P(x, y) is the sum of the moments of momentum of all the particles such as Q about point P. Taking the anti-clockwise moment as positive, we have:

Подпись: Am

Moment of momentum of the element

E

{ dv du.

AmYy — dx 1 SxSy.

For a circular disc, about its center, we have:

У ‘ Am SxSy = 0.

Therefore, the angular momentum of the disc becomes:

E

dv 2 X—> du 2

Am — (Sx) — У Am — (Sy) . dx ^’ dy

If the disc were a solid disc, its angular momentum would be Ію, where I is its polar moment of inertia about P and ю its angular velocity about P. Thus, assuming the fluid element as a solid disc, we have:

I = Am {(Sx)2 + (Sy)2}

and

YsAm (Sx)2 = E Am (Sy)2.

Thus, we have the angular momentum relation as:

m^2Am {(Sx)2 + (sy)2} = EAmdx(Sx)2—EAmdy(Sy)2.

This gives the angular velocity as:

^ dv du dx dy

The quantity 2w is the elemental spin, also referred to as vorticity, which is usually denoted as Z. Thus,

Подпись: dv du ^ dx dy(5.1)

Подпись: Z = 2ю

The units of Z are radian per second. From Equation (5.1) and the angular velocity relation, it is seen that:

that is, the vorticity is twice the angular velocity.