Two-Dimensional Flow

Fluid motion is said to be two-dimensional when the velocity at every point is parallel to a fixed plane, and is the same everywhere on a given normal to that plane. Thus, in Cartesian coordinates, if the fixed plane is the $x$-$y$ plane then we can express a general two-dimensional flow pattern in the form

$${\bf v} = v_x(x,y,t)\,{\bf e}_x + v_y(x,y,t)\,{\bf e}_y.$$ (468)

Figure 23: Two-dimensional flow.

Let $A$ be a fixed point in the $x$-$y$ plane, and let $ABP$ and $ACP$ be two curves, also in the $x$-$y$ plane, that join $A$ to an arbitrary point $P$. See Figure 23. Suppose that fluid is neither created nor destroyed in the region, $R$ (say), bounded by these curves. Since the fluid is incompressible, which essentially means that its density is uniform and constant, fluid continuity requires that the rate at which the fluid flows into the region $R$, from right to left across the curve $ABP$, is equal to the rate at which it flows out the of the region, from right to left across the curve $ACP$. Now, the rate of fluid flow across a surface is generally termed the flux. Thus, the flux (per unit length parallel to the $z$-axis) from right to left across $ABP$ is equal to the flux from right to left across $ACP$. Since $ACP$ is arbitrary, it follows that the flux from right to left across any curve joining points $A$ and $P$ is equal to the flux from right to left across $ABP$. In fact, once the base point $A$ has been chosen, this flux only depends on the position of point $P$, and the time $t$. In other words, if we denote the flux by $\psi$ then it is solely a function of the location of $P$ and the time. Thus, if point $A$ lies at the origin, and point $P$ has Cartesian coordinates $x$, $y$, then we can write

$$\psi= \psi(x,y,t).$$ (469)

The function $\psi$ is known as the stream function. Moreover, the existence of a stream function is a direct consequence of the assumed incompressible nature of the flow.

Consider two points, $P_1$ and $P_2$, in addition to the fixed point $A$. See Figure 24. Let $\psi_1$ and $\psi_2$ be the fluxes from right to left across curves $AP_1$ and $AP_2$. Now, using similar arguments to those employed above, the flux across $AP_2$ is equal to the flux across $AP_1$ plus the flux across $P_1 P_2$. Thus, the flux across $P_1 P_2$, from right to left, is $\psi_2-\psi_1$. Now, if $P_1$ and $P_2$ both lie on the same streamline then the flux across $P_1 P_2$ is zero, since the local fluid velocity is directed everywhere parallel to $P_1 P_2$. It follows that $\psi_1=\psi_2$. Hence, we conclude that the stream function is constant along a streamline. The equation of a streamline is thus $\psi=c$, where $c$ is an arbitrary constant.

Figure 24: Two-dimensional flow.

Let $P_1 P_2=\delta s$ be an infinitesimal arc of a curve that is sufficiently short that it can be regarded as a straight-line. The fluid velocity in the vicinity of this arc can be resolved into components parallel and perpendicular to the arc. The component parallel to $\delta s$ contributes nothing to the flux across the arc from right to left. The component perpendicular to $\delta s$ contributes $v_\perp\,\delta s$ to the flux. However, the flux is equal to $\psi_2-\psi_1$. Hence,

$$v_\perp = \frac{\psi_2-\psi_1}{\delta s}.$$ (470)

In the limit $\delta s\rightarrow 0$, the perpendicular velocity from right to left across $ds$ becomes

$$v_\perp = \frac{d\psi}{ds}.$$ (471)

Thus, in Cartesian coordinates, by considering infinitesimal arcs parallel to the $x$- and $y$-axes, we deduce that

$$v_x = -\frac{\partial \psi}{\partial y},$$ (472)

$$v_y = \frac{\partial\psi}{\partial x}.$$ (473)

These expressions can be combined to give

$${\bf v} = {\bf e}_z\times \nabla\psi= \nabla z\times \nabla \psi.$$ (474)

Note that when the fluid velocity is written in this form then it immediately becomes clear that the incompressibility constraint $\nabla\cdot{\bf v}=0$ is automatically satisfied [since $\nabla\cdot(\nabla A\times \nabla B)= 0$--see Equations (1480) and (1481)]. It is also clear that the stream function is undefined to an arbitrary additive constant.

The vorticity in two-dimensional flow takes the form

$$\mbox{\boldmath$\omega$}= \omega_z\,{\bf e}_z,$$ (475)

where

$$\omega_z = \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}.$$ (476)

Thus, it follows from Equations (472) and (473) that

$$\omega_z=\frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial y^2}=\nabla^2\psi.$$ (477)

Moreover, irrotational two-dimensional flow is characterized by

$$\nabla^2\psi = 0.$$ (478)

When expressed in terms of cylindrical coordinates (see Section C.3), Equation (474) yields

$${\bf v} = v_r(r,\theta,t)\,{\bf e}_r + v_\theta(r,\theta,t)\,{\bf e}_\theta,$$ (479)

where

$$v_r = - \frac{1}{r}\,\frac{\partial \psi}{\partial\theta},$$ (480)

$$v_\theta = \frac{\partial\psi}{\partial r}.$$ (481)

Moreover, the vorticity is $\mbox{\boldmath$\omega$}= \omega_z\,{\bf e}_z$, where

$$\omega_z =\frac{1}{r}\,\frac{\partial}{\partial r}\!\left(r\... ...rac{1}{r^{\,2}}\,\frac{\partial^2\psi}{\partial \theta^{\,2}}.$$ (482)