Streamline and acceleration of a flow field

Given: $u=2x$, $v=-2y$, steady 2D flow.

(a) Incompressibility

$$\nabla\cdot\vec V=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=2+(-2)=0$$

Since the divergence is zero, the continuity equation for incompressible flow is satisfied.

(b) Streamline through (2, 1)

$$\frac{dx}{u}=\frac{dy}{v}\ \Rightarrow\ \frac{dx}{2x}=\frac{dy}{-2y}\ \Rightarrow\ \frac{dx}{x}+\frac{dy}{y}=0$$

Integrating gives $\ln x+\ln y=\ln C$, so $xy=C$.

At (2, 1): $C=2\times1=2$.

Streamline: $xy = 2$ (a rectangular hyperbola). This is a stagnation-point flow: fluid approaches the origin along the y-axis and leaves along the x-axis.

(c) Acceleration

The flow is steady, so only the convective part remains:

$$a_x=u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=(2x)(2)+0=4x$$ $$a_y=u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=0+(-2y)(-2)=4y$$

At (2, 1): $a_x=8\ \text{m/s}^2$, $a_y=4\ \text{m/s}^2$.

$$|a|=\sqrt{8^2+4^2}=8.94\ \text{m/s}^2$$

Direction: $\tan^{-1}(4/8)=26.6^\circ$ above the x-axis.

(d) Rotation

$$\omega_z=\frac12\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=\frac12(0-0)=0$$

The flow is irrotational.

Answer: (a) $\nabla\cdot\vec V=0$; (b) $xy=2$; (c) $a=8.94\ \text{m/s}^2$ ($a_x=8$, $a_y=4$ m/s²); (d) irrotational.