Question

What is the circulation $\Gamma$ around a rectangle defined by

$x=1,y=1,x=5andy=4$

for the velocity field

$v=2x=3yandv=2y$

• A. - 18 units
• B. - 36 units
• C. 36 units
• D. 18 units

Answer 1

The correct option is B - 36 units


Circulation is defined as the line integral of velocity around a closed contour

$\Gamma ={\int }_{c}uds={\int }_{ABCD}(udx+vdy)$

$={\int }_{ABCD}\left[\right(2x+3y)dx-(-2y\left)dy\right]$

$={\int }_5^1(2x+3y)dx+{\int }_3^1(2x+3y)dx-{\int }_1^4\left(2y\right)dy-{\int }_4^2\left(2y\right)dy$

We can observe that y=1 for the first integral and y=4 for the second integral,'

$\Gamma ={|x^2+3x|}_1^5+{|x^2+12x|}_1^5={\left|y^2\right|}_2^4-{\left|y^2\right|}_4^2$

=$36-72=-36units$

Alternative Solution

We know the fact that circulation equals vorticity time the area vorticity

=$2{\omega }_2=\frac{\partial V}{\partial x}-\frac{\partial u}{\partial y}=0-3=-3$

$Areaoftherectangle=(5-1)\times (4-1)=4\times 3=12units$

$Circulation=Vorticity\times Area=-3\times =-36units$