# Question about source flow rate across line AB.

tracker890 Source h
Homework Statement:
Determine flow rate per unite width in the line
Relevant Equations:
flow rate equation
Q：Please hlep me to understand which ans is correct.

To determine the flow rate in Line AB.
$$\mathrm{Known}：V_A,q,r_A = constant.$$

so/
select：## A,{B}^{\text{'}},B,A,## is control volume

$${Q}_{AB}={Q}_{A{B}^{\text{'}}}=\iint _{A}^{}({V}_{A})dA={\int }_{{\theta }_{A}}^{{\theta }_{B}}({V}_{A}){r}_{A}d\theta$$

$$\overset\rightharpoonup{V}=\triangledown \phi =<\frac{\partial \phi }{\partial r},\frac{1}{r}\frac{\partial \phi }{\partial \theta }>=<\frac{1}{r}\frac{\partial \psi }{\partial \theta },-\frac{\partial \psi }{\partial r}>=<{V}_{r},{V}_{\theta }>$$

$$\therefore V_A=\frac1{r_A}\frac{\partial\psi}{\partial\theta}\;$$

$$Q_{AB}=\int_{\theta_A}^{\theta_B}{(V_A)}r_Ad\theta\;=\;\int_{\theta_A}^{\theta_B}{(\frac1{r_A}\frac{\partial\psi}{\partial\theta})}r_Ad\theta=\int_{\theta_A}^{\theta_B}{(\frac{\partial\psi}{\partial\theta})}d\theta=\psi_B-\psi_A$$

to find ##\psi##,
$$F(z)=\frac q{2\pi}\ln(z)=\frac q{2\pi}ln(re^{i\theta})=\frac q{2\pi}\ln r+i\frac q{2\pi}\theta=\phi+i\psi$$

so $$\psi=\frac q{2\pi}\theta$$
$$Q_{AB}=\psi\left(\theta_B\right)\mathit-\psi\left(\theta_A\right)\mathit=\frac q{2\pi}(\theta_B-\theta_A)$$

$$\theta_A=\tan^{-1}\left(\frac11\right)=0.7854\;rad,$$
$$\theta_B=\frac\pi2+\tan^{-1}\left(\frac1{0.5}\right)=2.6779\;rad$$
So ans by myself is
$$\therefore Q_{AB}=\frac q{2\pi}{(2.6779-0.7854)}=0.3012q............(Ans(1))$$

$$////////////////////////$$
But book say：
$$\theta_A=\tan^{-1}\left(\frac yx\right)=\tan^{-1}\left(\frac{\mathit1}{\mathit1}\right)=0.7854\;rad$$

$$\theta_B=\tan^{-1}\left(\frac yx\right)=\tan^{-1}\left(\frac{0.5}{-1}\right)\;=\;-0.4636\;rad$$

$$Q_{AB}=\psi\left(\theta_A\right)-\psi\left(\theta_B\right)=\frac q{2\pi}{(0.7854+0.4636)}=0.19878q........(Ans(2))$$

Last edited:

Homework Helper
Gold Member
2022 Award
You may have noticed that the two values for θB differ by π.
The result of arctan is of course multi-valued, with the values at intervals of π. So it is always necessary to make sure that the right value is selected. Question is, which of you selected the right value? (I'm with you.)

tracker890 Source h
You may have noticed that the two values for θB differ by π.
The result of arctan is of course multi-valued, with the values at intervals of π. So it is always necessary to make sure that the right value is selected. Question is, which of you selected the right value? (I'm with you.)
I think the flow rate in book is ## {Q}_{A{B}^{*}} ## not ## {Q}_{A{B}^{\text{'}}}##.
So the book answer is not correct.
Am I right ?

Last edited:
Homework Helper
Gold Member
2022 Award
I think the flow rate in book is ## {Q}_{A{B}^{*}} ## not ## {Q}_{A{B}^{\text{'}}}##.
So the book answer is not correct.
Am I right ?
View attachment 318535
I think so.

tracker890 Source h
Mentor
I agree with your result. The included angle is ##\tan^{-1}2+\frac{\pi}{4}##

tracker890 Source h