Question about source flow rate across line AB.

• tracker890 Source h
In summary, the conversation is about determining the flow rate in Line AB using the known variables V_A, q, and r_A. The selected control volume is A, {B}', B, A and the equation for the flow rate is Q_AB = Q_A{B}', which is equal to the integral of V_A over the included angle between θ_A and θ_B. The vector V can be expressed as a gradient of a potential function ϕ and ψ, and the flow rate can be calculated using the potential function ψ. The question arises about the correct value of θ_B, with the speaker believing that the correct value is 2.6779 rad, while the book claims it is
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:
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.)

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

tracker890 Source h

1. What is source flow rate?

Source flow rate refers to the rate or amount of fluid or material that is being supplied or generated by a source, such as a pump or a natural spring.

2. What does "across line AB" mean in this context?

In this context, "across line AB" refers to the flow rate of the source measured at a specific point or location along a line connecting point A and point B.

3. How is source flow rate measured?

Source flow rate can be measured using various methods such as flow meters, pressure gauges, or by calculating the volume of fluid passing through a specific point over a given time period.

4. What factors can affect the source flow rate across line AB?

The source flow rate across line AB can be affected by various factors such as the type and size of the source, the pressure and temperature of the fluid, the diameter and length of the line, and any obstructions or restrictions in the flow path.

5. Why is it important to monitor the source flow rate across line AB?

Monitoring the source flow rate across line AB is important for various reasons, such as ensuring the proper functioning of the source and maintaining a consistent flow of material, identifying any potential issues or changes in flow rate, and determining the efficiency and effectiveness of the system.

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