# A two-dimensional potential flow is produced entirely by three sources.docx

3 de Feb de 2023
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### A two-dimensional potential flow is produced entirely by three sources.docx

• 1. A two-dimensional potential flow is produced entirely by three sources that are located on the y- axis. They are (i) a source of strength 1 at y 3; (ii) a source of strength 2 at y 2; (iii) a source of strength 3 at y 1. (a) There are two stagnation points on the y-axis. Find their locations (b) Give a simple argument to justify the fact that there are no other stagnation points in the (z,y) plane. Solution The equaiton for the source is velocity = Q/(2 pi r) So for this we have v=Q1/(2 pi r)+Q2/(2 pi (1-r))+Q3/(2 pi (2-r)), we have normalized the r distance. substitute 1,2, and 3 for Q1 Q2 and Q3 respectively, 0=1/(2 pi (r))-2/(2 pi (1-r))+3/(2 pi (1+(1-r)) we know that the stangation points will take place inbetween Q1 and Q2 plus Q3 and Q2. so adjust the equation so that our q1 is negative. 0=-1/(2 pi (r))-2/(2 pi (1-r))+3/(2 pi (1+(1-r)), solve for r, r= .2324. meaning at 3-.2324 so y=2.7676 is a stangation point. re-normalize the r's to find the next point. 0=-1/(2 pi (1+r))-2/(2 pi (r))+3/(2 pi ((1-r)), r= .7139 so 2-.7129 so y= 1.2861 there are no other points since above and below all the velocities would be heading in the same direction.