## Need help with understanding water flow through branching pipes of different diameter

For a simple problem, assume a 4" ID pipe has a smaller ID pipe branching of of it at a right angle. I want to know about what percentage of flow is going to divert through the small side line versus continue down the 4" main line. Assume the 4" inlet has fluid velocity v_1, Area A_1, and Flow Q_1; the 4" outlet has fluid velocity v_2, Area A_2=A_1, and Flow Q_2, The side outlet has velocity v_3, Area A_3=(A_1)/4, and Flow Q_3.

Intuitively, I felt that Q_3 should be about 25% of Q_1 since A_3 is a quarter the area of A_1 and Q_2 should be about 75% of Q_1, but am struggling to prove it mathematically.

I was thinking it could be as simple as using the continuity formula A_1*v_1=A_2*v_2+A_3*v_3 or Q_1=Q_2+Q_3. By substituting and solving for Q_3 I get: Q_3=(Q_1+Q_2)/4

Can someone please check my work, and help me prove/disprove my hypothesis? I'm not looking for a rigorous application of fluid mechanics so I'm assuming laminar flow and ideal conditions to make the calculation simple. I just want to see if my intuition is correct.

Thanks very much!