Question

Three capillary tubes of the same radius r but of length `l_1, l_2 and l_3` are fitted horizontally to the bottom of a long cylinder containing a liquid at constant head and flowing through these tubes. Find the length of a single overflow tube of the same radius r, which can replaced the three capillaries.

Answer 1

Let A, B and C be three capillaries of elght `l_1, l_2 and l_3` connected in parallel as shown in

r be the radius of each capillary. Let p be the pressure difference across the ends of each capillary tube Let `V_1, V_2 and V_3` be the rates of flow of liquid through A, B and C, then `V_1 = (pi p r^4)/(8 eta l_1) , V_2 = (pi p r^4)/(8 eta l_2) and V_3 = (pi p r^4)/(8 eta l_3)`
total rate of flow of liquid is given by `V = V_1 + V_2 + V_3 =(pi p r^4)/(8 eta) [ (1)/(l_1) + (1)/(l_2) + (1)/(l_3)] .... (i)`
Let l be the length of a single tube through which the same rate of flow of liquid (i.e. V) is to take place under a pressrue difference p, then `V = (pi p r^4)/(8 eta l) .... (ii)` From (i) and (ii) we get `(pi pr^4)/(8eta l) =(pi p r^4)/(8 eta) [ (1)/(l_1) + (1)/(l_2) +(1)/(l_3)] or (1)/(l) = (1)/(l_1) + (1)/(l_2) + (1)/(l_3) or l =(l_1l_2l_3)/(l_2 l_3 + l_3l_1 + l_1l_2)`