Rate of flow through a single capillary tube:
The rate of steady flow of liquid through a capillary tube can be determined using Poiseuille's Law, which states that the rate of flow (v) is directly proportional to the pressure difference (P) and the fourth power of the radius (r) of the tube, and inversely proportional to the viscosity (η) and the length (L) of the tube.

Mathematically, the equation can be written as:
v = (πr^4ΔP) / (8ηL)

Connecting two capillary tubes in series:
When two capillary tubes are connected in series, the total length (L) of the combination remains the same, but the radius (r) of one of the tubes is halved.

Rate of flow through the combination:
To find the rate of flow through the combination of two capillary tubes connected in series, we need to consider the equivalent length and radius of the combination.

1. Equivalent length (Leq):
Since the two tubes are connected in series, the equivalent length of the combination is the sum of the lengths of the individual tubes.
Leq = L + L = 2L

2. Equivalent radius (req):
To find the equivalent radius of the combination, we use the fact that the total volume flow rate through the combination must be the same as the flow rate through each individual tube.

Let v1 be the flow rate through the first tube (with radius r) and v2 be the flow rate through the second tube (with radius r/2).

Since the pressure difference (P) across the combination is the same, we have:
v1 = v2

Using Poiseuille's Law, we can write the following equation for v1 and v2:
v1 = (πr^4ΔP) / (8ηL)
v2 = (π(r/2)^4ΔP) / (8ηL)

Equating v1 and v2, we get:
(πr^4ΔP) / (8ηL) = (π(r/2)^4ΔP) / (8ηL)

Simplifying the equation, we find:
r^4 = (r/2)^4

Taking the fourth root on both sides, we get:
r = r/2

Simplifying further, we find:
r = 2r/2

Therefore, the equivalent radius of the combination is r.

Using Poiseuille's Law with the equivalent length (Leq) and radius (req), we can calculate the rate of flow (vcomb) through the combination:
vcomb = (πreq^4ΔP) / (8ηLeq)

Substituting r for req and 2L for Leq, we get:
vcomb = (πr^4ΔP) / (8η(2L))

Simplifying the equation, we find:
vcomb = v

Hence, the rate of flow through the combination of two capillary tubes connected in series is the same as the rate of flow through each individual tube when the same pressure difference is maintained across the combination.