The volume of fluid flowing in a single time in the capillary tube V depends on the following factors- i. Tube radius, ii. Viscosity, iii. Pressure gradient. So evaluate the formula by analyzing the dimension.
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Answers
The flow of liquid through a tube
The rate at which fluid flows through a tube is likely to depend on
(a) the viscosity of the fluid,
(b) the dimensions of the tube, and
(c) the pressure difference between its ends.
This flow rate is of great importance in our lives since it governs things like the flow of blood round our bodies and the transmission of gas, water or oil through long distances in pipelines.
The proof of the relationship was first produced by Poiseuille in 1844 using dimensional analysis (a different proof based on the mechanics of fluids is available, but is outside the scope of this work at this level).
Consider a fluid of viscosity η flowing through a tube of length L and radius r due to a pressure difference Δp between its ends (Figure 1).
The formula for the rate of flow of fluid down the tube is:
Volume per second (V) = πr4Dp/8hL
This formula only applies to laminar flow, not to turbulent motion. It is interesting that warm-blooded animals regulate the heat loss from their bodies by changing the diameter of their blood vessels (varying r) and hence controlling the rate of blood flow.
A simple proof of this formula using dimensional analysis is given below:
Assume that the volume V passing through the tube per second is given by the equation:
V = kηxryDpz/L
where the quantity Dp/L is called the pressure gradient down the tube. Using dimensional analysis this gives x = -1, y = 4 and z = 1.
Therefore the volume per second is:
Volume per second = kr4Δp/ηL
The value of k can be shown to be π/8 and therefore:
Volume per second (V) = pr4Dp/8ηL
governs things like the flow of blood round our bodies and the transmission of gas, water or oil through long distances in pipelines.
The proof of the relationship was first produced by Poiseuille in 1844 using dimensional analysis (a different proof based on the mechanics of fluids is available, but is outside the scope of this work at this level).
Consider a fluid of viscosity η flowing through a tube of length L and radius r due to a pressure difference Δp between its ends (Figure 1).
The formula for the rate of flow of fluid down the tube is:
Volume per second (V) = πr4Dp/8hL
This formula only applies to laminar flow, not to turbulent motion. It is interesting that warm-blooded animals regulate the heat loss from their bodies by changing the diameter of their blood vessels (varying r) and hence controlling the rate of blood flow.
A simple proof of this formula using dimensional analysis is given below:
Assume that the volume V passing through the tube per second is given by the equation:
V = kηxryDpz/L
where the quantity Dp/L is called the pressure gradient down the tube. Using dimensional analysis this gives x = -1, y = 4 and z = 1.
Therefore the volume per second is:
Volume per second = kr4Δp/ηL
The value of k can be shown to be π/8 and therefore