Glycerine flows steadily through a horizontal tube of length...

Glycerine flows steadily through a horizontal tube of length $1.5 \mathrm{~m}$ and radius $1.0$ $\mathrm{cm}$. If the amount of glycerine collected per second at one end is $4.0 \times 10^{-3} \mathrm{~kg} \mathrm{~s}^{-1}$, what is the pressure difference between the two ends of the tube ? (Density of glycerine $=1.3 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ and viscosity of glycerine $\left.=0.83 \mathrm{~Pa} \mathrm{~s}\right) .$ [You may also like to check if the assumption of laminar flow in the tube is correct].

Key Concepts

Viscosity

Viscosity is a measure of a fluid's resistance to deformation and flow. It plays a central role in fluid mechanics by determining how much internal friction a fluid exhibits, which directly affects the pressure drop and flow rate in pipe systems.

Laminar Flow

Laminar flow describes a flow regime where the fluid moves in smooth, orderly layers with minimal mixing between them. This concept is critical because many theoretical analyses and formulas, like Poiseuille's law, assume a laminar flow condition, which typically occurs at lower flow velocities or for fluids with high viscosity.

Reynolds Number

The Reynolds number is a dimensionless quantity used to predict the flow regime within a fluid. It is the ratio of inertial forces to viscous forces, and low Reynolds numbers indicate laminar flow. Checking the Reynolds number ensures the correct application of laminar flow equations like Poiseuille's Law.

Mass and Volumetric Flow Rate Conversion

Mass flow rate refers to the mass of fluid passing through a cross-section per unit time, while volumetric flow rate is the volume of fluid that passes per unit time. The two are related by the density of the fluid, which is essential when converting between these rates in problem solving.

Pressure Difference

Pressure difference is the driving force for fluid flow in a system. In pipe flow problems, it represents the difference in pressure between two ends of the tube that overcomes viscous resistance, and it is calculated using laws like Poiseuille's Law when flow is laminar.

Poiseuille's Law

This law, also known as the Hagen-Poiseuille equation, relates the pressure drop across a pipe to the viscosity of the fluid, the length and radius of the tube, and the volumetric flow rate. It applies strictly to laminar flow in circular pipes and is fundamental for calculating pressure differences in viscous flows.

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