Hagen-Poiseuille Equation Derivation from Basics

Understanding Pozoy's Law

Introduction to Pozoy's Law

• Pozoy's Law, discovered by French scientist Pozoy, pertains to the behavior of incompressible liquids flowing through pipes.

• The law can be applied not only to pipes but also to biological systems, such as blood flow in arteries.

Viscosity and Flow Dynamics

• Viscosity is a key property that creates friction between liquid layers moving at different velocities and against the pipe walls.

• To maintain a steady flow despite resistance from viscosity, higher pressure must be present on the upstream side compared to the downstream side.

Volumetric Flow Rate

• The volumetric flow rate is defined as the volume of liquid passing through a cross-section of the pipe per unit time, typically measured in liters per second or cubic meters per second.

• The formula for volumetric flow rate involves parameters like pressure difference (ΔP), radius (r), length (L), and a constant factor related to viscosity.

Characteristics of Laminar Flow

• Laminar flow refers to smooth, orderly movement of fluid layers without turbulence; it contrasts with turbulent flow characterized by chaotic motion.

• In laminar conditions, fluid molecules adjacent to stationary tube walls have nearly zero velocity due to friction.

Velocity Distribution in Tubes

• At any point within the tube radius (r), velocity varies; it is maximum at the center and decreases towards the walls due to viscous effects.

• The no-slip condition states that fluid layers in contact with tube walls do not slip, resulting in zero velocity at those points.

Understanding Viscosity Further

• Viscosity describes how adjacent liquid layers interact when subjected to shear stress; this interaction influences overall fluid dynamics within pipes.

Understanding Viscosity and Fluid Flow

Definition of Viscosity

• The coordinate y is perpendicular to the flow direction, with velocity v at y and v + dv at y + dy, illustrating a differential change in velocity.

• Viscosity defines how faster-moving liquid layers exert stress (friction force), represented as tau (τ), which is force per unit area.

• The stress τ is directly proportional to the velocity gradient (dv/dy), indicating that greater changes in velocity across layers result in higher friction forces.

Properties of Liquids

• Viscosity varies among liquids; it’s higher for substances like honey and glycerin compared to lower viscosity fluids like alcohol and water.

• A free body diagram of a cylindrical liquid section shows how pressure forces balance shear stress due to viscosity.

Analyzing Fluid Dynamics

• In radial coordinates, the negative velocity gradient indicates outer layers slow down inner layers by applying opposing friction forces.

• The cylindrical portion's length L matches the pipe's length, with pressure differences influencing flow dynamics through shear stress τ.

Balancing Forces in Steady Flow

• For steady flow, the extra pressure force must counterbalance total friction forces acting backward on the fluid cylinder.

• The surface area of the tube contributes to calculating net friction force using shear stress τ defined as η(dv/dr).

Integration and Flow Control Equation

• Integrating leads to an equation where negative gradients indicate decreasing velocities as radius increases; adjustments ensure positive delta P values.

• Limits of integration are set from small r (velocity v) to capital R (velocity zero), confirming no-slip conditions at the tube's surface.

Fluid Dynamics: Understanding Velocity and Flow Rate

Velocity as a Function of Radius

• The equation for velocity is derived as v(r) = Delta P/4 eta L (R^2 - r^2) , indicating that velocity varies with the radius r .

• At r = 0 , the maximum velocity v_0 occurs, while at r = R , the velocity becomes zero, illustrating the no-slip condition.

• The relationship shows a parabolic function due to the presence of r^2 , which affects how fluid flows within a cylindrical tube.

Volumetric Flow Rate Calculation

• To find volumetric flow rate, consider a ring element within the tube defined by radius r and width dr .

• The cross-sectional area of this ring is calculated as A = 2pi r,dr . This area is crucial for determining how much liquid flows through it.

• By multiplying this area by the velocity function, we can express volumetric flow rate ( Q ) in terms of these variables. Integration from 0 to capital R gives total flow rate.

Integration Process for Total Flow Rate

• Constants are factored out during integration; only variable terms remain inside.

• The integral simplifies to yield an expression involving pressure difference ( Delta P ), viscosity ( eta ), length ( L ), and powers of radius.

• Specifically, results show contributions from both linear and cubic terms in relation to radius, emphasizing their impact on overall flow dynamics.