5.2 Couette Flow and Poiseuille Flow

Couette and Poiseuille flows are fundamental examples of viscous fluid motion. They showcase how boundary conditions and driving forces shape fluid behavior, with Couette flow driven by a moving plate and Poiseuille flow by pressure differences.

These flows illustrate key concepts in viscous fluid dynamics. Couette flow's linear velocity profile and Poiseuille flow's parabolic profile demonstrate how fluid properties and external forces interact to create distinct flow patterns in different scenarios.

Couette vs Poiseuille Flow Characteristics

Fundamental Definitions and Principles

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• Couette flow represents laminar flow of viscous fluid between two parallel plates with one stationary and one moving at constant velocity
• Poiseuille flow describes steady, laminar flow of incompressible viscous fluid through constant circular cross-section pipe or channel
• Both flows exemplify simple shear flows governed by Navier-Stokes equations under specific boundary conditions
• No-slip condition applies at boundaries for both flows, fluid velocity at walls equals wall velocity (zero for stationary walls)
• Both flows assume steady-state conditions, flow properties remain constant over time at any given point in fluid

Driving Mechanisms and Velocity Profiles

• Couette flow motion driven by shear stress induced by moving plate, resulting in linear velocity profile
• Poiseuille flow driven by pressure gradient along pipe or channel length, leading to parabolic velocity profile
• Couette flow velocity profile expressed as $u(y) = (U/h)y$ (U represents moving plate velocity, h denotes distance between plates)
• Poiseuille flow velocity profile given by $u(r) = (1/4μ)(-dp/dx)(R^2 - r^2)$ (μ represents dynamic viscosity, dp/dx denotes pressure gradient, R signifies pipe radius, r indicates radial position)

Velocity Profiles and Pressure Gradients

Derivation of Velocity Profiles

• Couette flow linear velocity profile derived from shear stress balance and no-slip boundary conditions
• Poiseuille flow parabolic velocity profile obtained by solving Navier-Stokes equations with cylindrical coordinates and appropriate boundary conditions
• Integration constants in velocity profile derivations determined by applying specific boundary conditions (stationary wall, moving wall, centerline symmetry)
• Velocity gradient in Couette flow remains constant across fluid layer, resulting in uniform shear stress

Pressure Gradients and Flow Characteristics

• Pressure gradient in Couette flow equals zero in flow direction, pressure remains constant along channel
• Poiseuille flow pressure gradient calculated using $dp/dx = -8μQ/(πR^4)$ (Q represents volumetric flow rate)
• Maximum velocity in Poiseuille flow occurs at centerline, expressed as $u_{max} = (-1/4μ)(dp/dx)R^2$
• Shear stress distributions derived using $τ = μ(du/dy)$ for Couette flow and $τ = -μ(du/dr)$ for Poiseuille flow
• Average velocity in Poiseuille flow equals half of maximum velocity, while in Couette flow it equals half of moving plate velocity

Viscosity's Effect on Velocity Profiles

Viscosity Impact on Flow Characteristics

• Increasing fluid viscosity in Couette flow decreases slope of linear velocity profile, reducing overall fluid velocity
• Higher viscosity in Poiseuille flow leads to steeper parabolic velocity profile, decreasing maximum centerline velocity
• Wall shear stress in both flows directly proportional to fluid viscosity, calculated using respective shear stress equations
• Viscosity changes affect Reynolds number, influencing transition from laminar to turbulent flow (lower viscosity leads to higher Reynolds number)

Viscosity Relationships with Flow Parameters

• Pressure gradient required to maintain constant flow rate in Poiseuille flow increases linearly with viscosity
• No-slip condition interacts with fluid viscosity to shape velocity profiles, creating velocity gradients near walls
• Viscosity determines velocity gradient magnitude, affecting momentum transfer between fluid layers
• In Couette flow, viscosity influences force required to maintain plate motion against fluid resistance

Couette vs Poiseuille Flow Comparison

Driving Forces and Velocity Profiles

• Couette flow driven by shear stress from moving plate, Poiseuille flow driven by pressure gradient
• Couette flow exhibits linear velocity profile, Poiseuille flow displays parabolic velocity profile
• Pressure gradient remains constant (zero) in Couette flow, constant non-zero in Poiseuille flow
• Couette flow involves moving upper plate, Poiseuille flow features stationary walls

Applications and Practical Considerations

• Couette flow applied in lubrication systems (journal bearings), rheometers, and viscometers
• Poiseuille flow models blood flow in vessels, fluid transport in microfluidic devices, and pipeline systems
• Couette flow more sensitive to changes in plate velocity, Poiseuille flow more affected by pipe diameter changes
• Poiseuille flow exhibits higher centerline velocity compared to average velocity, Couette flow maintains constant velocity gradient