💧Fluid Mechanics Unit 9 – Viscous Flow in Pipes

Key Concepts and Definitions

• Viscous flow involves fluids with internal resistance to flow due to friction between fluid layers
• Pipe flow is the movement of fluids through closed conduits or pipes
• Laminar flow occurs when fluid moves in parallel layers without mixing between the layers
• Turbulent flow is characterized by chaotic and irregular motion of fluid particles with mixing between layers
• Velocity profile represents the variation of fluid velocity across the pipe cross-section
• Shear stress is the force per unit area acting parallel to the pipe wall, caused by friction between fluid layers
• Pressure drop is the decrease in pressure along the length of the pipe due to friction and energy losses
• Friction factor is a dimensionless quantity that relates the pressure drop to the flow velocity and pipe roughness

Laminar vs Turbulent Flow

• Laminar flow occurs at low velocities and is characterized by smooth, parallel fluid layers (honey)
  • Fluid particles move in straight lines parallel to the pipe axis
  • Velocity profile is parabolic, with maximum velocity at the center and zero velocity at the pipe wall
• Turbulent flow occurs at high velocities and is characterized by chaotic and irregular motion of fluid particles (fast-flowing river)
  • Fluid particles have random fluctuations and mixing between layers
  • Velocity profile is flatter, with more uniform velocity distribution across the pipe cross-section
• Transition between laminar and turbulent flow depends on the Reynolds number, which considers fluid properties, velocity, and pipe diameter
• Laminar flow typically occurs at Reynolds numbers below 2300, while turbulent flow occurs at Reynolds numbers above 4000
• The transition region between laminar and turbulent flow (2300 < Re < 4000) exhibits intermittent and unstable flow behavior

Velocity Profiles in Pipes

• Velocity profile represents the variation of fluid velocity across the pipe cross-section
• In laminar flow, the velocity profile is parabolic, with maximum velocity at the center and zero velocity at the pipe wall
  • The velocity distribution is given by: $v(r) = v_{max}[1 - (\frac{r}{R})^2]$, where $v_{max}$ is the maximum velocity at the center, $r$ is the radial distance from the center, and $R$ is the pipe radius
• In turbulent flow, the velocity profile is flatter, with more uniform velocity distribution across the pipe cross-section
  • The velocity near the pipe wall is reduced due to the presence of a thin boundary layer
• The average velocity ($v_{avg}$) is related to the maximum velocity ($v_{max}$) by a factor that depends on the flow regime
  • For laminar flow: $v_{avg} = \frac{1}{2}v_{max}$
  • For turbulent flow: $v_{avg} \approx 0.8v_{max}$

Shear Stress Distribution

• Shear stress in pipe flow is caused by friction between fluid layers and between the fluid and the pipe wall
• The shear stress distribution varies across the pipe cross-section and depends on the flow regime
• In laminar flow, the shear stress varies linearly from zero at the center to a maximum value at the pipe wall
  • The maximum shear stress at the wall is given by: $\tau_{w} = \frac{4\mu v_{avg}}{D}$, where $\mu$ is the fluid viscosity, $v_{avg}$ is the average velocity, and $D$ is the pipe diameter
• In turbulent flow, the shear stress distribution is more complex due to the chaotic motion of fluid particles
  • The shear stress near the pipe wall is higher than in laminar flow due to increased mixing and momentum transfer
• The shear stress at the pipe wall is related to the pressure drop and the friction factor, which depends on the flow regime and pipe roughness

Friction Factor and Pressure Drop

• Friction factor ($f$) is a dimensionless quantity that relates the pressure drop to the flow velocity and pipe roughness
• The pressure drop ($\Delta P$) in a pipe is caused by friction and energy losses and can be calculated using the Darcy-Weisbach equation: $\Delta P = f\frac{L}{D}\frac{\rho v_{avg}^2}{2}$, where $L$ is the pipe length, $D$ is the pipe diameter, $\rho$ is the fluid density, and $v_{avg}$ is the average velocity
• For laminar flow, the friction factor depends only on the Reynolds number and is given by: $f = \frac{64}{Re}$
• For turbulent flow, the friction factor depends on both the Reynolds number and the relative pipe roughness ($\varepsilon/D$), where $\varepsilon$ is the absolute pipe roughness
  • The friction factor can be determined using the Moody diagram or empirical correlations such as the Colebrook-White equation
• Pressure drop increases with increasing flow velocity, pipe length, and pipe roughness, and decreases with increasing pipe diameter

Reynolds Number and Flow Regimes

• Reynolds number ($Re$) is a dimensionless quantity that characterizes the flow regime and the ratio of inertial forces to viscous forces
• The Reynolds number is defined as: $Re = \frac{\rho v_{avg}D}{\mu}$, where $\rho$ is the fluid density, $v_{avg}$ is the average velocity, $D$ is the pipe diameter, and $\mu$ is the fluid viscosity
• Laminar flow occurs at low Reynolds numbers (Re < 2300), where viscous forces dominate, and the flow is smooth and predictable
• Turbulent flow occurs at high Reynolds numbers (Re > 4000), where inertial forces dominate, and the flow is chaotic and irregular
• The transition region between laminar and turbulent flow (2300 < Re < 4000) exhibits intermittent and unstable flow behavior
  • In this region, the flow may alternate between laminar and turbulent states
• The critical Reynolds number ($Re_{cr}$) marks the onset of turbulence and is approximately 2300 for pipe flow
• The flow regime affects various flow characteristics such as velocity profile, shear stress distribution, and pressure drop

Energy Loss in Pipe Systems

• Energy loss in pipe systems is primarily due to friction between the fluid and the pipe wall and between fluid layers
• The total energy loss ($h_L$) in a pipe system consists of major losses ($h_f$) due to friction and minor losses ($h_m$) due to fittings, valves, and other components
  • $h_L = h_f + h_m$
• Major losses can be calculated using the Darcy-Weisbach equation: $h_f = f\frac{L}{D}\frac{v_{avg}^2}{2g}$, where $f$ is the friction factor, $L$ is the pipe length, $D$ is the pipe diameter, $v_{avg}$ is the average velocity, and $g$ is the gravitational acceleration
• Minor losses can be estimated using loss coefficients ($K$) specific to each component: $h_m = K\frac{v_{avg}^2}{2g}$
  • Loss coefficients for common fittings and valves can be found in engineering handbooks or manufacturer data
• Energy loss in pipe systems can be minimized by reducing flow velocity, using smooth pipes, minimizing the number of fittings and valves, and optimizing pipe layout and sizing
• The energy grade line (EGL) and hydraulic grade line (HGL) are used to visualize the energy loss along a pipe system
  • The EGL represents the total energy (elevation, pressure, and velocity) at each point along the pipe
  • The HGL represents the sum of elevation and pressure head at each point along the pipe

Real-World Applications and Examples

• Viscous flow in pipes is encountered in various engineering applications, such as water distribution networks, oil and gas pipelines, and industrial process piping
• In water distribution networks, understanding viscous flow is essential for designing efficient and reliable systems that deliver water to homes and businesses
  • Proper sizing of pipes, selection of materials, and layout optimization help minimize energy losses and ensure adequate water pressure and flow rates
• Oil and gas pipelines transport fluids over long distances, and viscous flow principles are crucial for predicting pressure drop, flow rates, and pumping requirements
  • Pipeline designers must consider factors such as fluid properties, flow regimes, and pipe roughness to optimize pipeline performance and minimize energy consumption
• Industrial process piping systems, such as those found in chemical plants and refineries, involve the transport of various fluids with different properties and flow requirements
  • Understanding viscous flow is essential for designing and troubleshooting these systems, ensuring efficient and safe operation
• Heating, ventilation, and air conditioning (HVAC) systems rely on viscous flow principles to distribute air and regulate temperature and humidity in buildings
  • Proper sizing and layout of ducts, selection of fans and blowers, and control of flow rates are critical for achieving optimal thermal comfort and energy efficiency
• Biomedical applications, such as blood flow in the cardiovascular system, involve viscous flow in complex geometries
  • Understanding the flow behavior of blood in arteries, veins, and capillaries is essential for diagnosing and treating cardiovascular diseases and designing medical devices such as stents and artificial heart valves