Fluid Mechanics: Laminar vs. Turbulent Flow

• Fluid mechanics is the study of fluids (liquids and gases) at rest and in motion.

Laminar Flow

• Laminar flow features smooth, orderly movement of fluid in parallel layers, lacking disruption.
• Lower velocities and higher viscosities characterize it.
• Adjacent fluid layers do not mix.
• It is predictable.
• Fluid particles travel in straight lines.
• Streamline flow or viscous flow are alternate names.

Turbulent Flow

• Turbulent flow has chaotic, disordered fluid motion and velocity fluctuations.
• Higher velocities and lower viscosities are characteristics.
• Mixing of fluid layers occurs.
• It is unpredictable.
• Fluid particles travel in irregular paths.

Reynolds Number

• The Reynolds number (Re) is a dimensionless quantity predicting flow type, whether laminar or turbulent.
• It is the ratio of inertial forces to viscous forces.
• Laminar flow (viscous forces dominate) is indicated by a low Reynolds number.
• Turbulent flow (inertial forces dominate) is indicated by a high Reynolds number.
• ( Re = \frac{\rho V L}{\mu} ) is the formula, where:
  • ( \rho ) is fluid density
  • ( V ) is fluid velocity
  • ( L ) is a characteristic length
  • ( \mu ) is dynamic viscosity

Viscosity

• Viscosity measures a fluid's resistance to flow.
• It describes the internal friction of a fluid.
• High viscosity describes a thick fluid resisting flow.
• Low viscosity describes a thin fluid that flows easily.
• Dynamic viscosity ((\mu)) is the tangential force per unit area needed to move one fluid layer past another at unit velocity, separated by a unit distance.
• Kinematic viscosity ((\nu)) is the ratio of dynamic viscosity to density: ( \nu = \frac{\mu}{\rho} )
• Pascal-seconds (Pa·s) or Poise (P) are units of dynamic viscosity.
• Square meters per second (m²/s) or Stokes (St) are units of kinematic viscosity.

Newtonian Fluids

• Newtonian fluids have a linear relationship between shear stress and shear rate.
• Viscosity remains constant, regardless of applied shear force.
• Water, air, and thin oils are examples.

Non-Newtonian Fluids

• Non-Newtonian fluids have a non-linear relationship between shear stress and shear rate.
• Viscosity changes under applied force.
• Paint, blood, and ketchup are examples.
• Shear-thinning (pseudoplastic) fluids show decreased viscosity as shear stress increases.
• Shear-thickening (dilatant) fluids show increased viscosity as shear stress increases.

Bernoulli's Equation

• Bernoulli's equation relates pressure, velocity, and elevation in flowing fluids.
• It is derived from the principle of energy conservation.
• It assumes incompressible, inviscid (no viscosity), and steady flow.
• ( P + \frac{1}{2} \rho V^2 + \rho g h = \text{constant} ) is the equation, where:
  • ( P ) is pressure
  • ( \rho ) is fluid density
  • ( V ) is fluid velocity
  • ( g ) is the acceleration caused by gravity
  • ( h ) is elevation
• The sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline.
• An increase in velocity corresponds with a decrease in pressure or a decrease in the fluid's potential energy.

Applications of Bernoulli's Equation

• Aircraft lift generation occurs from faster air movement over the wing creating lower pressure.
• Carburetors draw fuel into the air stream due to pressure differences.
• Venturi meters and orifice plates measure flow rates by using pressure differences.

Limitations of Bernoulli's Equation

• It assumes inviscid flow (no viscosity), untrue for real fluids.
• It assumes incompressible flow, untrue for high-speed gas flows.
• It assumes steady flow, untrue for turbulent flow.
• Energy losses from friction are not accounted for.

Fluid Statics

• Fluid statics studies fluids at rest.
• No shear stress exists in fluid statics.
• Pressure at one point in a static fluid is the same in all directions (Pascal's Law).

Pressure

• Pressure is the force exerted per unit area.
• Pressure increases with depth because of the weight of the fluid above.
• The formula ( P = \rho g h ) calculates pressure at depth ( h ), where:
  • ( \rho ) is fluid density
  • ( g ) is the acceleration caused by gravity
  • ( h ) is the depth from the surface

Absolute Pressure

• Absolute pressure is the pressure relative to a perfect vacuum.

Gauge Pressure

• Gauge pressure is the pressure relative to atmospheric pressure.
• ( P_{\text{absolute}} = P_{\text{gauge}} + P_{\text{atmospheric}} )

Buoyancy

• Buoyancy is the upward force exerted by a fluid opposing an immersed object's weight.
• Archimedes' principle states that the buoyant force equals the weight of the fluid displaced by the object.
• ( F_B = \rho_f V_d g ) defines buoyant force, where:
  • ( F_B ) is the buoyant force
  • ( \rho_f ) is the fluid density
  • ( V_d ) is the volume of fluid displaced
  • ( g ) is the acceleration caused by gravity
• An object floats if the buoyant force and the object's weight are equal.
• An object sinks if the buoyant force is less than the object's weight.

Stability of Floating Objects

• The center of buoyancy is the centroid of the displaced fluid volume.
• The center of gravity is where the entire weight of the object is considered to act.
• For stable equilibrium, the center of buoyancy must be above the center of gravity.

Continuity Equation

• The continuity equation expresses the principle of mass conservation in fluid flow.
• For steady flow, the mass flow rate remains constant along a flow path.
• The equation is ( \rho_1 A_1 V_1 = \rho_2 A_2 V_2 ), where:
  • ( \rho ) is fluid density
  • ( A ) is the cross-sectional area of the flow
  • ( V ) is fluid velocity
• Subscripts 1 and 2 refer to two different points along the flow path.
• For incompressible flow ((\rho_1 = \rho_2)), the equation simplifies to ( A_1 V_1 = A_2 V_2 )
• ( A V ) represents the volumetric flow rate, which is constant in incompressible steady flow.

Applications of the Continuity Equation

• Pipe flow: Fluid velocity increases as pipe diameter decreases to keep flow rate constant.
• Nozzles decrease area, increasing fluid velocity.
• Diffusers increase area, decreasing fluid velocity.