💧Fluid Mechanics Unit 5 – Fluid Kinematics

Key Concepts and Definitions

• Fluid kinematics studies the motion of fluids without considering the forces causing the motion
• Lagrangian description tracks individual fluid particles as they move through space and time
• Eulerian description focuses on specific locations in the fluid flow and the fluid properties at those locations over time
• Velocity field represents the velocity vector at each point in a fluid flow at a given instant
• Streamlines are curves that are tangent to the velocity vector at each point in a steady flow
  • Streamlines cannot cross each other in a steady flow
• Pathlines trace the actual path traveled by a fluid particle over a period of time
• Streaklines are the locus of fluid particles that have passed through a particular point in the flow

Fluid Properties and Behavior

• Fluids are substances that deform continuously under the application of shear stress (liquids and gases)
• Fluid density $(\rho)$ is the mass per unit volume of a fluid and varies with temperature and pressure
  • Incompressible fluids have constant density throughout the flow (water at low speeds)
  • Compressible fluids have density that varies significantly with pressure changes (air at high speeds)
• Viscosity $(\mu)$ is a measure of a fluid's resistance to deformation under shear stress
  • Newtonian fluids have a constant viscosity that is independent of the applied shear stress (water, air)
  • Non-Newtonian fluids have a viscosity that depends on the applied shear stress (blood, ketchup)
• Surface tension is the result of cohesive forces between fluid molecules at the surface of a liquid
• Capillary action occurs when adhesive forces between a liquid and a solid surface exceed the cohesive forces within the liquid, causing the liquid to rise in narrow spaces

Methods of Describing Fluid Motion

• Lagrangian method tracks individual fluid particles as they move through space and time
  • Useful for understanding the behavior of individual particles in a flow
  • Difficult to apply in complex flows with many particles
• Eulerian method focuses on specific locations in the fluid flow and the fluid properties at those locations over time
  • Useful for analyzing flow properties at fixed points in space
  • Commonly used in fluid mechanics and computational fluid dynamics (CFD)
• Substantial derivative $(\frac{D}{Dt})$ describes the rate of change of a fluid property (velocity, temperature) as a fluid particle moves through space and time
  • Includes both local $(\frac{\partial}{\partial t})$ and convective $(\vec{V} \cdot \nabla)$ rates of change
• Reynolds Transport Theorem relates the rate of change of a fluid property in a control volume to the net flux of that property across the control surface and any sources or sinks within the control volume

Velocity Fields and Flow Patterns

• Velocity field $\vec{V}(x, y, z, t)$ represents the velocity vector at each point in a fluid flow at a given instant
• Uniform flow has a constant velocity magnitude and direction throughout the entire flow field
• Non-uniform flow has a velocity that varies in magnitude and/or direction throughout the flow field
• Steady flow has fluid properties (velocity, pressure, density) that do not change with time at any point in the flow
• Unsteady flow has fluid properties that vary with time at one or more points in the flow
• Laminar flow is characterized by smooth, parallel layers of fluid with no mixing between layers (low Reynolds numbers)
• Turbulent flow is characterized by chaotic, irregular motion with mixing between fluid layers (high Reynolds numbers)
  • Eddies and vortices are common features of turbulent flows

Acceleration in Fluid Flow

• Acceleration in fluid flow can be caused by changes in velocity magnitude (speed) and/or direction
• Local acceleration $(\frac{\partial \vec{V}}{\partial t})$ is the rate of change of velocity with respect to time at a fixed point in space
• Convective acceleration $(\vec{V} \cdot \nabla \vec{V})$ is the rate of change of velocity due to the movement of a fluid particle to a new location with a different velocity
• Total acceleration $(\frac{D\vec{V}}{Dt})$ is the sum of local and convective accelerations experienced by a fluid particle
  • $\frac{D\vec{V}}{Dt} = \frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla) \vec{V}$
• Streamline curvature can cause centripetal acceleration in a fluid flow
• Pressure gradients and body forces (gravity) can also contribute to acceleration in fluid flows

Streamlines, Streaklines, and Pathlines

• Streamlines are curves that are tangent to the velocity vector at each point in a steady flow
  • Streamlines represent the instantaneous direction of fluid motion at each point
  • Streamlines cannot cross each other in a steady flow
• Streaklines are the locus of fluid particles that have passed through a particular point in the flow
  • Streaklines are formed by injecting a continuous stream of dye or smoke into a flow
  • In unsteady flows, streaklines can differ from streamlines and pathlines
• Pathlines trace the actual path traveled by a fluid particle over a period of time
  • Pathlines are obtained by tracking the motion of individual fluid particles
  • In steady flows, pathlines, streaklines, and streamlines coincide
• Timelines are a set of fluid particles that form a line at a given instant in time
  • Timelines are useful for visualizing the deformation and stretching of fluid elements over time

Conservation Laws in Fluid Kinematics

• Conservation of mass (continuity equation) states that mass cannot be created or destroyed in a fluid flow
  • For incompressible flows: $\nabla \cdot \vec{V} = 0$
  • For compressible flows: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$
• Conservation of momentum (Newton's second law) relates the acceleration of a fluid particle to the forces acting on it
  • Navier-Stokes equations are the fundamental equations of fluid motion based on conservation of momentum
• Conservation of energy (first law of thermodynamics) states that energy cannot be created or destroyed, only converted from one form to another
  • Bernoulli's equation is a simplified form of the energy conservation equation for steady, inviscid, incompressible flows along a streamline
• Circulation $(\Gamma)$ is a measure of the rotation of a fluid in a closed loop
  • Kelvin's circulation theorem states that circulation is conserved in an inviscid, barotropic fluid with conservative body forces

Applications and Real-World Examples

• Aerodynamics: Study of the motion of air and its interaction with solid objects (aircraft wings, cars)
  • Lift and drag forces on airfoils and vehicles
  • Boundary layer separation and wake formation
• Hydrodynamics: Study of the motion of liquids and their interaction with solid boundaries (ships, submarines)
  • Propulsion systems and hull design for marine vessels
  • Cavitation in pumps and propellers
• Meteorology: Study of atmospheric motion and weather patterns
  • Wind velocity fields and pressure gradients
  • Cyclones, anticyclones, and jet streams
• Cardiovascular system: Flow of blood through the heart, arteries, and veins
  • Pulsatile flow and pressure waves in arteries
  • Atherosclerosis and stenosis in blood vessels
• Environmental fluid mechanics: Study of fluid motion in natural systems (oceans, rivers, atmosphere)
  • Dispersion of pollutants in air and water
  • Sediment transport and erosion in rivers and coastal areas