💨Mathematical Fluid Dynamics Unit 2 – Kinematics of Fluid Motion

Key Concepts and Definitions

• Fluid dynamics studies the motion and behavior of fluids, both liquids and gases
• Kinematics describes the motion of fluids without considering the forces causing the motion
• Fluid properties include density, viscosity, compressibility, and surface tension
• Velocity field represents the velocity of fluid particles at each point in space and time
• Acceleration field describes the acceleration of fluid particles at each point in space and time
• Streamlines are curves tangent to the velocity field at a given instant in time
• Pathlines trace the path of individual fluid particles over time
• Streaklines are the locus of fluid particles that have passed through a particular point in the flow

Fluid Properties and Characteristics

• Density $(\rho)$ is the mass per unit volume of a fluid and affects its inertia and buoyancy
• Viscosity $(\mu)$ measures a fluid's resistance to deformation and is responsible for internal friction
  • Dynamic viscosity relates shear stress to velocity gradients in the fluid
  • Kinematic viscosity $(\nu = \mu/\rho)$ is the ratio of dynamic viscosity to density
• Compressibility describes how much a fluid's density changes with pressure (ideal gases, liquids)
• Surface tension arises from intermolecular forces and affects the formation of droplets and bubbles
• Newtonian fluids have a linear relationship between shear stress and strain rate (water, air)
• Non-Newtonian fluids have a nonlinear or time-dependent relationship between shear stress and strain rate (blood, paint)

Fundamental Equations of Fluid Motion

• Continuity equation expresses the conservation of mass in a fluid: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$
  • For incompressible fluids, the continuity equation simplifies to $\nabla \cdot \vec{u} = 0$
• Momentum equation (Navier-Stokes equations) describes the conservation of momentum in a fluid: $\rho \left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g}$
  • Includes pressure gradients, viscous forces, and body forces (gravity)
• Energy equation represents the conservation of energy in a fluid, considering heat transfer and work done
• Constitutive equations relate stress and strain in the fluid, depending on its rheological properties
• Boundary conditions specify the behavior of the fluid at the boundaries of the domain (no-slip, free-slip)

Eulerian vs. Lagrangian Descriptions

• Eulerian description focuses on fluid properties at fixed points in space as time passes
  • Observes fluid motion from a fixed reference frame
  • Uses spatial coordinates $(x, y, z)$ and time $(t)$ as independent variables
• Lagrangian description tracks individual fluid particles as they move through space and time
  • Follows fluid particles along their trajectories
  • Uses initial position $(\vec{X})$ and time $(t)$ as independent variables
• Eulerian and Lagrangian descriptions are related through the material derivative: $\frac{D}{Dt} = \frac{\partial}{\partial t} + \vec{u} \cdot \nabla$
• Eulerian description is more common in fluid dynamics due to its simplicity and ease of measurement

Streamlines, Pathlines, and Streaklines

• Streamlines are curves tangent to the velocity field at a given instant in time
  • Represent the direction of fluid motion at each point
  • Do not intersect each other in steady flows
• Pathlines trace the path of individual fluid particles over time
  • Show the trajectory of a fluid particle from its initial position
  • Can intersect each other in unsteady flows
• Streaklines are the locus of fluid particles that have passed through a particular point in the flow
  • Formed by injecting dye or smoke into the fluid at a fixed point
  • Reveal the history of fluid motion and can highlight unsteady flow features
• In steady flows, streamlines, pathlines, and streaklines coincide
• In unsteady flows, streamlines, pathlines, and streaklines can differ significantly

Conservation Laws in Fluid Dynamics

• Conservation of mass ensures that mass is neither created nor destroyed in a fluid: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$
• Conservation of momentum (Newton's second law) relates the change in momentum to the forces acting on a fluid: $\rho \left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g}$
• Conservation of energy (first law of thermodynamics) states that energy is conserved in a fluid, considering heat transfer and work done
• Conservation of angular momentum is important in rotating flows and leads to the concept of vorticity
• Conservation laws form the basis for the governing equations of fluid motion (continuity, Navier-Stokes, energy equations)

Vorticity and Circulation

• Vorticity $(\vec{\omega})$ is a measure of the local rotation in a fluid, defined as the curl of the velocity field: $\vec{\omega} = \nabla \times \vec{u}$
  • Vorticity is a vector quantity, with its direction indicating the axis of rotation
  • Vorticity magnitude represents the strength of the local rotation
• Circulation $(\Gamma)$ is a scalar measure of the total rotation in a fluid, defined as the line integral of velocity along a closed curve: $\Gamma = \oint_C \vec{u} \cdot d\vec{l}$
• Kelvin's circulation theorem states that circulation is conserved in inviscid, barotropic flows
• Helmholtz's vorticity theorems describe the behavior of vorticity in inviscid flows:
  • Vortex lines move with the fluid and maintain their strength
  • Vortex tubes have constant circulation along their length
• Vorticity and circulation are important in understanding rotational flows, such as vortices and turbulence

Applications and Real-World Examples

• Aerodynamics: Study of fluid motion around aircraft, vehicles, and buildings (lift, drag, wind loading)
  • Streamlined shapes reduce drag and improve efficiency (airfoils, bullet trains)
  • Vortex shedding can cause vibrations and structural damage (bridges, towers)
• Hydrodynamics: Study of fluid motion in water bodies and hydraulic systems (rivers, oceans, pipes)
  • Pressure gradients drive flow in pipes and channels (water supply, hydropower)
  • Tides and currents transport heat, nutrients, and pollutants in oceans and lakes
• Cardiovascular system: Blood flow through the heart, arteries, and veins
  • Pulsatile flow and non-Newtonian fluid properties affect blood circulation
  • Vortices and turbulence can occur in heart valves and aneurysms
• Meteorology: Study of atmospheric fluid dynamics (weather patterns, climate)
  • Pressure gradients, Coriolis force, and buoyancy drive atmospheric circulation
  • Vorticity plays a key role in the formation and evolution of cyclones and hurricanes
• Environmental fluid mechanics: Study of fluid motion in natural systems (rivers, estuaries, atmosphere)
  • Mixing and dispersion of pollutants in air and water bodies
  • Sediment transport and morphodynamics in rivers and coastal areas