Fluid Mechanics Course

Essential Foundation: Critical prerequisite for plasma physics (MHD), astrophysics, and engineering

Fluid Kinematics

The geometry and description of fluid motion without considering forces

Introduction

Fluid kinematics studies the motion of fluids without considering the forces that cause the motion. It describes how fluids move - velocity fields, acceleration, deformation rates - but not why they move.

This geometric description is essential for deriving conservation laws and understanding flow patterns before tackling the full dynamics problem.

Lagrangian vs Eulerian Description

Lagrangian Description

Follow individual fluid particles as they move through space:

$$\vec{x} = \vec{x}(\vec{x}_0, t)$$
  • โ€ข Track each particle's trajectory
  • โ€ข Natural for solid mechanics
  • โ€ข Useful for particle-laden flows
  • โ€ข Computationally expensive for fluids

Eulerian Description

Describe flow properties at fixed points in space:

$$\vec{v} = \vec{v}(\vec{x}, t)$$
  • โ€ข Field description (velocity field)
  • โ€ข Natural for fluid mechanics
  • โ€ข Fixed measurement locations
  • โ€ข Standard approach for Navier-Stokes

Material Derivative

The connection between descriptions - the rate of change following a fluid particle:

$$\frac{D\phi}{Dt} = \frac{\partial \phi}{\partial t} + (\vec{v} \cdot \nabla)\phi$$
โˆ‚ฯ†/โˆ‚t - Local (Eulerian) rate of change at fixed point
(vยทโˆ‡)ฯ† - Convective change due to motion through varying field

Flow Visualization Lines

Streamlines

Curves tangent to the velocity field at a given instant:

$$\frac{dx}{v_x} = \frac{dy}{v_y} = \frac{dz}{v_z}$$

No flow crosses a streamline (by definition). In steady flow, streamlines are fixed in space.

Pathlines

Trajectories traced by individual fluid particles over time:

$$\frac{d\vec{x}}{dt} = \vec{v}(\vec{x}, t)$$

Obtained by integrating the velocity field along particle paths (Lagrangian).

Streaklines

The locus of all particles that have passed through a given point:

Like dye injection - shows the current positions of all particles that passed through the injection point. In experiments, smoke or dye traces produce streaklines.

Important Note

For steady flow, streamlines, pathlines, and streaklines all coincide. For unsteady flow, they differ and must be distinguished carefully.

Vorticity and Circulation

Vorticity

The curl of the velocity field - measures local rotation:

$$\vec{\omega} = \nabla \times \vec{v} = \begin{pmatrix} \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \\ \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \\ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \end{pmatrix}$$

Vorticity equals twice the angular velocity of a fluid element: ฯ‰ = 2ฮฉ

Circulation

Line integral of velocity around a closed curve:

$$\Gamma = \oint_C \vec{v} \cdot d\vec{l} = \int_S \vec{\omega} \cdot d\vec{A}$$

By Stokes' theorem, circulation equals the flux of vorticity through any surface bounded by C.

Irrotational Flow

ฯ‰ = 0 everywhere โ†’ potential flow exists: v = โˆ‡ฯ†

Rotational Flow

ฯ‰ โ‰  0 in some regions (boundary layers, wakes, vortices)

Reynolds Transport Theorem

The fundamental theorem relating system (Lagrangian) and control volume (Eulerian) analyses:

$$\frac{d}{dt}\int_{sys} \phi \rho \, dV = \frac{\partial}{\partial t}\int_{CV} \phi \rho \, dV + \oint_{CS} \phi \rho (\vec{v} \cdot \hat{n}) \, dA$$

Left Side

Rate of change of property ฯ† for a system (material volume)

Right Side

Rate of change in CV + net flux through control surface

This theorem is the starting point for deriving integral forms of mass, momentum, and energy conservation.

Velocity Gradient and Deformation

The velocity gradient tensor describes how fluid elements deform:

$$\frac{\partial v_i}{\partial x_j} = \underbrace{\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)}_{e_{ij} \text{ (strain rate)}} + \underbrace{\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}\right)}_{\omega_{ij} \text{ (rotation)}}$$

Linear Strain

eii - elongation/compression along axes

Shear Strain

eij (iโ‰ j) - angular deformation

Volumetric Strain

โˆ‡ยทv = eii - rate of volume change

โ† Fluid StaticsConservation Laws โ†’