Kinematics

Before asking what forces do, describe how a fluid moves. A fluid is described by a velocity field, and its local motion is translation, rotation and strain.

1. Velocity fields, pathlines and streamlines (Def. 1.1, Example 1.2).
2. The material derivative (Prop. 1.3), Reynolds transport (Thm 1.4) and mass conservation (Cor. 1.5).
3. Stream functions (Prop. 1.6), vorticity (Def. 1.7), local rotation and strain (Prop. 1.8) and circulation (Def. 1.9).

In the Eulerian description, $\mathbf u(\mathbf x, t)$ is the velocity of whatever fluid particle is at $\mathbf x$ at time $t$. A pathline is the trajectory of one particle: $\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf u(\mathbf x, t)$. A streamline at a fixed time $t$ is a curve tangent to $\mathbf u(\cdot, t)$ everywhere: $\frac{\mathrm dx}u = \frac{\mathrm dy}v = \frac{\mathrm dz}w$. A streakline joins all particles that have passed through a fixed point (dye injected there). In steady flow all three coincide.

For the unsteady flow $\mathbf u = (1, t)$: at time $t$ the streamlines are straight lines of slope $t$. The particle starting at the origin at $t = 0$ follows $x = t$, $y = \frac{t^2}2$, a parabola $y = \frac{x^2}2$, which is not a streamline at any time.

For a quantity $F(\mathbf x, t)$, its rate of change following a fluid particle is

In particular a particle's acceleration is $\frac{D\mathbf u}{Dt} = \frac{\partial\mathbf u}{\partial t} + (\mathbf u\cdot\nabla)\mathbf u$.

For a material volume $V(t)$ (always made of the same particles),

Mass conservation, $\frac{\mathrm d}{\mathrm dt}\int_{V(t)}\rho\,\mathrm dV = 0$ for every material volume, is equivalent to the continuity equation $\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf u) = 0$, i.e. $\frac{D\rho}{Dt} + \rho\nabla\cdot\mathbf u = 0$. A flow is incompressible if $\frac{D\rho}{Dt} = 0$; then $\nabla\cdot\mathbf u = 0$ (volumes are preserved).

For a two-dimensional incompressible flow on a simply connected region there is a stream function $\psi$ with $u = \frac{\partial\psi}{\partial y}$, $v = -\frac{\partial\psi}{\partial x}$. Streamlines are the level curves $\psi =$ const, and the volume flux across any curve from $A$ to $B$ is $\psi(B) - \psi(A)$.

The vorticity is $\boldsymbol\omega = \nabla\times\mathbf u$. In two dimensions $\boldsymbol\omega = \omega\mathbf e_z$ with $\omega = v_x - u_y = -\nabla^2\psi$. A flow is irrotational if $\boldsymbol\omega = 0$; on a simply connected region this means $\mathbf u = \nabla\phi$ for a velocity potential $\phi$.

Near $\mathbf x_0$, $\mathbf u(\mathbf x_0 + \mathbf r) = \mathbf u(\mathbf x_0) + E\mathbf r + \frac12\boldsymbol\omega\times\mathbf r + O(|\mathbf r|^2)$, where $E = \frac12(\nabla\mathbf u + \nabla\mathbf u^{\mathsf T})$ is the rate-of-strain tensor. So locally a fluid element translates, rotates rigidly with angular velocity $\frac12\boldsymbol\omega$, and deforms by $E$ ($\operatorname{tr}E = \nabla\cdot\mathbf u$ is the rate of volume change).

The circulation around a closed curve $C$ is $\Gamma = \oint_C\mathbf u\cdot\mathrm d\mathbf l$. By Stokes' theorem, $\Gamma = \iint_S\boldsymbol\omega\cdot\mathbf n\,\mathrm dS$ for any surface $S$ spanning $C$. An irrotational flow can still have circulation around a curve enclosing a hole (e.g. a line vortex).

• Steady? Then pathlines = streamlines; solve $\frac{\mathrm dx}u = \frac{\mathrm dy}v$.
• Incompressible? Check $\nabla\cdot\mathbf u = 0$; in 2D find $\psi$.
• Irrotational? Check $\nabla\times\mathbf u = 0$; then find $\phi$.
• Accelerations: use $\frac{D\mathbf u}{Dt}$, not $\mathbf u_t$.