= Solution
\b[Following the moving <fluid>.] The Eulerian <velocity field> $\mathbf u(\mathbf x,t)$ assigns a <velocity> to each position. A <fluid> <particle> satisfies $\dot{\mathbf x}=\mathbf u$, so the <chain rule> gives the <material derivative>, also called the convective <derivative>,
$$
\boxed{\frac D{Dt}=\frac\partial{\partial t}+\mathbf u\cdot\nabla,
\qquad \mathbf a=\frac{D\mathbf u}{Dt}.}
$$
Even in steady flow a <particle> can accelerate because it moves between regions with different <velocity>. For example, flow around a curved obstacle changes direction despite $\partial_t\mathbf u=0$. <Mass> conservation is
$$
\partial_t\rho+\nabla\cdot(\rho\mathbf u)=0,
\qquad \frac{D\rho}{Dt}+\rho\nabla\cdot\mathbf u=0.
$$
For a constant-density incompressible <fluid>, $\nabla\cdot\mathbf u=0$.
\b[<Euler equations for an inviscid fluid> and <Bernoulli's equation>.] Neglect <viscosity> and take the body <force> per unit <mass> to be $-\nabla\Phi$. <Newton's second law> for a <fluid> element gives
$$
\rho\frac{D\mathbf u}{Dt}=-\nabla p-\rho\nabla\Phi.
$$
For steady flow, dot with <displacement> along a <streamline>. The tangential component of $\mathbf u\cdot\nabla\mathbf u$ is the <derivative> of $u^2/2$ along that line. For constant density, integration gives
$$
\boxed{\frac p\rho+\frac{u^2}{2}+\Phi=B,\qquad B\text{ constant along a streamline}.}
$$
In uniform gravity, $\Phi=gz$. Thus <pressure>, <kinetic energy> per unit <mass> and <gravitational potential energy> per unit <mass> exchange along a <streamline>. For a barotropic compressible <fluid>, replace $p/\rho$ by $\int dp/\rho(p)$. Different <streamlines> need not have the same Bernoulli constant.
\b[<Vorticity> and <potential flow>.] <Vorticity> is $\boldsymbol\omega=\nabla\times\mathbf u$. A small <fluid> element rotates locally with <angular velocity> $\boldsymbol\omega/2$ after its deformational part is separated. The <vector> identity
$$
(\mathbf u\cdot\nabla)\mathbf u=\nabla(u^2/2)-\mathbf u\times\boldsymbol\omega
$$
gives $\nabla B=\mathbf u\times\boldsymbol\omega$ for steady constant-density <inviscid flow>. Thus an irrotational region has one common Bernoulli constant on each connected region. In a simply connected irrotational domain, introduce a <velocity potential> $\mathbf u=\nabla\phi$. <Incompressibility> then gives <Laplace's equation>, $\nabla^2\phi=0$. For unsteady <irrotational flow>,
$$
\partial_t\phi+\frac12|\nabla\phi|^2+\frac p\rho+\Phi=C(t).
$$
An impermeable stationary wall fixes $\partial_n\phi=0$, but ideal <potential flow> does not enforce no slip. In a multiply connected domain, <irrotational flow> may still have nonzero <circulation> around an excluded vortex core, so a globally single-valued potential requires care.
For example, uniform two-dimensional flow past a circular cylinder has $\phi=U(r+a^2/r)\cos\theta$ outside $r=a$. Its normal <velocity> vanishes at the cylinder, and its surface tangential <velocity> is $-2U\sin\theta$. <Bernoulli's equation> predicts a symmetric <pressure> distribution and zero <drag>. This is <D'Alembert's paradox>: the assumptions omit the viscous <boundary layers>, separation and wake responsible for real <drag>.
For an incompressible <fluid> of constant <kinematic viscosity> $\nu$, the <curl> of the motion equation gives
$$
\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf u+\nu\nabla^2\boldsymbol\omega.
$$
The first term stretches and tilts <vortex lines>; the second diffuses <vorticity>. In two-dimensional incompressible <inviscid flow> the stretching term is absent and <vorticity> is carried with the <fluid>. In a three-dimensional <inviscid flow> it can be amplified by stretching. Density gradients not parallel to <pressure> gradients provide an additional source in a general compressible <fluid>.
\b[Viscous flow.] A <Newtonian fluid>'s shear <stress> is proportional to rate of <strain>. For constant <viscosity> and <incompressibility> the <Navier-Stokes equation> is
$$
\boxed{\rho\frac{D\mathbf u}{Dt}=-\nabla p-\rho\nabla\Phi+\eta\nabla^2\mathbf u,}
$$
with <dynamic viscosity> $\eta$ and <kinematic viscosity> $\nu=\eta/\rho$. At an ordinary <solid> wall, the <no-slip boundary condition> sets <fluid> <velocity> equal to the wall <velocity>. <Viscosity> transfers <momentum> between neighbouring layers and converts mechanical <energy> into <heat>; unlike the <pressure> <force>, its shear contribution depends on <velocity> gradients.
For plane <Couette flow> between plates $y=0,h$, with the upper plate moving at $U$, the steady solution is $u_x=Uy/h$, with shear <stress> $\eta U/h$. For pressure-driven flow through a pipe of radius $a$, a constant <pressure> gradient gives the parabolic Poiseuille profile
$$
u_z(r)=\frac{-dp/dz}{4\eta}(a^2-r^2),\qquad
Q=\frac{\pi a^4}{8\eta}\left(-\frac{dp}{dz}\right).
$$
The <Reynolds number> $\mathrm{Re}=\rho UL/\eta$ compares convective inertia with viscous <force>. Small <Reynolds number> often permits the Stokes-flow approximation; large <Reynolds number> can produce thin <boundary layers> and, under suitable conditions, <turbulence>. It does not justify neglecting the wall conditions. Bernoulli's inviscid constant is generally not conserved along a viscous <streamline>, and pipe flow instead shows a <pressure> loss corresponding to dissipation.
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