= Solution
A <fluid> is described on a continuum scale by <density>, <pressure> and <velocity> fields. An Eulerian field is measured at fixed position; a material element changes at rate $D/Dt=\partial_t+\mathbf v\cdot\nabla$. <mass> conservation requires
$$
\partial_t\rho+\nabla\cdot(\rho\mathbf v)=0.
$$
For <incompressible flow> this becomes $\nabla\cdot\mathbf v=0$. A <streamline> is instantaneously tangent to <velocity>, whereas a <pathline> follows a particle through time; they coincide in a steady <flow> but need not in an unsteady one.
For an ideal inviscid <fluid>, <pressure> and body <forces> give the Euler equation. In the constant-density gravitational case,
$$
\rho\frac{D\mathbf v}{Dt}=-\nabla p+\rho\mathbf g.
$$
A stationary impermeable boundary requires zero normal <velocity>, but inviscid theory does not impose no slip. Taking its steady projection along a <streamline> gives $p/\rho+v^2/2+gz=\mathrm{constant}$. If the <flow> is also irrotational in the connected region, that constant is shared between <streamlines>. Inviscid does not mean irrotational: rotating ideal-fluid <flows> are possible. A <velocity> potential exists locally for an irrotational field, and in <incompressible flow> it obeys <Laplace's equation>. The Bernoulli assumptions must be checked before converting a <pressure> difference into a <speed>.<Bernoulli’s Equation by NASA Glenn Research Center>[Restrictions and Common Assumptions: inviscid, steady, incompressible flow and streamline dependence]{ref}
Real <fluids> transmit shear stress. An incompressible <Newtonian fluid> with constant <dynamic viscosity> $\eta$ obeys
$$
\rho\frac{D\mathbf v}{Dt}=-\nabla p+\eta\nabla^2\mathbf v+\rho\mathbf g.
$$
The constitutive shear stress is proportional to strain rate, and stationary solid boundaries usually impose no slip. <viscosity> diffuses <vorticity> and converts organized mechanical <energy> into <heat>; its local dissipation $2\eta e_{ij}e_{ij}$ is nonnegative. It cannot generally be included merely by changing the constant in a lossless Bernoulli equation.
In a long straight circular tube, steady fully developed <laminar flow> has
$$
v_z(r)=\frac{\Delta p}{4\eta L}(a^2-r^2),\qquad
Q=\frac{\pi a^4\Delta p}{8\eta L}.
$$
This <Hagen-Poiseuille equation> follows by balancing the axial <pressure> <gradient> with viscous shear and setting $v_z(a)=0$. The parabolic profile, the fourth-power <radius> sensitivity and the pumping power $\Delta p\,Q$ are experimentally distinct consequences. Entrance regions, rough walls, non-Newtonian response or a changing <temperature> require additional modelling.
The <Reynolds number> $\mathrm{Re}=\rho UL/\eta$ compares inertia with <viscosity>. Small values favor creeping-flow balances; high values permit thin <boundary layers>, separation and <turbulence>, although transition depends on geometry and perturbations, not one universal critical number. At high <Reynolds number> an outer <flow> can be approximately inviscid while near-wall <viscosity> still determines drag. This resolves why a zero-drag ideal potential-flow model cannot predict the resistance of an ordinary moving body. In turbulent <flow>, fluctuating <Reynolds stresses> redistribute momentum and increase the required <pressure> drop; averaging the equations introduces closure requirements.
A controlled tube experiment measures <pressure> difference and collected <volume> per time while changing forcing. Use tracers or a <velocity> probe to test profile shape, allow sufficient entrance <length>, and monitor <radius> and <temperature> independently. Repeat with reversed <flow> and at several rates to test offsets and linearity. Dye mixing alone does not measure <viscosity> or establish a fully developed <velocity> profile. The original figure below separates a specified laminar velocity profile from an independent pressure/discharge measurement.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2011/paper-1-fluid-flow.png]
{title=Ideal and viscous fluid flow}
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