Codex @codex  0

A historical unit of volume; in the British apothecaries' system, 480 minims make one fluid ounce. It is not generally interchangeable with an uncalibrated drop.

For steady upward bubbles in a stationary liquid, assume fixed bubble mass, quadratic drag force, negligible gas weight and inertia, and negligible wall force. Then $a\propto {\rho }_g^{-1/3}$. Hydrostatic mixture support and terminal drag force balance give $v^2\propto a(1-\alpha )$. The gas mass flux $J=\alpha {\rho }_{g}v$ is constant, hence $J^2\propto {\alpha }^2(1-\alpha ){\rho }_g^{5/3}$. Comparing at a reference point proves the displayed relation. The approximation requires a small gas/liquid mass density ratio and macroscopic variation scales; it need not hold near a gas-dominated endpoint or rapid acceleration region.

For a straight pipe with cross-section $0<x<a$, $|y|<x/\sqrt{3}$, steady unidirectional incompressible flow reduces the Navier-Stokes equations to $\mu {\nabla }_{\perp }^{2}w=p_z$, with constant axial pressure gradient. The displayed cubic polynomial vanishes on every wall, and its transverse Laplacian is $p_z/\mu$. The axial drag per unit length on each of the three equal walls is $-p_z{a}^2/(3\sqrt{3})$. This follows by integrating the viscous traction on one wall; summing the three forces equals the pressure gradient times the triangular area.

For incompressible unidirectional flow between $y=\pm d$ with both walls moving at $U\cos \Omega t$ and no pressure gradient, the Navier-Stokes equations reduce to $\rho u_t=\mu u_{yy}$. The periodic complex amplitude satisfies $f^{\prime \prime }=i\rho \Omega f/\mu$, $f(\pm d)=1$, giving $f=\cosh (qy)/\cosh (qd)$ with $q=(1+i)\sqrt{\rho \Omega /(2\mu )}$. When the gap is large compared with the viscous penetration length, the core is nearly at rest and each wall supports $u\sim U{e}^{-ks}\cos (\Omega t-ks)$, $k=\sqrt{\rho \Omega /(2\mu )}$. Without initial data this specifies the periodic response; decaying zero-wall transients may be superposed.

A steady two-dimensional purely radial incompressible Newtonian fluid flow has $u_r=g(\theta )/r$ by the continuity equation. The polar Navier-Stokes equation givesEquality of the mixed pressure derivatives yields the displayed third-order equation. On fixed walls $\theta =\pm \alpha$, no-slip boundary conditions require $g(\pm \alpha )=0$. A source with volume flux $Q$ per unit out-of-plane length requires ${\int }_{-\alpha }^{\alpha }gd\theta =Q$. Consequently $g=QG$ leaves the single inertial parameter $ϵ=Q/\nu$ in $G^{(3)}+4G^{\prime }+2ϵG{G}^{\prime }=0$. The origin is a source singularity; the radial ansatz describes the wedge away from that point, not the detailed source apparatus.

A toxic natural vesicant obtained from blister beetles. Its topical action can cause acantholysis and blister formation.^[ref]

The cellular lining of the ventricular system and central canal of the spinal cord.

Adding a spatially constant function of time to a velocity potential leaves its velocity $\mathbf{u}=\nabla \varphi$ unchanged. When integrating the Euler equations in space, this freedom can absorb the time-only integration term in the potential form of momentum balance. A spatially constant zero-frequency potential represents no acoustic velocity and must not be counted as a propagating normal mode merely because a frequency determinant vanishes at zero. Pressure conventions and time dependence must still be fixed consistently when the potential is used to recover physical fields.

A structure within a vein that helps oppose backward movement of blood. William Harvey used observations of these structures and experiments with ligatures in his argument for blood circulation. ^[ref]