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For a clean bubble of size $a$ and speed $U$ at large Reynolds number, the outer potential flow violates the stress-free condition by a strain of order $U/a$. A boundary layer of thickness $\delta \sim a{\mathrm{Re}}^{-1/2}$ needs a velocity correction only of order $U\delta /a$, so its dissipation is smaller than the outer-flow dissipation by order $\delta /a$. The leading drag power is therefore obtained from

A common high-Reynolds number closure for a deep-ambient gravity current is $\dot{L}=\mathrm{Fr}\sqrt{g^{\prime }h}$, where $L$ is front position and the order-one Froude number depends on the chosen front model.

A microswimmer is a microscopic body that propels itself through a fluid. At small Reynolds number its motion is governed by Stokes flow, and self-propulsion must respect the force-free and torque-free conditions.

For $A=\left(\begin{array}{cc}-i{\omega }_1& a\\ c& -i({\omega }_1-b)\end{array}\right)$ with real parameters,Thus the displayed conditions are necessary and sufficient for a normal matrix. In the family $a=1$, $b=c=q$, normality requires $q=-1$, not $q=1$. If $q=10/Re$, this is the formal value $Re=-10$ and no positive Reynolds number qualifies. At that value $A$ is a skew-Hermitian matrix and its exponential is unitary, so every initial direction has optimal energy amplification of a linear system equal to one.

The Orr-Sommerfeld equation governs the wall-normal velocity of normal mode disturbances to an incompressible viscous parallel shear flow. Here $D=d/dy$, ${\kappa }^2={\alpha }^2+{\beta }^2$, and $Re$ is the Reynolds number. It follows by eliminating pressure from the linearized Navier-Stokes equations. At a rigid no-slip wall, $\hat{v}=D\hat{v}=0$. In the inviscid limit it reduces to the Rayleigh equation for inviscid shear flow.

A Reynolds number based on particle diameter and relative speed in the carrier fluid. Small values justify Stokes drag law; a high-Reynolds-number regime with nearly constant drag coefficient instead gives quadratic drag. The drag law must be evaluated in the appropriate regime for every species.