= Dynamic viscosity
{title2=$\mu$}
= Viscosity
{synonym}
= Shear viscosity
{synonym}
The <dynamic viscosity> $\mu$ is the coefficient relating <shear stress> to <shear rate> in a <Newtonian fluid>. For a <simple shear flow> $\mathbf u=(u(y),0,0)$,
$$
\tau_{xy}=\mu\frac{du}{dy}=\mu\dot\gamma,\qquad
[\mu]=\mathrm{Pa\,s}=\mathrm{kg\,m^{-1}\,s^{-1}}.
$$
Here $\tau_{xy}$ is the $x$ component of viscous <traction> on a plane with <normal vector> $+\mathbf e_y$. The <shear rate> $\dot\gamma=du/dy$ has units $\mathrm{s^{-1}}$. Thus <dynamic viscosity> measures the <stress> required to sustain a given rate of deformation. It is the shear coefficient; the separate coefficient for <volume> change is <volume viscosity>. https://www.homepages.ucl.ac.uk/~uceseug/Fluids2/Slides_1.pdf[UCL, Fluid Mechanics II, Forces in Fluids, slides 4–5]{ref}
For a worked <Couette flow> example, place a <Newtonian fluid> of constant <dynamic viscosity> $\mu=0.10\,\mathrm{Pa\,s}$ between parallel plates separated by $h=0.0010\,\mathrm m$. Hold the lower plate fixed and move the upper plate at $U=0.020\,\mathrm{m\,s^{-1}}$. Assume <steady flow>, no streamwise <pressure gradient>, and the <no-slip boundary condition>. Then
$$
u(y)=\frac{Uy}{h},\qquad
\dot\gamma=\frac Uh=20\,\mathrm{s^{-1}},\qquad
\tau_{xy}=\frac{\mu U}{h}=2.0\,\mathrm{Pa}.
$$
The fluid pulls backward on the moving plate with <force> magnitude $F=\tau_{xy}A$, where $A$ is its wetted <area>. For $A=0.010\,\mathrm{m^2}$ this is $0.020\,\mathrm N$. Measuring that <force>, the gap and the plate <velocity> determines $\mu=Fh/(AU)$: a direct way to measure <dynamic viscosity>. The linear profile and opposing wall <tractions> are the standard plane <Couette flow> solution. https://ocw.mit.edu/courses/2-25-advanced-fluid-mechanics-fall-2013/1a114d602956fa0dd328155f9b45f93d_MIT2_25F13_Couet_and_Pois.pdf[MIT, Couette and Poiseuille Flows, p. 1]{ref}
\Image[/viscosity-couette-shear.png]
{title=Couette flow: a constant velocity gradient produces a constant shear stress}
{description=Blue arrows show the linear velocity profile between the plates. Orange arrows show viscous traction on the upper and lower faces of a fluid slice; reversing the surface normal reverses the traction. The slice has zero net viscous force although each face carries stress.}
The <velocity gradient> matters, rather than the <velocity> itself. Adding the same constant <velocity> to both plates and the fluid leaves $du/dy$, and hence <shear stress>, unchanged. In <uniform flow>, every layer translates together and $du/dy=0$. More generally, even a nonzero <velocity gradient> need not generate <viscous stress>: <solid-body rotation> has zero <strain-rate tensor>, as the <viscous stress tensor> example shows. Thin gaps amplify <shear stress> for a fixed plate-speed difference, making <dynamic viscosity> useful for predicting the <force> needed to move lubricated surfaces.
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