Flow of Stokes

When a fluid viscous runs out slowly in a narrow place or around a small object, the viscous effects dominate over the inertial effects. Its flow is then called flow of Stokes sometimes (and one speaks about Fluide of true Stokes in opposition to Fluid). It is indeed governed by a simplified version of the Navier-Stokes equation: the equation of Stokes , in which the inertial terms are absent. The Reynolds number measurement the relative weight viscous and inertial terms in the Navier-Stokes equation. The flow of Stokes corresponds thus to weak a Reynolds number (much smaller than 1).

The equation of Stokes makes it possible in particular to describe the flows of liquid in the microfluidic devices. The flows of Feather bed and Poiseuille are also described by this equation.

Equation of Stokes

The equation of Stokes, which describes the flow of a Newtonian fluid incompressible in permanent mode and with weak Reynolds number, is written:

$\backslash \; eta\; \backslash \; Delta\; \backslash \; vec\; \{v\}\; =\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash ,\; p\; -\; \backslash \; rho\; \backslash \; vec\; \{F\}$,

where:

• $\backslash \; vec\; \{v\}\; (\backslash \; vec\; \{R\})$ is the speed of the fluid;
• $p\; (\backslash \; vec\; \{R\})$ is the Pression in the fluid;
• $\backslash \; rho$ is the density of the fluid
• $\backslash \; eta$ is the dynamic Viscosité fluid;
• $\backslash \; vec\; \{F\}$ is a mass force being exerted in the fluid (for example: gravity);
• $\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}$ and $\backslash \; Delta$ is respectively the differential operators gradient and Laplacian.

Conditions for application

The equation of Stokes is a simplified form of the Navier-Stokes equation. For a Newtonian fluid, this one is written:

$\backslash \; rho\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; vec\; \{v\}\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; rho\; \backslash \; left\; (\backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash \; right)\; \backslash \; vec\; \{v\}$

\ rho \ vec {F} - \ overrightarrow {\ mathrm {grad}} \, p + \ eta \ Delta \, \ vec {v}

+ \ left (\ zeta + \ frac {\ eta} {3} \ right) \ overrightarrow {\ mathrm {grad}} (\ mathrm {div} \, \ vec {v}) ,

where:

• $\backslash \; vec\; \{v\}\; (\backslash \; vec\; \{R\},\; T)$ is the speed of the fluid;
• $p\; (\backslash \; vec\; \{R\},\; T)$ is the Pression in the fluid;
• $\backslash \; rho$ is the density of the fluid
• $\backslash \; eta$ is the dynamic Viscosité in shearing of the fluid;
• $\backslash \; zeta$ is the dynamic Viscosité in compression of the fluid;
• $\backslash \; vec\; \{F\}$ is a mass force being exerted in the fluid (for example: gravity);
• $T$ represents the Temps;
• $\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}$, $\backslash \; text\; \{div\}$ and $\backslash \; Delta$ is respectively the differential operators gradient, divergence and Laplacian.

NB: to establish this formula, one must suppose that the space variations of $\backslash \; eta$ and $\backslash \; zeta$ are negligible.

So moreover the fluid is incompressible (good approximation for the liquids), then $\backslash \; text\; \{div\}\; \backslash ,\; \backslash \; vec\; \{v\}\; =\; 0$ and the equation is simplified:

$\backslash \; rho\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; vec\; \{v\}\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; rho\; \backslash \; left\; (\backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash \; right)\; \backslash \; vec\; \{v\}$

\ rho \ vec {F} - \ overrightarrow {\ mathrm {grad}} \, p + \ eta \ Delta \, \ vec {v}

.

One can evaluate the order of magnitude of the inertial and viscous terms in this equation. If speed characteristic of the liquid is $U$, and if the typical scale of variation speed is $L$ (this one could be imposed by dimensions of the channel in which the liquid runs out, dimensions of the object around whose the liquid runs out, etc), then:

$\backslash \; rho\; \backslash \; left\; (\backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash \; right)\; \backslash \; vec\; \{v\}\; \backslash \; sim\; \backslash \; frac\; \{\backslash \; rho\; U^2\}\; \{L\}$ and $\backslash \; eta\; \backslash \; Delta\; \backslash ,\; \backslash \; vec\; \{v\}\; =\; \backslash \; rho\; \backslash \; naked\; \backslash \; Delta\; \backslash ,\; \backslash \; vec\; \{v\}\; \backslash \; sim\; \backslash \; frac\; \{\backslash \; rho\; \backslash \; naked\; U\}\; \{L^2\}$,

where $\backslash \; nu$ is the kinematic viscosity liquid.

The inertial term $\backslash \; rho\; \backslash \; left\; (\backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash \; right)\; \backslash \; vec\; \{v\}$ will be thus negligible in front of the viscous term $\backslash \; eta\; \backslash \; Delta\; \backslash ,\; \backslash \; vec\; \{v\}$ if:

$\backslash \; frac\; \{\backslash \; rho\; U^2\}\; \{L\}\; \backslash \; L\; \backslash \; frac\; \{\backslash \; rho\; \backslash \; naked\; U\}\; \{L^2\}$, is $\backslash \; frac\; \{UL\}\; \{\backslash \; naked\}\; \backslash \; L\; 1$,

where one recognizes the expression of the Reynolds number.

Properties of the solutions of the equation of Stokes

Contrary to the Navier-Stokes equation, the equation of Stokes is linear (the term inertial, non-linear, is indeed negligible). The flows solutions of this equation have consequently quite particular properties:

• unicity : for Boundary conditions given (value speed on the level of the walls and/or ad infinitum), there exists one and only one flow checking the equation of Stokes;
• additivity : the solutions of the equation of Stokes check the Principe of superposition: if $\backslash \; vec\; \{v\}\; \_1$ and $\backslash \; vec\; \{v\}\; \_2$ are solutions, then any linear combination $\backslash \; lambda\_1\; \backslash \; vec\; \{v\}\; \_1\; +\; \backslash \; lambda\_2\; \backslash \; vec\; \{v\}\; \_2$ will be to it also (this is not incompatible with the property of unicity: only the flow checking the good conditions in extreme cases will be observed);
• reversibility : if a field speed $\backslash \; vec\; \{v\}\; (\backslash \; vec\; \{R\})$ is solution of the equation, then $-\; \backslash \; vec\; \{v\}\; (\backslash \; vec\; \{R\})$ is also, on the condition of changing the sign of the gradients of pressure, as well as speeds with the walls and ad infinitum (this is a direct consequence of the principle of superposition); this property is contrary with our intuition, founded on our experiment of the macroscopic flows: the reversibility of the flows to low Reynolds number thus pushed the living beings of very small size to develop original means of propulsion.

References

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