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| [[File:Stokes boundary layer.gif|frame|right|Stokes boundary layer in a viscous fluid due to the harmonic oscillation of a plane rigid plate (bottom black edge). Velocity (blue line) and particle excursion (red dots) as a function of the distance to the wall.]]
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| In [[fluid dynamics]], the '''Stokes boundary layer''', or '''oscillatory boundary layer''', refers to the [[boundary layer]] close to a solid wall in [[oscillation|oscillatory]] flow of a [[viscosity|viscous]] [[fluid]]. Or, it refers to the similar case of an oscillating plate in a viscous fluid at rest, with the oscillation direction(s) [[parallel (geometry)|parallel]] to the plate.
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| For the case of [[laminar flow]] at low [[Reynolds number]]s over a smooth solid wall, [[George Gabriel Stokes]] – after whom this boundary layer is called – derived an [[analytic solution]], one of the few exact solutions for the [[Navier–Stokes equations]].<ref>{{cite journal | journal=Annual Review of Fluid Mechanics | volume=23 | pages=159–177 | year=1991 | doi=10.1146/annurev.fl.23.010191.001111 | title=Exact solutions of the steady-state Navier-Stokes equations | first=C. Y. | last=Wang |bibcode = 1991AnRFM..23..159W }}</ref>
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| <ref>Landau & Lifshitz (1987), pp. 83–85.</ref> In [[turbulent]] flow, this is still named a Stokes boundary layer, but now one has to rely on [[flow measurement|experiments]], [[Computational fluid dynamics|numerical simulations]] or [[approximation|approximate methods]] in order to obtain useful information on the flow.
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| The [[boundary-layer thickness|thickness]] of the oscillatory boundary layer is called the '''Stokes boundary-layer thickness'''.
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| ==Vorticity oscillations near the boundary==
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| An important observation from Stokes' solution for the oscillating [[Stokes flow]] is, that [[vorticity]] oscillations are confined to a thin boundary layer and damp [[Exponential decay|exponentially]] when moving away from the wall.<ref name=Phil46> Phillips (1977), p. 46.</ref> This observation is also valid for the case of a turbulent boundary layer. Outside the Stokes boundary layer – which is often the bulk of the fluid volume – the vorticity oscillations may be neglected. To good approximation, the flow velocity oscillations are [[irrotational]] outside the boundary layer, and [[potential flow]] theory can be applied to the oscillatory part of the motion. This significantly simplifies the solution of these flow problems, and is often applied in the irrotational flow regions of [[sound wave]]s and [[water wave]]s.
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| ==Stokes boundary layer for laminar flow near a wall==
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| The oscillating flow is assumed to be [[uni-directional]] and parallel to the plane wall. The only non-zero velocity component is called ''u'' ([[SI]] measure in meter/[[second]], or m/s) and is in the ''x''-direction parallel to the oscillation direction. Moreover, since the flow is taken to be [[incompressible flow|incompressible]], the velocity component ''u'' is only a function of time ''t'' (in seconds) and distance from the wall ''z'' (in meter). The [[Reynolds number]] is taken small enough for the flow to be laminar. Then the [[Navier–Stokes equations]], without additional forcing, reduce to:<ref name=Batch179>Batchelor (1967), p. 179.</ref>
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| :<math>\frac{\partial u}{\partial t} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \frac{\partial^2 u}{\partial z^2},</math>
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| with:
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| *''ρ'' the [[mass density]] of the fluid ([[kilogram|kg]]/m<sup>3</sup>), taken to be a [[Constant (mathematics)|constant]],
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| *''p'' the fluid [[pressure]] (SI: [[Pascal (unit)|Pa]]),
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| *''ν'' the [[kinematic viscosity]] of the fluid (m<sup>2</sup>/s), also taken constant.
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| and
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| *''u'' the velocity of the fluid along the plate (m/s)
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| *''x'' the position along the plate (m)
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| *''z'' the distance from the plate (m)
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| *''t'' the time (s)
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| Because the velocity ''u'' is not a function of position ''x'' along the plate, the pressure gradient ''∂p/∂x'' is also independent of ''x'' (but the pressure ''p'' varies [[linear]]ly with ''x''). Moreover, the Navier–Stokes equation for the velocity component perpendicular to the wall reduces to ''∂p/∂z'' = 0, so the pressure ''p'' and pressure gradient ''∂p/∂x'' are also independent of the distance to the wall ''z''. In conclusion, the pressure forcing ''∂p/∂x'' can only be a function of time ''t''.<ref name=Batch179/>
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| The only non-zero component of the [[vorticity]] [[Euclidean vector|vector]] is the one in the direction [[perpendicular]] to ''x'' and ''z'', called ''ω'' (in s<sup>-1</sup>) and equal to:<ref name=Phil46/>
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| :<math>\omega = \frac{\partial u}{\partial z}.</math>
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| Taking the ''z''-derivative of the above equation, ''ω'' has to satisfy<ref name=Phil46/>
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| :<math>\frac{\partial \omega}{\partial t} = \nu \frac{\partial^2 \omega}{\partial z^2}.</math>
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| As usual for the vorticity dynamics, the pressure drops out of the vorticity equation.<ref>Since the vorticity equation is obtained by taking the [[Curl (mathematics)|curl]] of the Navier–Stokes equations, and the curl of the pressure [[gradient]] equals zero, see [[vector calculus identities]].</ref>
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| ===Oscillation of a plane rigid plate===
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| [[Harmonic motion]], parallel to a plane rigid plate, will result in the fluid near the plate being dragged with the plate, due to the [[Shear stress#Shear stress in fluids|viscous shear stresses]]. Suppose the motion of the plate is
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| :<math>u_0(t) = U_0\, \cos\left( \Omega\, t \right),\,</math>
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| with
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| *''U''<sub>0</sub> the velocity [[amplitude]] of the plate motion (in m/s), and
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| *''Ω'' the [[angular frequency]] of the motion (in [[radian|rad]]/s).
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| The plate, located at ''z = 0'', forces the viscous fluid adjacent to have the same velocity ''u''<sub>1</sub>( ''z'', ''t'' ) resulting in the [[no-slip condition]]:
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| :<math>u_1(0,t) = u_0(t) = U_0\, \cos\left( \Omega\, t \right) \quad \text{ at }\; z = 0.</math>
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| Far away from the plate, for ''z → ∞'', the velocity ''u''<sub>1</sub> approaches zero. Consequently, the pressure gradient ''∂p/∂x'' is zero at infinity and, since it is only a function of time ''t'' and not of ''z'', has to be zero everywhere:<ref>Batchelor (1967), p. 190.</ref>
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| :<math>\frac{\partial u_1}{\partial t} = \nu \frac{\partial^2 u_1}{\partial z^2}.</math>
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| Such an equation is called a one-dimensional [[heat equation]] or [[diffusion equation]].
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| {| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
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| !Details on the derivation of the solution
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| |-
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| |This equation can be solved for harmonic motion using [[complex number]]s and [[separation of variables]]:<ref name=Batch192>Batchelor (1967), p. 192.</ref>
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| :<math>u_1 = \Re\left\{ F(z)\; \text{e}^{-i\, \Omega\, t} \right\},</math>
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| with ''i''<sup>2</sup> = -1 and ℜ{•} denoting the [[real part]] of the quantity between brackets. Then from the flow equation, ''F''(''z'') is required to satisfy:
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| :<math>-i\, \Omega\, F = \nu \frac{\text{d}^2 F}{\text{d} z^2}</math>
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| with the boundary conditions:
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| :<math>
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| \begin{align}
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| F &= U_0 & \qquad & \text{ for } z = 0,
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| \\
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| F &\to 0 & \qquad & \text{ for } z \to +\infty.
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| \end{align}
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| </math>
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|
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| The solution for ''F''(''z'') becomes:
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| :<math>F(z) = U_0\, \text{e}^{(-1+i)\, \sqrt{\frac{\Omega}{2\nu}}\, z}.</math>
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| |}
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| As a result, the solution for the flow velocity is<ref name=Batch192/>
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| :<math>
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| u_1(z,t) = U_0\, \text{e}^{-\kappa\, z}\, \cos\left( \Omega\, t\, -\, \kappa\, z\right)
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| \quad \text{ with }\; \kappa\, =\, \sqrt{\frac{\Omega}{2\nu}}.
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| </math>
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| Here, ''κ'' is a kind of [[wavenumber]] in the ''z''-direction, associated with a length<ref name=Batch192/>
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| :<math>\delta = \frac{2\pi}{\kappa} = 2\pi\, \sqrt{\frac{2\nu}{\Omega}}</math>
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| which is called the '''Stokes boundary-layer thickness'''. At a distance ''δ'' from the plate, the velocity amplitude has been reduced to e<sup>-2π</sup> ≈ 0.002 times its value ''U''<sub>0</sub> at the plate surface. Further, as can be seen from the phase changes ''Ω t - κ z'' in the solution ''u''<sub>1</sub>, the velocity oscillations propagate as a damped [[wave]] away from the wall, with [[wavelength]] ''δ'' and [[phase speed]] ''Ω / κ''.
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| The vorticity ''ω''<sub>1</sub> is equal to
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| :<math>
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| \omega_1(z,t)
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| = \frac{\partial u_1}{\partial z}
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| = -\kappa\, U_0\, \text{e}^{-\kappa\, z}\,
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| \Bigl[\,
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| \cos\left( \Omega\, t\, -\, \kappa\, z \right)\,
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| -\,
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| \sin\left( \Omega\, t\, -\, \kappa\, z \right)\,
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| \Bigr]
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| </math>
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| and, as ''u''<sub>1</sub>, dampens exponentially in amplitude when moving away from the plate surface.
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| ===Flow due to an oscillating pressure gradient near a plane rigid plate===
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| The case for an oscillating [[far-field]] flow, with the plate held at rest, can easily be constructed from the previous solution for an oscillating plate by using [[linear superposition]] of solutions. Consider a uniform velocity oscillation ''u<sub>∞</sub>'':
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| :<math>u_\infty(z,t) = U_0\, \cos\left( \Omega\, t \right), \,</math>
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| which satisfies the flow equations for the Stokes boundary layer, provided it is driven by a pressure gradient
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| :<math>\frac{\partial p_2}{\partial x} = \rho\, \Omega\, U_0\, \sin\left( \Omega\, t \right).</math>
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| Subtracting the solution ''u''<sub>1</sub>( ''z'', ''t'' ) from ''u''<sub>∞</sub>( ''z'', ''t'' ) gives the desired solution for an oscillating flow near a rigid wall at rest:<ref name=Phil46/>
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| :<math>
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| u_2(z,t) = U_0\, \Bigl[\, \cos\left( \Omega\, t \right)\, -\, \text{e}^{-\kappa\, z}\, \cos\left( \Omega\, t\, -\, \kappa\, z \right)\, \Bigr],
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| </math>
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| which is zero at the wall ''z = 0'', corresponding with the no-slip condition for a wall at rest. Further the velocity ''u''<sub>2</sub> oscillates with amplitude ''U''<sub>0</sub> far away from the wall, ''z → ∞''. This situation is often encountered in [[sound waves]] near a solid wall, or for the fluid motion near the sea bed in [[water waves]].
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| The vorticity, for the oscillating flow near a wall at rest, is equal to the vorticity in case of an oscillating plate but of opposite sign: ''ω''<sub>2</sub> = - ''ω''<sub>1</sub>.
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| ==See also==
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| *[[Basset force]]
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| *[[Stokes flow]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book | first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 }}
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| *{{cite book | first=H. | last=Lamb | authorlink=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th| isbn=978-0-521-45868-9 }} Originally published in 1879, the 6th extended edition appeared first in 1932.
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| *{{Cite book | author1-link=Lev Landau | last1=Landau | first1=L.D. | author2-link=Evgeny Lifshitz | last2=Lifshitz | first2=E.M. | year=1987 | title=Fluid Mechanics | publisher=Pergamon Press | edition=2nd | isbn=0-08-033932-8 | series=Course of theoretical physics | volume=6 }}
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| *{{cite book| first=O.M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}
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| {{physical oceanography}}
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| [[Category:Fluid dynamics]]
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| [[Category:Boundary layers]]
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