In fluid dynamics, **Stokes problem** also known as **Stokes second problem** or sometimes referred to as **Stokes boundary layer** or **Oscillating boundary layer** is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations^{[1]}^{[2]}. In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.

Consider an infinitely long plate which is oscillating with a velocity in the direction, which is located at in an infinite domain of fluid, where is the frequency of the oscillations. The incompressible Navier-Stokes equations reduce to

where is the kinematic viscosity. The pressure gradient does not enter into the problem. The initial, no-slip condition on the wall is

and the second boundary condition is due to the fact that the motion at is not felt at infinity. The flow is only due to the motion of the plate, there is no imposed pressure gradient.

The initial condition is not required because of periodicity. Since both the equation and the boundary conditions are linear, the velocity can be written as the real part of some complex function

because .

Substituting this into the partial differential equation reduces it to ordinary differential equation

with boundary conditions

The solution to the above problem is

The disturbance created by the oscillating plate travels as the transverse wave through the fluid, but it is highly damped by the exponential factor. The depth of penetration of this wave decreases with the frequency of the oscillation, but increases with the kinematic viscosity of the fluid.

The force per unit area exerted on the plate by the fluid is

There is a phase shift between the oscillation of the plate and the force created.

An important observation from Stokes' solution for the oscillating Stokes flow is that vorticity oscillations are confined to a thin boundary layer and damp exponentially when moving away from the wall.^{[7]} 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 waves and water waves.

If the fluid domain is bounded by an upper, stationary wall, located at a height , the flow velocity is given by

where .

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 far away from the plate and a vanishing velocity at the plate . Unlike the stationary fluid in the original problem, the pressure gradient here at infinity must be a harmonic function of time. The solution is then given by

which is zero at the wall *z = 0*, corresponding with the no-slip condition for a wall at rest. This situation is often encountered in sound waves near a solid wall, or for the fluid motion near the sea bed in water waves. 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.

Consider an infinitely long cylinder of radius exhibiting torsional oscillation with angular velocity where is the frequency. Then the velocity for the steady state (i.e. neglecting the transient time) is given by^{[8]}

where is the modified Bessel function of the second kind.

This solution can be expressed with real argument^{[9]} as:

where

and is to the dimensionless oscillatory Reynolds number defined as , being the kinematic viscosity.

If the cylinder oscillates in the axial direction with velocity , then the velocity field is

where is the modified Bessel function of the second kind.

In the Couette flow, instead of the translational motion of one of the plate, an oscillation of one plane will be executed. If we have a bottom wall at rest at and the upper wall at is executing an oscillatory motion with velocity , then the velocity field is given by

The frictional force per unit area on the moving plane is and on the fixed plane is .

**^**Wang, C. Y. (1991). "Exact solutions of the steady-state Navier-Stokes equations".*Annual Review of Fluid Mechanics*.**23**: 159–177. Bibcode:1991AnRFM..23..159W. doi:10.1146/annurev.fl.23.010191.001111.**^**Landau & Lifshitz (1987), pp. 83–85.**^**Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.**^**Lagerstrom, Paco Axel. Laminar flow theory. Princeton University Press, 1996.**^**Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.**^**Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. "Fluid mechanics." (1987).**^**Phillips (1977), p. 46.**^**Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.**^**Rivero, M.; Garzón, F.; Núñez, J.; Figueroa, A. "Study of the flow induced by circular cylinder performing torsional oscillation".*European Journal of Mechanics - B/Fluids*.**78**: 245–251. doi:10.1016/j.euromechflu.2019.08.002.**^**Landau, L. D., & Sykes, J. B. (1987). Fluid Mechanics: Vol 6. pp. 88

Alexander G. Ramm (born 1940 in St. Petersburg, Russia) is an American mathematician. His research focuses on differential and integral equations, operator theory, ill-posed and inverse problems, scattering theory, functional analysis, spectral theory, numerical analysis, theoretical electrical engineering, signal estimation, and tomography.

Geneviève RaugelGeneviève Raugel (27 May 1951 – 10 May 2019) was a French mathematician working in the field of numerical analysis and dynamical systems.

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Navier–Stokes existence and smoothnessThe Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.

Even more basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy. This is called the Navier–Stokes existence and smoothness problem.

Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. It offered a US $1,000,000 prize to the first person providing a solution for a specific statement of the problem:

Prove or give a counter-example of the following statement:In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

Rayleigh problemIn fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of an infinitely long plate from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations. The impulse movement of semi-infinite plate was studied by Keith Stewartson.

Sir George Stokes, 1st BaronetSir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Anglo-Irish physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the Lucasian Professor of Mathematics from 1849 until his death in 1903. As a physicist, Stokes made seminal contributions to fluid mechanics, including the Navier-Stokes equations, and to physical optics, with notable works on polarization and fluorescence. As a mathematician, he popularised "Stokes' theorem" in vector calculus and contributed to the theory of asymptotic expansions. Stokes, along with Felix Hoppe-Seyler, first demonstrated the oxygen transport function of hemoglobin and showed color changes produced by aeration of hemoglobin solutions.

Stokes was made a baronet (hereditary knight) by the British monarch in 1889. In 1893 he received the Royal Society's Copley Medal, then the most prestigious scientific prize in the world, "for his researches and discoveries in physical science". He represented Cambridge University in the British House of Commons from 1887 to 1892, sitting as a Tory. Stokes also served as president of the Royal Society from 1885 to 1890 and was briefly the Master of Pembroke College, Cambridge.

Stokes approximation and artificial timeThis article provides an error analysis of time discretization applied to spatially discrete approximation of the stationary and nonstationary Navier-Stokes equations. The nonlinearity of the convection term is the main problem in solving a stationary or nonstationary Navier-Stokes equation or Euler equation problems. Stoke incorporated ‘the method of artificial compressibility’ to solve these problems.

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