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Couette flowIn fluid dynamics, Couette flow refers to the laminar flow of a viscous liquid in the space between two surfaces, one of which is moving relative to the other. The flow is driven by virtue of viscous drag force acting on the fluid. This type of flow is named in honor of Maurice Marie Alfred Couette, a Professor of Physics at the French university of Angers in the late 19th century. Additional recommended knowledge
Simple Conceptual Configuration
Mathematical DescriptionCouette flow is frequently used in undergraduate physics and engineering courses to illustrate sheardriven fluid motion^{[1]}. The simplest conceptual configuration finds two infinite, parallel plates separated by a distance h. One plate, say the top one, translates with a constant velocity u_{0} in its own plane. Neglecting pressure gradients, the NavierStokes equations simplify to
where y is a spatial coordinate normal to the plates and u (y) is the velocity distribution. This equation reflects the assumption that the flow is unidirectional. That is, only one of the three velocity components (u,v,w) is nontrivial. If y originates at the lower plate, the boundary conditions are u(0) = 0 and u(h) = u_{0}. The exact solution
can be found by integrating twice and solving for the constants. Constant ShearA notable aspect of this model is that shear stress is constant throughout the flow domain. In particular, the first derivative of the velocity, u_{0} / h, is constant. (This is implied by the straightline profile in the figure.) According to Newton's Law of Viscosity (Newtonian fluid), the shear stress is the product of this expression and the (constant) fluid viscosity. Taylor's Idealized ModelThe configuration shown in the figure cannot actually be realized, as the two plates cannot extend infinitely in the flow direction. Sir Geoffrey Taylor was interested in sheardriven flows created by rotating coaxial cylinders. He reported a mathematical result in 1923 that accounts for curvature in the flow direction having the form^{[2]}
where C_{1} and C_{2} are constants that depend on the rotation rates of the cylinders. (Note that r has replaced y in this result to reflect cylindrical rather than rectangular coordinates.) It is clear from this equation that curvature effects no longer allow for constant shear in the flow domain, as shown above. This model is incomplete in that it does not account for nearwall effects in finitewidth cylinders, although it is a reasonable approximation if the width is large compared to the space between the cylinders. Generalizations of Taylor's basic model have also been examined. For example, the solution for the timedependent "startup" process can be expressed in terms of Bessel functions^{[3]}. FiniteWidth ModelTaylor's solution accounts for the curvature inherent in the cylindrical devices typically used to create Couette flows, but not the finite nature of the width. A complimentary idealization accounts for finiteness, but not curvature. In the figure above, we might think of the "boundary plate" and the "moving plate" as the edges of two cylinders having large radii, say R_{1} and R_{2}, respectively, where R_{2} is only slightly greater than R_{1}. In this case, curvature can be neglected locally. The physicist/mathematician Ratip Berker reported a mathematical solution for this configuration in terms of a trigonometric expansion^{[4]} Wendl's Result for Physical DevicesActual coaxial cylinder devices used to create Couette flows have both curvature and finite geometry. The latter gives rise to increased drag in the wall region. A mathematical result that accounts for both of these aspects was given only recently by Michael Wendl^{[5]}. His solution takes the form of an expansion of modified (hyperbolic) Bessel functions of the first kind. References
See also


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Couette_flow". A list of authors is available in Wikipedia. 