Plane-parallel Couette flow

V.A. Gorodtsov and A.I. Leonov, On a linear instability of a plane parallel Couette flow of a viscoelastic fluid, J. Appl. Math. Mech. (PMM), 31 (1967) 310-319. 

V.A. Romanov, Stability of plane-parallel Couette flow, Funct. Anal. AppL 7 (1993), 137-146. 

Simple shear (also known as planar Couette flow) is achieved when fluid is contained between two plane parallel plates in relative in-plane motion. If the velocity direction is taken to be x, one has x = y, all other xa 3 zero and... 

The first mode may occur when a droplet is subjected to aerodynamic pressures or viscous stresses in a parallel or rotating flow. A droplet may experience the second type of breakup when exposed to a plane hyperbolic or Couette flow. The third type of breakup may occur when a droplet is in irregular flow patterns. In addition, the actual breakup modes also depend on whether a droplet is subjected to steady acceleration, or suddenly exposed to a high-velocity gas stream.[2701[2751... 

Renardy and Renardy [66,73] have investigated the stability of plane Couette flows for Maxwell-type models involving the derivative (2). The flow lies between parallel plates at a = 0 euid x = 1, which are moving in the j/-direction with velocities 1, such as in Figure 6. 

In this case, G is called the gradient of the flow rate or the shear rate. The Couette flow occurs between two parallel moving planes or in the gap between coaxial cylinders rotating at different... 

Note that in [51] a nonisothermal flow of a power-law fluid between two parallel planes one of which moves at a constant velocity (a Couette flow) was studied, as well as a gravity flow in an annular gap and a flow between two rotating cylinders of a fluid with exponential consistence factor (6.6.5) under constant wall temperature. 

G5e. Goldman, A. J., Cox, R. G., and Brenner, H, Slow viscous motion of a sphere parallel to a plane wall—II. Motion in a Couette flow. J. Fluid Mech. (in press) see also Goldman, A. J., Investigations in low Reynolds number fluid-particle dynamics. Ph.D. Dissertation, New York University, New York, 1966. 

Hinze also discussed various well-defined flow forms and the types of droplet deformation associated with them. The flow patterns described are parallel flow, plane hyperbolic flow, rotating flow, axisymmetric hyperbolic flow, Couette flow, and irregular flow (turbulent). 

Fluid inertia effects have been found to be very small for the cone-and-plate geometries typically supplied with these instruments. While inertial corrections are foimd to be imimportant for the parallel plate geometries, for shearing gaps of the order of 2 mm or less (except possibly for very thin fluids), they must be taken into accoimt in the concentric cylinder geometry (especially for high-density, mobile fluids). Evaluation methods are available for p, in the case of cylindrical and plane Couette flow, taking into account fluid inertia [Aschoff and Schummer, 1993]. 

Another example is plane Couette flow with variable viscosity [13,14]. Consider the steady flow of an incompressible Newtonian fluid between two infinite parallel plates separated by a distance, a, as shown in Eigure 6.2. Each plate is maintained at a temperature Tq. The upper plate is allowed to move... 

For the Poiseuille flow presented in the previous two sections, the flow was induced by a pressure difference applied between the two ends of a straight pipe. The plane Couette flow makes it possible to study the way how a force is applied to move a solid wall, without pushing the fluid in contact with the wall, and can generate a flow due to viscosity. The fluid is held between two parallel plates of infinite length (Figure 1.5). One of the plates, at z = 0, is kept stationary. The other... 

Example 6.6. Mixing in Plane Couette Flow (Parallel Plate Geometry PCF)... 

Flow birefringence of polymer solutions is, in general, measured with the aid of an apparatus of the Couette type, containing two coaxial cylinders. One of these cylinders is rotated at constant speed, the other is kept in a fixed position. The light beam for the birefringence measurement is directed through the annular gap between these cylinders, in a direction parallel with the axis of the apparatus. In this way, the difference of principal refractive indices An is measured just in the above defined plane of flow (1—2 plane). 

Figure 2-9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate, and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement of the fluid viscosity (see, e.g., Fig. 3-5). In the present discussion, we approximately represent the flow in these devices as the flow between two plane boundaries as described in the text and sketched in this figure.

The Couette arrangement shown in Fig. 3.22 is to allow the three-dimensional nature of the shear flow to be explored. Details of the design are described elsewhere (Mitchell et al. 2015). As we can see in Fig. 3.24, the graphene nanoflakes have a lateral extent of more than 1 pm. Thus although we might anticipate the plane of the flake to lie parallel to the flow direction, it must lie normal to the velocity gradient, otherwise it will rotate and probably breakup. 

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