Proceso de gestión de cliente

Each of the problems derived above share the fact that the differential equations to be solved were linear. As I have mentioned, this is important because it means that the solution to problems which can be deconstructed into two or more of these simple problems is just the sum of the individual solutions. Several such superpositional solutions form simple models which will be of use in describing cilia-driven flows in my system, and so I will briefly mention those here.

2.6.1

Poiseuille-Couette flow

This type of flow is in the same geometry as plane Poiseuille and plane Couette flow, parallel plates with a separation h, and combines the boundary conditions of Couette flow with the pressure gradient of Poiseuille flow. Although I previously derived Couette flow (equation2.13) with the moving boundary at the top, it will be of more relevance to my discussion to have the moving boundary be at the bottom. This is accomplished with the simple transformationy_→h₋y, which leaves Poiseuille flow the same because it is symmetric about the midpoint separation. The solution of the combined flows is

−20 −1 0 1 2 3 0.2 0.4 0.6 0.8 1

Poiseuille−Couette Flow Velocity Profiles

Velocity Height ∇p 2η = 0 ∇p 2η =±4 ∇p 2η =±9 ∇p 2η =±1

Figure 2.3: Family of velocity profiles of Poiseuille-Couette flow, equation 2.39, with

various pressure gradients. The velocity of the sliding plane (the floor) is set to u0 = 1

and the height ish= 1.

then called Poiseuille-Couette flow, or also general Couette flow, and given by

u(y) = u0 h (h−y) + ∇p 2η y 2 −hy . (2.39)

A family of these velocity profiles is shown in Figure 2.3 for a given plane velocity u0

and a number of different pressure gradients ∇p.

In many situations of relevance the pressure gradient drives flow in the direction opposite of the flow driven by the boundary. In this case, the net flow rate can be in either direction, as suggested by the set of curves plotted in Figure2.3.

Driven cavity flow

Poiseuille-Couette flow will be of relevance in Chapter 5, but is also applicable to a number of industrial systems as the limiting solution for the flow in a ‘driven cavity’. The geometry is a completely enclosed cavity with height h, width w, and length l.

Just as in Couette flow, the top or bottom boundary is translated along the length of the cavity, and the major feature is a large rotational flow. The fluid near the translating boundary is dragged by the plane’s motion, then encounters one of the end walls and is forced to recirculate around the boundary opposite the translating boundary. The problem has been widely used as a computational benchmark because, at a well-known Reynolds number, counter vortices begin to develop in the corners of the cavity (Bye, 1966; Koseff and Street, 1984). Of most relevance to the later discussion is that Poiseuille-Couette flow is the velocity profile of a driven cavity in the limiting case where the length and width of the cavity are much larger than its height (Albensoeder et al., 2001).

2.6.2

Modified Stokes 2nd problem plus Couette flow

Another example of a superposition of two exact solutions which will be of use in Chapter 5 is a combination of modified Stokes 2nd problem with Couette flow. This describes a geometry in which the lower boundary is translated at a constant velocity, but with an oscillatory motion superimposed onto the constant velocity motion. Again, the solution is just the sum of equations 2.38 and 2.13. In this case, however, it will be useful to further generalize this problem so that the oscillatory motion is in both dimensions within the plane such that the plane exhibits a circular oscillation. In terms of the coordinates I have been using, the plane translates in x with constant velocity u0, and oscillates in y and z with a given frequency ω, a maximum velocity

u(y, t) =uxxˆ+uzzˆ, the solution of Navier-Stokes with these boundary conditions is ux(y, t) = Re " U eiωtsinh _δ0 A(h−y) sinh_δ0h A # + u0 h (h−y), uz(y, t) = Re " U ei(ωt−π/2)sinh _δ0 A(h−y) sinh_δ0h A # . (2.40)

The fluid motion in this problem is what I will define as ‘epicyclic transport’ in Chap- ter 5, the combination of oscillatory movement with an overall constant velocity which is reminiscent of the planetary epicycles hypothesized within the Aristotelian model of the solar system. Such epicycles have also been observed in theoretical and experimen- tal studies of the low Reynolds number flow driven by a single rotating rod (Bouzarth et al., 2007).

The velocity profiles of this superposition are depicted in Figure 2.4. These also serve as good visualizations of the flow in Stokes 2nd problem in general, as it is easy to envision the velocity profile without the offset produced by the Couette model. In addition, the penetration layer derived as equation 2.37 can be visually compared for two frequencies an order of magnitude different.

To foreshadow the epicyclic transport I will present for cilia-driven flow, I present in Figure 2.5 the displacement of a fluid element along the direction of the Couette boundary, at a given height, for the solutions2.40, as well as the trajectory of the fluid element in thex₋z plane.