### 2.1 Steady pressure- and boundary-driven flow through long channels

- 2.1.1 Couette flow
- 2.1.2 Poiseuille flow

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### 2.2 Startup and development of unidirectional flows

VIDEO: Startup of Couette, Poiseuille, and electroosmotic flows.

The solutions above are for steady flows through infinitely long channels. We are also interested in flows with unsteady boundary conditions, for example the startup ofthe flow or the flow in response to oscillatory boundary conditions. For example, startup of a Couette flow corresponds to the case where two plates and the fluid are motionless for time t < 0, but the plates move for t > 0. Figure 2.3 shows the startup of Couette flow, both in terms of velocity as well as local force and acceleration. The startup and steady-state of a Couette flow is a useful illustration of the forces, velocities, and accelerations for a simple flow. Consider first the steady-state solution. Since this is at equilibrium, the net force on a control volume is zero. In fact, each term in the Navier-Stokes equation is zero. In contrast, upon startup, the fluid near the top wall feels a net viscous force pulling it forward, and thus the fluid near the wall accelerates. The viscous force and thus the acceleration diffuses down toward the other plate, until eventually the steady-state solution is reached.

Figure 2.3: Startup of a Couette flow. Elapsed time increases from top to bottom.

Similarly, the startup of Poiseuille flow (acceleration from rest of quiescent fluid caused by a pressure gradient applied at t = 0) initially involves uniform acceleration as the pressure gradient is applied. This is then counteracted by a viscous force, first at the walls, and then throughout the fluid, until steady-state is reached. The startup of a Poiseuille flow is shown in Figure . Startup of Couette and Poiseuille flows (and, in fact, all time-varying Couette and Poiseuille flows) can be solved analytically with separation of variables, with solutions given by eigenfunction expansions—sine series for Couette flow and Bessel series for Poiseuille flow. This is possible because the governing equation for each flow is linear, since the (nonlinear) convection term is zero for these flows.

We ignore startup effects for flow through a channel of radius or half-depth R at all elapsed times t ≫, or for changes with acharacteristic frequency ω ≪.

Figure 2.4: Startup of a Poiseuille flow. Elapsed time increases from top to bottom.

In contrast to startup, which refers to a temporal change in a fluid system, we refer to development ofa flow to mean the spatial evolution of the flow profile from an inlet profile to the final profile that the channel would have if it were infinitely long. Near the inlet of a channel, we can not assume that the velocity gradients are normal to the flow direction, and thus the convective terms are nonzero and the Navier-Stokes equations cannot be solved analytically. Solutions for the flow near an inlet to a channel are typically performed numerically. We refer to a region of a flow as fully-developed if the velocity profile in that region is equal to that which would be observed if the channel were infinitely long. The laminar flow in a channel of radius or half-depth R can be assumed fully-developed when the distance from the inlet ℓ satisfies the relation ℓ∕R ≫ Re. When this is satisfied, we can use the results from this chapter directly to analyze flows in channels of finite length.

### 2.3 Summary

The Navier-Stokes equations can be solved analytically if certain simplifications were made. In particular, the convection term ⋅∇ is the most mathematically difficult term in the Navier-Stokes equations to handle, and this chapter examines flows for which the geometry was so simple that we can assume away any spatial dependence that leads to nonzero net convective flux. This approach leads to steady solutions for Couette flow, i.e., flow between two infinite parallel plates:

and Hagen-Poiseuille flow, i.e., pressure-driven flow through a circular channel:

The startup of these flows highlights differences between forces, velocities and accelerations. During startup, the acceleration is proportional to the force, which is a sum of the pressure and viscous forces. At equilibrium, the acceleration is by definition zero, and the concavity of the velocity distribution (which is proportional to the viscous force) is proportional to the local pressure gradient. Thus steady Couette flow, which has no pressure gradient, has no concavity in the velocity distribution, while steady Poiseuille flow, which has a uniform pressure gradient, has uniform concavity in the velocity profile. Development of these flows includes nonzero convective terms, which are nonlinear and preclude general analytical solution.

2.4 Supplementary reading

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2.5 Exercises

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