2.1 Steady pressure- and boundary-driven flow through long channels [Couette/Poiseuille top]

The Navier-Stokes equations can be simplified when flow proceeds through an infinitely long channel of uniform cross-section, owing to the geometric elimination of the nonlinear term in the equation. Couette and Poiseuille flows, in turn, describe flow driven by boundary motion or pressure gradients.

2.1.1 Couette flow

Couette flow is the flow of fluid in between two infinite parallel flat plates driven by the motion of one or more of the plates. Consider steady flow between two infinite parallel moving plates with a uniform pressure field. Assume one plate is located at y = h and moves in the x-direction with speed u_H, and the other plate is located at y = -h and moves in the x-direction with speed u_L. A schematic of this flow is shown in Figure 2.1. Ignoring body forces, the Navier-Stokes equations are

(2.1)

Simplified equation We solve this flow by making two simplifying approximations. The resulting solution satisfies the boundary condition and governing equation and thus validates these approximations. We begin by assuming that the fluid motion is only in the x-direction. Since the flow is only in the x-direction, we can replace  with u, resulting in

(2.2)

Our next simplifying approximation is that the velocity profile is independent of x. The first three terms of the equation can therefore be assumed zero, in turn, because (a) the flow is assumed steady, (b) u is assumed independent of x, and (c) the pressure is assumed uniform. Thus, the simplified governing equation for this flow is

(2.3)

In this case, we have chosen a problem statement and geometry that allows much of the Navier-Stokes equation to be ignored; in so doing, we have a system that is much simplier mathematically.

Solution Equation 2.3 can be integrated directlyto give a linear distribution:

(2.8)

Applying boundary conditions at the top plate:

(2.9)

as well as the bottom:

(2.10)

we find

Physical interpretation The Couette flow has no acceleration, no net pressure forces, and no net convective transport of momentum. Because of this, the governing equation also says that the net viscous force on any control volume is also zero. For spatially-uniform viscosity, this means that the concavity (i.e., the second derivative with respect to y) of the flow distribution is zero. If we draw a control volume around the flow, we can see that the top surface applies a force per unit area to the fluid given by η, and the bottom surface applies a force per unit area to the fluid given by -η. The velocity profile of a Couette flow is not a function of the viscosity, but the force required to move the plates is.

Flow kinematics and viscous stresses The Couette flow has a simple functional form. While a tensor description is not necessary for this flow, our Couette flow is a good opportunity to use the tensor description in a simple case.

Evaluating the strain rate and viscous stress tensor, we obtain

(2.12)

or

(2.13)

The only nonzero viscous stress terms are the τ_xy and τ_yx terms, each of which has a value equal to the viscosity times the velocity gradient:

This expression has no x- or y-dependence. The viscous stress is uniform, consistent with the governing equation, which can also be written as = 0. This flow exhibits both shear strain (off-diagonal elements of ) and rotation (off-diagonal elements of ω).

Reynolds number The Reynolds numberis defined as Re = ρUℓ∕η, where U and ℓ are the characteristic velocity and length, respectively, which come from the boundary conditions (see the appendix, Section E.2.1). By characteristic, we mean that the velocity and length are representative of (a) the inertial forces, as represented by the dynamic pressure (1∕2)ρU², and (b) the viscous forces, as represented by the viscous stress ηU∕ℓ. In both cases, we care most precisely about velocity differences.

For Couette flow, we define the characteristic velocity U as U = , which is a measure of the total difference between the fastest part of the flow and the slowest part of the flow. The dynamic pressure (1∕2)ρU² tells us the amount the pressure would increase if the fast fluid was slowed down along a streamline to the speed of the slow fluid, and thus gives a measure of the inertia of the fast fluid as compared to the slow fluid.

For Couette flow, we define the characteristic length ℓ as ℓ = h, since the viscous stresses are τ_xy = η everywhere in the flow.

Substituting these into the definition of the Reynolds number, we get that the Reynolds number for Couette flows is:

(2.15)

For Couette flows, the Reynolds number indicates whether the laminar solution derived above is observed experimentally, or if a turbulent solution is observed. For Re < 100, the solution derived above is the one observed. In microsystems, this is typically satisfied, owing to the small h values used.

2.1.2 Poiseuille flow

Poiseuille flow describes the flow resulting from pressure gradients in a tube, and Hagen-Poiseuille flow refers specifically to pressure-driven flow through a tube of circular cross-section. This type of flow is characteristic of pressure-driven flow through channels in microdevices, and this solution (Figure 2.2) is the fundamental source for all of the results in Chapter 3.

To solve for the Hagen-Poiseuille flow, we assume that all flow is in the z-direction (u_r = 0, u_θ = 0) and that gradients in the θ- and z-directions are zero. The Navier-Stokes equations are:

(2.26)

Simplified equation Because of the geometric simplifications, the convection and unsteady terms are all zero, leaving:

(2.27)

For radially symmetric flow, we write this as:

(2.28)

Solution To solve, we assume that  is uniform, and integrate the r terms:

(2.29)

rearrange,

(2.30)

and integrate again to find

(2.31)

By requiring that (a) u_z be bounded at r = 0 and (b) u_z = 0 at r = R, we solve and find

We can integrate this distribution spatially (Exercise 2.5) to find the flowrate Q:

and normalize by the area (Exercise 2.6) to generate the area-averaged velocity u_z:

Physical interpretation The steady Poiseuille flow has no convective transport of momentum, nor acceleration. The governing equation highlights the balance between the net pressure forces and the net viscous force. Since the pressure gradient (and thus the net pressure force) is uniform, the derivative of the shear stress is uniform, and thus the concavity (i.e., 2nd derivative with respect to y) of the distribution is uniform. If we draw a control volume around the fluid in its entirety, we can see that the net pressure force on the control volume is dz(πR²), while the viscous force per unit area that the wall applies to the fluid is -, and the net viscous force in total is -dz(πR²).

Flow kinematics and viscous stresses There are only two nonzero viscous stress terms for Poiseuille flow— the τ_rz and τ_zr terms. Each is given by the viscosity times the velocity gradient:

(2.35)

Evaluating the derivative, we obtain

For Poiseuille flows, the viscous stress varies linearly, with a minimum of zero at r = 0 and a maximum of at r = R. The velocity distribution of a Poiseuille flow is parabolic.

Reynolds number As was the case with the Couette flow, we need a characteristic velocity and a characteristic length to use to define theReynolds number.

Unlike for the Couette flow, the problem statement does not prescribe any velocities; rather, it specifies a pressure gradient. We therefore use the velocities from the solution to guide definition of the Reynolds number. We typically define U = for Poiseuille flow and define ℓ = 2R for this flow. With these definitions,

As was the case for Couette flow, Re for Poiseuille flow tells us whether the laminar flow distribution derived above is observed—Re < 2300, the flow is laminar.

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