1. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The basic equations of incompressible Newtonian fluid mechanics are the incompressible forms of the Navier-Stokes equations and the continuity equation:
These equations specify four equations (continuity is a scalar equation, Navier-Stokes is a vector equation) in four unknowns ui (i = 1..3) and p.

2. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The physical meaning of the terms in the Navier-Stokes equations can be interpreted as follows. Multiplying by  and using continuity, the equations can be rewritten as
A B C D E
Term A ~ time rate of change of momentum
Term B ~ pressure force
Term C ~ net convective inflow rate of momentum ~ inertial force
Term D ~ viscous force ~ net diffusive inflow rate of momentum
Term E ~ gravitational force

3. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
We make the transformations (u1, u2, u3) = (u, v, w) and (g1, g2, g3) = (gx, gy, gz). Expanding out the equations we then obtain the following forms for the Navier-Stokes equations:
and the following form for continuity:

4. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The simplest flow we can consider is constant rectilinear flow. For example, consider a flow with constant velocity U in the x direction and vanishing velocity in the other directions, i.e. (u, v, w) = (U, 0, 0). This flow is an exact solution of the Navier-Stokes equations and continuity.
Thus for any constant rectilinear flow, all that needs to be satisfied is the hydrostatic pressure distribution (even though there is flow): or

5. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
For plane Couette flow we make the following assumptions:
the flow is steady (/t = 0) and directed in the x direction, so that the only velocity component that is nonzero is u (v = w = 0);
the flow is uniform in the x direction and the z direction (out of the page), so that /x = /z = 0;
the z direction is upward vertical;
the plate at y = 0 is fixed; and
the plate at y = H is moving with constant speed U
For such a flow the only component of the viscous stress tensor is  y x u
moving with velocity U
fixed
fluid H

6. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
That is, the components of the viscous stress tensor are
Here we abbreviate 
moving with velocity U u
fluid H y x 
fixed

7. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Thus u = u(y) only, and v = w = 0. This result automatically satisfies continuity:
Momentum balance in the x, y and z directions (z is upward vertical)

8. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Momentum balance in the z direction (out of the page):
That is, the pressure distribution is hydrostatic. Recall that the general relation for a pressure distribution ph obeying the hydrostatic relation is:  y x u
moving with velocity U
fixed
fluid H

9. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Momentum balance in the x (streamwise) direction:
The no-slip boundary conditions of a viscous fluid apply:
the tangential component of fluid velocity at a boundary = the velocity of the boundary (fluid sticks to boundary)  y x u
moving with velocity U
fixed
fluid H

10. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Integrate once:
Thus the shear stress  must be constant on the domain.
Integrate again:
Apply the boundary conditions to obtain
C2 = 0, C1 = U/H and thus  y x u
moving with velocity U
fixed
fluid H

11. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
For open-channel flow in a wide channel we make the following assumptions:
the channel has streamwise slope angle ;
x denotes a streamwise (not horizontal) coordinate, z denotes an upward normal (not vertical) coordinate and y denotes a cross- stream horizontal coordinate;
the flow is steady (/t = 0) and directed in the x direction, so that the only velocity component that is nonzero is u (v = w = 0);
the flow is uniform in the x direction and the y direction (out of the page), so that /x = /y = 0;
the bottom of the channel at z = 0 is fixed;
there is no applied stress at the free surface where z = H.

12. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The channel width is denoted as B. It is assumed that the channel is sufficiently wide (B/H << 1) so that sidewall effects can be ignored.
Thus streamwise velocity u is a function of upward normal distance z alone, i.e. u = u(z). H B
The vector of gravitational acceleration is (gx, gy, gz) = (gsin, 0, -gcos)

13. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Continuity is satisfied if u = u(z) and v = w = 0.
The equations of conservation of streamwise and upward normal momentum reduce to:

14. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The equations thus reduce to:
Since
The first equation can thus be rewritten as
where  is an abbreviation for 13 = 31.

15. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Assuming that a) pressure is given in gage pressure (i.e. relative to atmospheric pressure) and there is no wind blowing at the liquid surface, the boundary conditions on
are
viscous fluid sticks to immobile bed
no applied shear stress as free surface
gage pressure at free surface = 0 (surface pressure = atmospheric)

16. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Now the condition
states that the hydrostatic relation prevails perpendicular to the streamlines (which are in the x direction). Integrating the relation with the aid of the boundary condition
yields a pressure distribution that varys linearly in z:

17. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The equation
subject to
similarly yields a linear distribution for shear stress  in the z direction:
Note that the bed shear stress b at z = 0 is given as

18. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Thus
subject to
Integrates to give the following parabolic profile for u in z:

19. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The maximum velocity Us is reached at the free surface, where z = H and  = 1);
Thus
Depth-averaged flow velocity U is given as
Thus

20. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
A dimensionless bed friction coefficient Cf can be defined as
Here Cf = f/8 where f denotes the D’arcy-Weisbach friction coefficient. Between the above relation and the relations below
it can be shown that
Here Re denotes the dimensionless Reynolds No. of the flow, which scales the ratio of inertial forces to viscous forces.

21. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Now suppose that there is a wind blowing upstream at the free surface, exerting shear stress w in the – x direction. The governing equations of the free surface flow remain the same as in Slide 15, but one of the boundary conditions changes to
The corresponding solution to the problem is
where r is the dimensionless ratio of the wind shear stress pushing the flow upstream to the force of gravity per unit bed area pulling the flow downstream:

22. LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The solution for velocity with the case of wind can be rewritten as
where und is a dimensionless velocity equal to 2u/(gsinH2).
A plot is given below of und versus  for the cases r = , 0.5, 1 and 1.5.
r = 0
(no wind)
r = 0.5
r = 0.25
r = 1.5
r = 1