As R = A/P, eq 58 can be rewritten as
αi fi Pi u - υ 2
b fb Pb u
where αb and αi are the bed and ice-affected portions of the flow area. The total
shear stress is τb + τi and the total shear force on a unit length of channel is τbPb +
τiPi , which leads to the following expression for Sf
τ P + τi Pi
The time integral of the friction force is
∫ Ff dt = ∫ ∫ ρgASf dxdt = ∫ ∫ [τ bPb + τi Pi ] dxdt.
Equations 55 to 57, rearranged in terms of the unknown variables u, υ, d, and η,
ρfbu2 Pb ρfi (u - υ) Pi
τ b Pb + τi Pi =
fi (u - υ) Pi ρfbu2 Pb
An expression for Sf results when eq 59 and 62 are combined
fi Pi u - υ 2
τ b Pb + τi Pi fbu2Pb
fb Pb u
The effects of flow and ice velocities on τiPi are evident in Figure 28. As ice veloc-
ity increases beyond water velocity, the shear stress caused by the jam's presence
reverses to the downstream direction and the portion of the flow area affected by
this force increases.
The full momentum equation for the water flow can be written as
∆M - Mf = ∫ Fp1dt + ∫ Fg1dt - ∫ Fg2dt - ∫ Ff dt.
Equations 39, 40, 44, 48, 51, and 61 combined with eq 64 give
Figure 28. Shear force on the ice
jam underside vs. ice velocity
(τiPi vs. υ).