As *R *= *A/P*, eq 58 can be rewritten as

αi *f*i *P*i *u *- υ 2

α =

(59)

b *f*b *P*b *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 +

τ *P *+ τi Pi

τ

τ*P*

=

= bb

.

(60)

ρ*gR * ρ*gA*

ρ*gA*

The time integral of the friction force is

∫ *F*f dt = ∫ ∫ ρ*gAS*f dxdt = ∫ ∫ [τ bPb + τi Pi ] dxdt.

(61)

Equations 55 to 57, rearranged in terms of the unknown variables *u*, υ, *d, *and η,

become

ρ*f*bu2 Pb ρ*f*i (u - υ) Pi

2

τ b Pb + τi Pi =

+

8

8

ρ*f*bu2 Pb

2

αi

(62)

=

1 +

=

1 +

.

αb

8

8

τ b Pb + τi Pi fbu2Pb

=

1 +

.

(63)

ρ*gA*

8*gA*

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 *- *M*f = ∫ *F*p1dt + ∫ *F*g1dt - ∫ *F*g2dt - ∫ *F*f dt.

(64)

Equations 39, 40, 44, 48, 51, and 61 combined with eq 64 give

ρfiu2 Pi

υ=u

8

τi P

i

υ

36