$2.$ Shear stress $(\_\_/25$ puntos$)$

Imagine a small river flowing down a gentle slope in a rural area. The slope of the river has an

angle of $=5\backslash$deg with respect to the horizontal. The average depth of the water in the river is $1$

meter. The water has a dynamic viscosity of $=0\mathrm{.}001$ and a density of $=$

$1000/3$

$.$ The velocity profile of the water as a function of depth $($measured from the

riverbed$)$ is parabolic and given by the following equation:

$()=$

$()$

$2$

$2$

$(2$

$$

$$

$$

$$

$2$

$$

$2$

$)$

where:

$()$ is the velocity of the water at a depth $$ from the free surface,

$=9\mathrm{.}81$ m$/$s

$2$

is the acceleration due to gravity,

$$ is the depth of the river,

$$ is the angle of the slope.

If a fish is swimming upstream at a depth of $0\mathrm{.}4$ meters from the free surface of the river, calculate

the shear stress that the fish must overcome to swim against the current at that point in the river.