Liquid flow (river), viscosity, force, height etc.

[ApeXaviour]

Hi. This is a two part question, I'm pretty sure about the first bit but in the second part my answer seems a bit weird. Not the order of magnitude I was expecting. Just wondering can anyone confirm or deny my suspician? Point me in more correct direction if I am indeed wrong. This isn't for anything, its just a problem sheet given out and this last question has been bugging me :)

(i)Water flows down a broad channel at an angle $$\theta$$ to the horizontal. $$\theta$$ is sufficiently small that the flow is laminar. By balancing the pressure from the gradient with the shear force due to viscosity (i.e. constant velocity flow) find an expression for the velocity of the surface of the water. Recall that the velocity adjacent to the channel bed is zero.

(ii)If the Liffey[A river in Dublin ] is 5m deep, the surface water velocity is 0.3m/s and the viscosity of the water is 10^-3 kg/ms, find the height of the river surface between heuston stn and talbot bridge, a distance of 2km.


This is what I've got so far:

$$u$$ is the velocity
$$\rho$$ density of the water
$$z$$ is the height off the river bed (water depth when at surface)
$$\tau$$ is the shear stress
$$\eta$$ the viscosity
$$w l$$ arbitrary width and lengthThe shear stress between planes in a liquid is: $$\tau = \eta \frac{du}{dz}$$

So the total force per unit area is

$$\frac{F_\tau}{A} = w l \eta \frac{du}{dz}$$

Which must be equal to the gravitational force per unit area for steady flow:
$$\frac{F_g}{A} = m g Sin(\theta) = w l z \rho g Sin(\theta)$$

so:
$$w l \eta \frac{du}{dz}= w l z \rho g Sin(\theta)$$

$$\int_0^u \,du = \int_0^z \frac{z \rho g Sin(\theta)}{\eta} \,dz$$
And hence the surface velocity is given by:
$$u = \frac{z^{2} \rho g Sin(\theta)}{2 \eta}$$

So when I put in the figures I get
$$Sin(\theta) = \frac{2 u \eta}{z^{2} \rho g} = 2.4 \times 10^{-9} \cong \theta$$

Then from triginometric relations, H the height change is given by:

$$(2 \times 10^{3})(Tan(\theta)) \cong (\theta)(2 \times 10^{3}) = 4.8 \times 10^{-6}m$$ Which to me seems - and maybe my intuition is out of whack - just a little ridiculous.

Thanks
-Dec

 

[Tony Elston]

Hi
Sorry I can't help with your problem, maybe you could help with mine?
I would like to calculate the force on a 1X1 m plate placed in a river against the flow. The river flows at about 5 kmph and has a flow rate of 320 cubic metres per second. I don't have the width or depth of the river it is a big river. I hope you guys can help.

Regards

Tony Elston