- #1
Particle Head
Homework Statement
Problem is given in this image,
https://gyazo.com/454370ff9549dcd7c53604ebfe5df105
Homework Equations
Continuity or conservation of mass equation:
[tex] \frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} = 0 [/tex]
Where u is the horizontal velocity and w is the vertical velocity
The Attempt at a Solution
Firstly I integrated the conservation of mass equation with respect to x between the two points:
[tex] \int_ {x_1} ^ {x_2} \frac{\partial u}{\partial x} \mathrm{d}x + \int_ {x_1} ^ {x_2} \frac{\partial w}{\partial z} \mathrm{d}x = 0
[/tex]
Which after evaluating I get,
[tex] u(x_2) - u(x_1) + x_2 \frac{\partial w}{\partial z} - x_1 \frac{\partial w}{\partial z} = 0 [/tex]
Firstly here I am not 100% sure I can assume w is just a function of z only, but I have yet to see it as a function of anything else other than t in 2D flow?
I then integrated the mass equation vertically first from [itex] z = 0 [/itex] to [itex] z = h_1 [/itex] and then from [itex] z=0 [/itex] to [itex] z= h_2 [/itex]
[tex] \int_{0}^{h_1} \frac{\partial u}{\partial x} dz + \int_{0}^{h_1} \frac{\partial w}{\partial z} dz = 0 [/tex]
which yields,
[tex] h_1 \frac{\partial u}{\partial x} - w(0) = 0 [/tex]
Since [tex] w(h_1) = 0 [/tex]
Similarly for [itex] z=0 [/itex] to [itex] z= h_2 [/itex] I get,
[tex] h_2 \frac{\partial u}{\partial x} - w(0) = 0 [/tex]
This is where I am kind of stuck, from these two equations it appears [itex] h_1 = h_2 [/itex] which doesn't make sense since this is a hydraulic jump and the nature of such is to increase the surface of the liquid.
Any hints where I may be going wrong here or missing something is appreciated.
Thanks.