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Mathinline
body\large{u = -\frac{1}{\rho}\frac{\delta p}{\delta y}}
                   (3.1)

The result is the following relation (Margules relation) for the slope of the frontal boundary:

(3.2)

where γ is the angle the dense fluid makes with the horizontal, 

Mathinline
bodyu_{2}
, 
Mathinline
body\rho_{2}
 are the velocity and density of the less dense fluid, respectively, 
Mathinline
bodyu_{1}
, 
Mathinline
body\rho_{1}
 are the velocity and density of the denser fluid, and 
Mathinline
body\Omega
 is the rate at which the system is rotating, in radians/second.

Assuming that angular momentum of the fluid is conserved (only a valid assumption when friction can be discounted), we get the following relation for the (approximate) deformation radius:

 (3.3)

where H is the height of the free surface with respect to the bottom of the tank.

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Initially, the shape of the front resembles a Gaussian, with a wide sloping surface and flat top.  As the experiment progresses, the frontal boundary gradually sinks relative to the free surface.  Its sides flatten and the width of the center peak narrows (see Figure 4.2).

Using the equator for equation for the deformation radius,

 

given in 3.3, we calculate the predicted radius of deformation to be around 20 centimeters, or half of the width of the tank.  Since the boundary of the front extends all the way to the outside of the tank, about 3/4 of the distance to the bottom of the tank, the actual radius of deformation for this front is slightly larger than this prediction.  Therefore, we see the slope of the front decrease towards the edge of the tank, as it has no more room to spread outwards.

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