A parameter \\(\beta\\) is used to capture the fluid-turning cost.

Here is the explanation.

The cost within the pipes of getting fluid from the root to \\(x\\) has already been accounted for above.

The cumulative cost of all the turns along the path to \\(x\\) is defined by a factor \\(m_\beta(x)\\).

\\(m_\beta(x)\\) is defined as the product of a sequence of values \\(f_\beta(y)\\) for all the nodes \\(y\\) along the path from the root to \\(x\\), where \\(f_\beta(y)\\) is a factor expressing the turning cost at \\(y\\).

Let \\(u\\) be the unit vector in the direction of the pipe leading into \\(y\\), and \\(v\\) be the unit vector in the direction of the pipe leading out of \\(y\\) along the path.

Then \\(f_\beta(y)\\) is defined to be \\(|u \cdot v|^{-\beta}\\) if \\(u \cdot v\\) is greater than zero, else infinity.

(We can interpret this later, after this exposition is finished.)