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.)