Pressure difference at pipe ends from dynamic viscosity and flow rate
=====================================================================

In non-ideal fluid mechanics, the Hagen—Poiseuille equation is a physical law that gives
the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing
through a long cylindrical pipe of constant cross section. It can be successfully
applied to air flow in lung alveoli, or the flow through a drinking straw.

**Conditions:**

#. The fluid is incompressible and Newtonian.
#. The flow of the fluid is laminar.
#. The pipe is long enough for the flow to be laminar.
#. The pipe has constant cross section.

**Links:**

#. `Wikipedia, first part of the first equation <https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation#Equation>`__.

..
    TODO: rename file to use descriptive name

.. py:currentmodule:: symplyphysics.laws.hydro.hagen_poiseuille_equation

.. py:data:: dynamic_viscosity

    :attr:`~symplyphysics.symbols.classical_mechanics.dynamic_viscosity` of the fluid.

Symbol:
    :code:`mu`

Latex:
    :math:`\mu`

Dimension:
    :code:`pressure*time`


.. py:data:: length

    :attr:`~symplyphysics.symbols.classical_mechanics.length` of the pipe.

Symbol:
    :code:`l`

Latex:
    :math:`l`

Dimension:
    :code:`length`


.. py:data:: flow_rate

    :attr:`~symplyphysics.symbols.classical_mechanics.volumetric_flow_rate` of the fluid through the pipe.

Symbol:
    :code:`Q`

Latex:
    :math:`Q`

Dimension:
    :code:`volume/time`


.. py:data:: radius

    :attr:`~symplyphysics.symbols.classical_mechanics.radius` of the pipe.

Symbol:
    :code:`r`

Latex:
    :math:`r`

Dimension:
    :code:`length`


.. py:data:: pressure_difference

    Difference in :attr:`~symplyphysics.symbols.classical_mechanics.pressure` between the two ends of the pipe.

Symbol:
    :code:`Delta(p)`

Latex:
    :math:`\Delta p`

Dimension:
    :code:`pressure`


.. py:data:: law

    :code:`Delta(p) = 8 * mu * l * Q / (pi * r^4)`


    Latex:
        .. math::
            \Delta p = \frac{8 \mu l Q}{\pi r^{4}}