### Engineering Motivation

In this Worked Problem we look at a potential flow into a bend with a
parametrization on the internal and the external radius. It is a
first (very preliminary) example of design of a parametrized
fluidic device as an element of a more complex modular fluidic
system (if we were to adopt a more complex fluid model). Flows in ducts,
channels, and pipelines are of great interest in fluid mechanics
applications above all if flows can be studied in a parametrized
geometrical configuration. This geometrical configuration could be
a first preliminary example for a fluidic device ``handbook'' made
up of parametrized elements. The aspects illustrated in this
example deal with the imposition of the velocity on the walls of
the bend or at the inlet, the calculation of pressure and velocity
by Bernoulli Theorem, and the curvy geometry parametrization. With
this problem we also illustrate some new features dealing with
potential flows: (*i*) an error bound on velocity and pressure (and
not only on the potential solution), and (*ii*) visualization of the flow
field (streamlines) and pressure contours.

### Physical Principles

The physical principles illustrated in this Worked Problem deal with ideal fluid mechanics (non-viscous fluids) modelled by potential flows and based on the Bernoulli Theorem. Velocity and pressure are influenced by the bend configuration (i.e., angle and radius of curvature). The major points are the curved streamlines, the pressure gradient that induces the curvature, and the classical irrotational flow field associated with zero vorticity. Gravitational effects or other force fields could be applied.