  We investigate the motion of closed, smooth non-self-intersecting curves that
evolve in space $\mathbb{R}^3$. The geometric evolutionary equation for the
evolution of the curve is accompanied by a parabolic equation for the scalar
quantity evaluated over the evolving curve. We apply the direct Lagrangian
approach to describe the geometric flow of 3D curves resulting in a system of
degenerate parabolic equations. We prove the local existence and uniqueness of
classical H\"older smooth solutions to the governing system of nonlinear
parabolic equations. A numerical discretization scheme has been constructed
using the method of flowing finite volumes. We present several numerical
examples of the evolution of curves in 3D with a scalar quantity. In this
paper, we analyze the flow of curves with no torsion evolving in rotating and
parallel planes. Next, we present examples of the evolution of curves with
initially knotted and unknotted curves.
