# Does fluid motion follow some periodic function?

I have heard of oscillations (i.e. simple harmonic motions) where a particle repeats its motion after a period of time, due to the restoring force acting opposite to the displacement and proportional to it. Does the same apply for a liquid (especially during streamline flow)?

Can we represent it as $$y= A \sin (k x - \omega t)$$

• Can you elaborate on the question? The question s unclear. Mar 8 '17 at 14:53

Deterministic motion falls in to two categories: periodic motion and chaotic motion; I'll try to provide a brief overview. In physics, we usually discuss the evolution of dynamical systems in terms of differential equations. Consider a simple pendulum of length $l$. For small oscillations, the angle of the pendulum obeys the equation

$$\ddot{\theta} + {g\over l}\theta=0.$$

The solution to this is a simple sine wave (i.e., $\theta(t) = \theta_0 \cos(\sqrt{g/l} t)$}.

Now, lets add a damping term $\beta \dot{\theta}$, and a driving force $\gamma \cos(\omega t)$, the differential equation to solve is now:

$$\ddot{\theta} + \beta \dot{\theta}+ {g\over l}\theta=\gamma \cos(\omega t).$$

Even when we complicate things to this point, the motion of the pendulum is still periodic. To stray from the realms of periodic behavior, we need to introduce a non-linearity.

Let us assume the pendulum has a stiffness to it that obeys a cubic version of Hooke's law. We then have the following equation $$\ddot{\theta} + \beta \dot{\theta}+ {g\over l}\theta + \alpha \theta^3=\gamma \cos(\omega t).$$

This equation goes by the name of the Duffing Equation and it is one of the simplest differential equations that exhibits chaotic behaviour. For certain parameters in the Duffing equation, the motion is very aperiodic. Although the motion is deterministic, the behavior can appear random.

The (incompressible) Navier-Stokes equations describing fluid flow can be written as $${\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \nabla ^{2}\mathbf {u} =-\frac{1}{\rho}\nabla p .}$$

While this equation does admit periodic solutions in particular expansions, solutions can exhibit chaotic behaviour as the second term is nonlinear in ${\bf u}$. It is believed that it is this term that is responsible for the phenomenon of turbulence.

So in conclusion, if the fluid obeys the Navier-Stokes equation, then there is no periodic function that can describe its chaotic behavior.

We can, however, perform perturbation theory and linearize the Navier-Stokes equation by assuming the form ${\bf u} = {\bf u}_0 +{\bf u}_1$, and neglecting terms of order $\mathcal{O}({\bf u}_1^2)$. The result is

$${\partial {\bf u}_1 \over \partial t} + ({\bf u}_0 \cdot \nabla){\bf u}_1 + ({\bf u}_1 \cdot \nabla){\bf u}_0 - \nu\nabla^2 {\bf u}_1 = {-1 \over \rho}\nabla p_{1}.$$

This linear differential equation will admit periodic solutions for ${\bf u}_1$. However, the validity of the solutions requires $|{\bf u}_1|/|{\bf u}_0| \ll 1$. The results of linear perturbation theory are often very useful for determining the properties of a fluid around an equilibrium.

• Can the nonlinear equation be approximately linearized in some limiting situation? Does that help our cause? Mar 11 '17 at 6:16

If the oscillations are small enough that the governing differential equation may be reasonably approximated with a wave equation, almost every system in equilibrium will exhibit the pattern of motion you've described. It's often a terrible approximation for fluid systems.

• In some situations it's fine, like the propagation of sound waves far from their source Mar 9 '17 at 15:17
• @kleingordon Indeed. That's why I qualified the statement with the word often. Should I re-write that? Mar 9 '17 at 16:54