DR HELEN J WILSON: PROBLEM PLASTICS - AND HOW MATHEMATICS CAN HELP
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PROBLEM PLASTICS - AND HOW MATHEMATICS CAN HELP
Have you ever wondered how plastic bags are made?

Dr. Helen Wilson from UCL's Mathematics department is working with colleagues around the country on the Microscale Polymer Processing project to make the process more reliable.

Plastics are made from very long molecules called polymers, thousands or millions of times bigger than a simple molecule like carbon dioxide, water or methane. They are usually processed in a liquid state, at high temperatures and pressures. Unstable flow of a molten polymer

These long molecules naturally tend to coil up in their liquid state, because of Brownian motion. But when the liquid is being processed - say, extruded into a thin film to make bags - these coils can get stretched out. Their natural tendency to retract back to a coiled shape gives the liquid an elasticity - like the stringiness of cheese on a pizza.

The equations that describe the flow of a melted plastic are complicated - so complicated, in fact, that there is no perfect set of equations for all plastics. Instead, a different set of equations applies to different shapes of molecules. Even for the simplest shape - a single strand - the first properly self-consistent model was only published in 2003.

You might ask, why do we need equations to model these molten plastics? One of the biggest problems facing the plastics industry today is flow instabilities: processes that should produce a nice, smooth shape of product instead produce something rough, and varying with time. In the photograph above, courtesy of Dr. Tim Gough from the Chemical Engineering Department at Bradford University, a molten plastic is being extruded to produce a uniform tape, but an instability has set in and the surface is visibly irregular. Sometimes this happens when the flow is too fast, which limits the usable speed; sometimes it happens with no warning and the product has to be scrapped, wasting precious resources.

Dr. Wilson and her colleagues are using these new models of polymers along with high-performance computing and the mathematical technique of linear stability theory (see right) to predict when flows will go unstable. In time, they hope to use this work to understand, both mathematically and physically, why instabilities happen, and develop methods for avoiding them - even linking right back to the shape of the original molecules! The pendulum

To get an idea of how linear stability theory works, let's look at an example - the pendulum. Not the perfect, ideal pendulum you might meet in A-level mechanics, but a real pendulum with its weight spread out along its length and a bit of air resistance slowing it down. Now suppose our pendulum is in a steady state - it's not moving and we're not holding it. What position must it be in?

You probably guessed that it must be at the bottom of its circle of possible positions. This hanging position is called a stable equilibrium. The word equilibrium simply means balance. Now say the pivot moves very slightly. This sort of "noise" - maybe vibration from traffic outside, or a breeze blowing through - is impossible to avoid in real life. Our pendulum may swing a little, but as air resistance damps the motion it will return to rest. That fact is what defines our equilibrium as stable.

In fact there is another position where in theory the pendulum can be in equilibrium - at the top of its circle of possible positions. It will only have gravity and the force at the pivot acting on it, and these are both vertical, so what would make it fall one way or the other? It is possible to balance a pendulum "upside down" - but it takes a lot of effort, as you'll know if you've ever tried to balance a pencil on your finger. The reason is that this position is an unstable equilibrium. Going back to the pendulum, if the pivot moves very slightly, the pendulum will tilt very slightly one way or the other. Then gravity pulls it further in that direction, and very quickly the pendulum is a long way away from the equilibrium point. Because we can't eliminate noise in real life, you would never expect to see a pendulum balancing heavy end up.

Linear stability theory is a five stage process.
  1. Write down the equations that govern our system.
    For the pendulum we just have one equation:
    d^2 theta/dt^2 = -k sin(theta) - lambda d theta/dt
    where theta is the angle measured from the bottom of the swing, k is a positive constant related to the shape and length of the pendulum, and lambda is another positive constant relating to the amount of air resistance.
  2. Solve these equations, looking for an equilibrium - that is, a solution that doesn't change in time.
    For the pendulum we will need
    d theta/dt = 0 and d^2 theta/dt^2 = 0
    so we are solving the equation
    sin(theta)=0
    which has solutions theta=0 and theta=pi: the bottom and top positions.
  3. Add a small "kick" delta to our system.
    We talked earlier about moving the hinge, but it's easier for us to move the pendulum: i.e. set
    theta=theta_eq + delta
    where delta is very small. We substitute this into the equation:
    d^2(theta_eq + delta)/dt^2 = -k sin(theta_eq + delta) - lambda d(\theta_eq + delta)/dt
    We can use the formula for sin(a+b) to expand this:
    d^2 theta_eq/dt^2 + d^2 delta/dt^2 = -k sin(theta_eq)cos(delta) -k cos(theta_eq)sin(delta) - lambda d theta_eq/dt - lambda d delta/dt
    Now remember that d theta_eq/dt=0 and d^2 theta_eq/dt^2=0, and also sin(theta_eq)=0, so
    d^2 delta/dt^2 = -k cos(theta_eq)sin(delta) - lambda d delta/dt
  4. This stage is what makes this linear stability theory. We expand all the terms as a power series in delta, and throw away anything which is the same size as delta^2 or smaller. When we've finished, our system is linear because every term contains exactly one power of delta or d delta/dt or d^2 delta/dt^2.
    In our case we know that sin(delta)=delta-delta^3/6+..., so we get:
    d^2 delta/dt^2 = -k cos(theta_eq) delta - lambda d delta/dt
  5. The final step is one of the reasons linear equations are so useful. We assume (rashly) that the solution to our equation is of the form exp(alpha t), and look to see which values of alpha work.
    In our example, we set
    delta = A exp(alpha t)
    for some (unknown) constants A and alpha. Let's try it. The derivative of exp(alpha t) is alpha exp(alpha t) so our equation is
    alpha^2 A exp(alpha t) = -k cos(theta_eq)A exp(alpha t) - lambda alpha A exp(alpha t)
    so, dividing by A exp(alpha t) and rearranging,
    alpha^2 + lambda alpha + k cos(theta_eq) = 0
    which is just a quadratic equation!

    Now let's look at our equilibrium points. Remember, the top is unstable and the bottom stable:
    At the top, theta_eq=pi and cos(theta_eq)=-1 so
    alpha = (-lambda +/- sqrt(lambda^2+4k))/2
    Since lambda-squared + 4k is bigger than lambda-squared, the square root is bigger than lambda and one of the alpha-values is positive. The disturbance grows exponentially in time.     		  At the bottom, theta_eq=0 so cos(theta_eq)=1 and
    alpha = (-lambda +/- sqrt(lambda^2-4k))/2
    Now lambda-squared - 4k is less than lambda-squared, so the size of -lambda means both the solutions are negative. The "kick" dies away exponentially.

So at the final stage, when we look for solutions like exp(alpha t), if alpha turns out to be positive (even for only one solution) then the equilibrium is unstable; if alpha is negative for all solutions then the noise causes little disturbances that fade away, and the equilibrium is stable.