Thank you, Alan. So there was a lot of meticulous planning that went into this so that everything would go nice and slick and as you can see, nothing has gone slick. But that's quite a nice way to start off, because now I don't have high expectations. I'm as Alan said, I'm an industrial mathematician, having quite a lot of us in here. And what we do is we work with lots of industries who bring along problems and then we work on these and
find solutions that can help to advance society or technology. And what I'd like to do in this talk today is tell you some stories about some of the industries that I've worked with and the kinds of techniques that we use to solve some of the problems that might present. So what I want to do to start with is motivate this idea of going backwards in time. That is in to title. So let me try and switch to the. Come on. OK. So what we have here is a
container cylinder container and it's full of glycerol. So this is the kind of stuff if you've ever had a sore throat and gone to the pharmacy, you get honey lemon in glycerine. It's that kind of thing. So it's a sugar solution and is quite discussed, quite sticky. And that's inside the container. And then inside the centre of here, we have a cylinder. And when I turn this dial here, the handle, it spins the list of all around. And hopefully you can see that this place of all is clear. So
you can't see anything going on at the moment. So what I'm gonna do is I'm going to inject some colour dye in so that when I move the handle, you can see how it how it smells. OK, so let's have a guy try and inject some blue dye. First of all, inject it and then have a look and see if it's in the shot on it. You can see it, it's a little bit faint, but you can see it. And then I'll try some Medina. Now, try and be a bit more creative with this. Well. That's supposed to be an aim. But
we we've got a little bit of a pattern there. OK. So what I'm going to do now is I'm going to mix this, Oola. As many. And we should hopefully see this, you see the camera biting and I mix and makes. And then I think we need to go too far before it's all the way come round and actually you can't see much here. You can still see me as on here. So now we ask the question, what if I go in the other direction?
So in some mix, as you mix in one direction, then you mix in the other direction and it makes things up even more. So the options are here. It's going to mix even more. It'll stay about the same or it'll completely on mix. So just and the blogs will go back to how they were again. So that's an option. So just to cheque hands. I'll give you three options. So mix more, stay the same or complete your mix. So 100. If you think you'll makes more. Quite a lot. Stay about the same.
And completely unmixed. OK. So we have a big, big number of people who say it's going to completely omic. So if I spin this round now, the other way turning. Tony. Turning like this, then, sure enough, a completely on mixes. And actually, you can. It's pretty robust that you can flip this man and go back with any and you can do this pretty fast and unmake his every single time. So this is perhaps perhaps not to this audience, but perhaps to the other audience,
quite unexpected. You know, when you give a mass talk in the maths department, you're always going to get people who know the answer. But this is kind of surprising. And you can have a go at this if you didn't believe me until I was doing a bit of a trick. You can either go yourself. You can just
take a break again by the UK. But another baker inside like a cylinder and get some glycerine from a pharmacist and then put the column in and turn it round and turn it back and you'll get this kind of feature. That's kind of interesting. And so that's the idea. And we want to kind of tap into this notion of going backwards in time to see if we can solve some industrial problems using this.
So if you're interested in looking this up, it's called the reversible Stokes flow experiment. If you look this up on Google and have a look on YouTube, there are some nice videos of this. So how is this going to help us with the kinds of industry problems that we're working on? Well, here's a box of Samuel's disco Nesquik cereal. And if any of you have young children, then you might be familiar with this because they might be making their name when you're trying to get them
off to school in the morning. So the idea with this is that you take you make these things in a similar way to a play doh fun factory. And again, if you have small children, you might be familiar with this thing as well. So what you have here is some play dough that you inject into this. This device here. And then you squish in tanks and you can make stars. We can make coarsest or you can make these wholely shapes.
And then this funny shape here. Now, the good thing about Plato is that it's quite it's almost solid, which means that when you squishy tanked the hole, when it comes out of this nozzle, it's pretty much the shape of the thing you get. Now, with Nesquik cereal, what you're squeezing out. It's the same kind of idea. You squeeze it out of a hole. But the Nesquik S.A.M. is a hot, bubbly liquid.
So when it squeeze it out, it doesn't behave lightly. So this is a simulation done by Michael Holmes there to give us an idea of exactly what does happen. So here Michael has been trying to make Mickey Mouse see that we start off with Mickey Mouse at the start here. But because it's a serial molton mixture, it uses eggs and you actually lose Mickey as you go along.
So this is no good. This is no good because it means that if you want to try and make particular letters, then you're not actually clear what hole you should. This scenario makes Jeff way because, for instance, if you pump it through an R shape because of this evolution, you don't end up with an R. In fact, one of the features you notice here is that the hole closes off and another feature is that they're not very regular.
All of the letters. So we have another experiment over here. This is where the overhead projector, this is very old school that hopefully we can rely on the old school kind of stuff. So what we've got here is a cookie cutter. And I understand to put some more, I like my sugary solution. This is Golden State. I'm just going to squeeze the Golden State into the cookie cutter. Like, say. And then I'm just going to take it out. So what you see here is that quite quickly, remember
what the thing looked like. It's like a star shape. It had quite pointy edges quite quickly. You lose the points and you get something like they said. This is exactly the kind of thing that's going on with the Nestlé problem. It's coming out and it's changing. So you don't get the star that you started with. Now, if we were to try and model this mathematically, this is what Michael did with this. He just took what are the Stokes flow experiments? That's exactly Stokes flow equations.
That's exactly what we used to model this here. Then if I give you an initial shape, you can tell me how it evolved. So I could say, what does it look like 10 seconds later? A minute later? That's called of well posed a problem. Now, if I take this shape here and if you hadn't watched me do that experiment and I said, what did that look like? When I started, you wouldn't be able to tell me it's lost all of the features. So if I
want to go backwards in time. This is what's called an imposed problem. And this means it's mathematically challenging and impossible to solve because we can't just one time backwards to work out what shape we need to start. But this is exactly the challenge that Nestlé won't say. They say we want to make Anar. What shape should
we start with? And you can't say, well, let's start with an arm and just one time backwards. So the kinds of mathematical tools that we're going to introduce in this talk will be to see how this how we can get around this problem and work out how we can make this kind of cereal's. So this is the idea that we have. We want to try and make better. Let us have. Now, this is quite a hard problem because, as I said, it's a bubble, see, and we mixture. So I'm going to take
a step back and consider a glass manufacturer instead. Now, glass manufacture is exactly the same process. It's an extrusion process. But glass is easier to work with than a molten say, a real mixture. So if you want to make all of these products here, anything from these optical fibres to test tubes and medicine vials here, they're all made in exactly the same process, these extrusion process.
And you take your molten glass and you pump it through the hole here and you pull vertically downwards. Now, notice, all of these are circular and that's good because that means that it doesn't matter how much you pull. It's never going to evolve from a star. Well, everything wants to end up as a circle. If you look at this here, it's going towards a circle. Now, this was working with a company called Shots. Shot makes glass for lots of different companies such as
Samsung in Hawaii Way. And they asked a very similar question. I said we don't want to make circular capillary tubes. We want to make square ones like this. I wanted to make square Maddieson vials and they wanted to make square a test tube so they don't want to weigh in on the table. So their question was exactly the same. What hole should we pump this through so that we end up with the square? Here's my schematic. But we're going to try and find out what that is.
Now, we use the Stokes equations again, which are exactly what would describe this, exactly what would describe Michael's problem. But the forward problem is well posed. I give you a shape. I go forwards in time. You can tell me what the shape looks like at any given light point. The inverse problem is ill posed. I give you a shape at the lifetime. You can't tell me how that evolves backwards in time. So this is a challenge that we want to try and overcome.
And the way in which we do this is by making various simplifying steps. And the first one is we look at this problem and this is a complicated, three dimensional problem of drawing. I choose. Now, the first thing we do is say, let's convert this into slices. So just like I showed you on here, this was on a plane. Two dimensional problem. And we looked at evolution of this blob with time. So let's say we'll consider a time evolution problem in two dimensions with the idea that each of these
snapshot in time corresponds to a different axial position. So we can imagine stacking these up like Coaster's to reconstruct a three dimensional shape. That's the first simplification that turns it from a 3-D problem into a Tudi. The next simplification is we say, let's just look at a little bit of this glass tubing here. This zoomed picture here and we model this using the angle data and the arc lens. So that's X as we go round and time to see if I tell you the angle
every single point on this glass and every single time. That's enough to reconstruct the problem. Now, what we've done here is we've taken a complicated, three dimensional problem that he's ill posed for negative time and turned it into a tiny little piece that we're going to reconstruct the evolution of this tiny little piece. We can run backwards in time that's well posed. So that is exactly what we do. We we
construct our picture by doing lots of tiny little pieces. And then we start with the square and run it backwards in time. And we get our two dimensional slices, which then allow us to reconstruct the three dimensional picture. And the interesting things about this are that you can do it for any shape you like. And I've done it for a square. You can do whatever you like. And what was a very complicated problem to solve mathematically
or to solve using experiments can be written in this little box here. This encapsulates all of the information about this. And they speak to as naughtier tells you the shape that you want to start with. Think of it whatever you like into this as an example. Here's a funny shape that we might want to start with the whole shape. What do you think that ends up evolving to? Does anybody know? Once the triangle and the other guest is Mickey Mouse again. Looks like Mickey Mouse.
It's the heart. It's Valentine's Day tomorrow and we have to have something. So for those of you who have not yet bought a Valentine's Day present for your loved one, you can come and talk to me and we can make you a Valentine's shaped, heart shaped tube. For those of you who didn't know, it's Valentine's Day tomorrow. I have just saved your relationship. But this generalises a lot more than just what we've shown here. It generalises into
multi structured optical fibres. And these are pretty big business nowadays as we're trying to transmit more and more data around. We need to transmit lots of signals in one fibre. So here are two examples. This one here looks a bit like a Mercedes Benz. It's quite hard to see, but it's got three struts. So you can send three times the amount of signals down this tube. This one here is surrounded by a honeycomb. And this prevents the signal from leaks.
And we can do exactly the same kind of idea with this. How do we make a Mercedes-Benz shape? Well, this one is a lot harder, actually. It's perhaps not what you would expect. It also doubles up as a really nice thing to keep your flowers in. So I'll keep those out so you can kind of see this stays in my office. Nice flowers. This has been 3D printed on our lands 3D printer. So this is a nice way to visualise these kinds of products when you've made them.
OK, so let's stick with the idea of glass manufacture for the moment. But what you can see and what I've shown you is that this all translates to making the Nesquik cereal. These kinds of ideas, these multi structured optical fibres are exactly what you need to make the Nesquik cereal, albeit in a more complicated fashion. Now, we're going to move on and we're going to ask a question. How do you make glass sheets for
mobile phones? This is a glass sheet that this guy has got in his hand here, and it's so thin. You're watching a minute and squish squishier and it flexes. And I find this incredible, this glass. You have this idea that glass is very brittle. This during F is glass, and we've got some here might be able to see at the back, but it really is flexible. Now, what we did once is my P actually finished a student handed this angrist for people to have
play around with. But the trouble is it has a limit of how much you can flex it. So you could hear this cracking sound as it was going round the audience and then what came back was dust or. Oh I guess sun. Sand is the more material of that glass. You can see just how flexible this stuff really is. And this is used for flexible electronics. Something released their first flexible device last year and Glena. And to release one. So we were interested in trying to
make this kind of glass sheet. And the way in which this is made is called the withdrawal process. The glass tubes were made using what's called the down draw process. This is called the withdrawal process. Now, what you do here is you start off with a rectangular block here and you feed down to this heat zone here and you pull it on the bottom with the moment. As you pull in, you stretch and gets thinner and thinner until
the product that comes out is the product that you just saw. Now, the downside of doing this is if you've ever had a go at making pizza, when you're trying to pull your structural pizza wanked, when you're done, you have a look at this and you've got Fat Ed June, the white man. That's your pizza crust. Pizza crust is nice on a pizza pizza crust. Not so nice on a piece of glass. So here you get exactly the same kind of thing when I pull my glass
downwards. You get these fat edges. This is a 3-D print of of this kind of thing. Perhaps you can't see. But I've started with a rectangle here. And when I pull, I get fat edges at either side. And if you're interested in looking at any of these things at the end, then you can come down equally if you want to play around with this at the end. So this is a similar kind of question we want to know. How do we make glass?
That is completely flat at the end. No fat edges. And these strategies are bad for two reasons. One is that if you have a mobile phone and got fat edges on it, it'll distort your picture. Two is if you have these fat edges, it won't flex. It'll fracture. This is called fracture. But the problem, just like the other one, the overclass problem, the forward problem is well posed. I can take a rectangle here and I can pull and stretch and I can tell you exactly what the shape will be at the end.
But if I give you a final shape, very hard to work out what shape I should have started with to get. So here's the question. Rectangle, Mike, flatted, rectangle. What shape should we start with? To make a flat rectangle. And we need to come up with another creative idea, because we can't just one time backwards. Same equations as we've used everywhere else to stokes equations. But we're now going to do something a bit different.
What we're going to do is say if we put this GLASSINE and then we pull it out faster, then it stretches. What happens if we put the GLASSINE ampoule at a lower speed than the glass is going in? So physically, that doesn't make sense. It's almost like you're trying to push the glass back up into the heat zone. But mathematically, we can choose our pulling speed to be whatever we like. So if we choose a pulling speed to be 20 times slower than the input speed
with a rectangle. And what we find is we get this funny shape coming out of the bottom. And what that means is if we were to start with this shape at the top, this type of shape and pull at 20 times the input speed, we should in principle recover the rectangle. So does this work? Let's have a try. It does. So here's another thing. So, again, you may not be able to say it back, but we start off with this type of rectangle now a pool. And what you get out to the bottom here is
perfectly flat glass. And so these ideas now are used on your Samsung or your Hawaiian way phone products. So this is how they make the flat glass that goes on the front and also goes on the back and covers your camera. So it's nice to see that the mathematical models that we develop are actually implemented in industry. OK, so that's been a lot about glass manufacture. I meant to switch topics now to filtration. Still keeping the idea of this. Can we go backwards
in time to what? Some guy. Is it made at home doing my vacuuming because I'm quite proud. But I'm frowning as I'm looking at the vacuum cleaner. And the reason I'm frowning is because it's not picking up the dust. So you might say always because the bags full Dyson have no banks and they have no loss of suction. That's my sales pitch for Dyson. That's what gives my half price. Dyson, I'm so the reason it's blocked is because the filter is blocked.
That's not picking up the dead. Now, what we're working on with Dyson is trying to make a film that lasts longer, ideally lasts the lifetime of vacuum cleaner, so you never need to clean it. In principle, you're supposed to change or clean your filter about every three months. How many of you actually change your filter? Everything. How many of you actually vacuum every three months? Well, that's the idea.
So if you are doing as you're told, then you should change or clean this filter every three months. If we could extend this so we never needed to change it, that's a good idea. Now to understand what's going on in a field. So this is an example of a Dyson film. They're quite fluffy. It's like hot and warm effectively. And I've illustrated this cotton movie. Film. In this picture here. So imagine all of the blue circles here
are fibres that are sticking out of the page. Then I'm going to put some dirt into my favourite coming in the top black. And see where it sticks. Here comes my dust. And what we say, I'm changing the colours here and the colour change corresponds to how much dust has been trapped by each of these fibres. And what you see is that the colours at the top change a lot
more than the colours at the bottom. And this is dynamic in filters. If you've ever actually looked at your filter in your column or your vacuum cleaner and you take it out, you'll find that it's really dirty on the one side, black. And you flip it over on the other side and it's a white. It's all clean. And actually, even if you look in the depth, it's own clean as well. So that's indicative of a filter that's not working to its full capacity, because when the top surface
is blocked, then it's all over. So we asked the question, could we make a porosity graded film? And by that I mean, could we open the top I to be so that it lets more of the way so that the filter down here is also trapping. And really what we're after is a filter that traps that uniformly for wanked so that when it blocks at the top, it blocks in the middle and it blocks at the bottom, it blocks everywhere at once and it's trapped all the dust. Now, how do we mount the leaks? Well,
it's a really complicated problem to model. All of these particles, all of these dust particles wiggling a man in your felt. But that's not what Dyson cares about anyway. They just care about what concentration of dust do we have? How dirty is your house and what concentration of dust is coming out of the other side, which we hope is zero. So what we want to do here is use a technique called homogenisation failing. And what we do in this is we take this complicated problem on the micro scale
and upscale it so that we look at this on the macro scale. This is a bit like looking at the picture go into the back of the room and screaming your eyes up a bit and getting a bit early picture out there. So that's the idea. And in doing that, what we can do is we can model the macro scale picture that we care about. That's for the fluid flow. In this case, the fluid is a gas and the dust transport within our filter. But then this is coupled to the micro scale, so we don't
lose that micro scale behaviour. So here we zoom in and we have the little fibres on the micro scale. And what we say is when this filter has blocked these fibres, I'm all touching one another. And ideally, we want to make sure that these fibres of all touching one another everywhere in AFL. Here and we have at Green Box again that we're becoming quite familiar with. Now the forward problem is well posed. I give you a filter. I say this is your dust concentration. You can run it.
You can see where the dust will stay. If I gave you a filter, you would not be able to tell me how this thing to go into this current state. But that's what we want. We want to say here we have a filter that is blocked everywhere at once. How should I run this backwards? What filter should I have started with to end up with a. So this is what we want. If I plot a graph and graph that I'm going to show, this is the filter depth here. And what we want to say
is that at the end point, every way it looks like this. So if I plot the function of the free volume, which is the whitespace hand compared with the total volume, then I want this to be uniform as I go from the top of the filter through to the bottom. And now I just want to run this backwards in time. And what helps us in this problem to go backwards in time is the fact that we have two distinct timescale.
We have the time scale over which a little piece of dust wiggles its way through the filter. And that's about a millisecond. No. And then we have the timescale over which the film, two blocks, that's about three months or so. So this disparity in timescales that allows us to work on these longer time scale of blocking, which is the one that we are actually interested in, and we can run time backwards on this timescale. So that's what we're going to
do one time backwards. And these are snapshots of the fraction at different points. And this is what we end up with. So sure enough, we want it to be quite separated at the top because that's where the concentration of dust is the highest. Less so in the middle. I'm quite close together at the bottom. And with this filter, if we went it forwards in time, then if a block everywhere at once, which is exactly what we want. So we
these kinds of filters, you can get about four or five times the lifetime after this. And this is pushing to the limit of the lifetime of a vacuum cleaner. So this is something that we're exploring with Dyson. OK. So everything on Shamone so far has been going backwards in time. In a quite nice way. We'll go back over to these RHP again. I'm going to do one final experiment here, which is kind of similar to the other experiments, and that is I'm going to blow up some more golden syrup.
On to this thing here. One of the things you notice is that it quite quickly forms a circle and which you sharing surface tension, tension. And then I'm going to drop another perspective plate on top. And I'm going to ask we. It's incredible how secular this is. Right now, I'm going to start to pull these plates apart again. I'm going to go back in time and you have to think, well, what is going to happen in this case? Is it going to just shrink? And we go backwards
in time again. Well, let's see what happens. I promise very carefully. It's quite tricky to do. I also don't want to get it on the overhead projector. So you get something completely different. You get this quite pretty pattern. And if I push against it, don't regularises again and you go back to the circle. I see. Repeatable. You can get some really quite nice patterns coming after this. She's got Becka's fingering. So this is an example. I can't see anything. Now I have to step in Diamond
Quest to the stage. So this is an example where where you can't go backwards in time. The inverse problem is very different to the forward problem. And this is a big issue in the oil industry. So let's imagine that I've got some oil down here. I'm actually we're running out because we've used up a lot of this. So this guy is looking by side because there's not so much oil. And then someone else comes along and says, I can help you. What I'm gonna
do is I'm going to pump some carbon dioxide down here. And the carbon dioxide is going to push the oil and you're going to get more oil out of here. That's the idea. So this guy starts doing that. But you get exactly the same picture that we got over here because this is a going into oil. Just like here, we had a pushing into golden syrup. So these air pushes CO2. And it just pushes straight past the oil and goes straight to pay. So you've got this one guy here,
one guy here. So this guy here is pumping CO2 down. This guy who is collecting all the CO2 that this guy is from Dan. And then after a day of work, I go home and think they've done a good job. But this is a really difficult issue in the oil industry because of this idea of this fiscus fingering, trying to push a less Becka's fluid, in this case, the CO2 into a more Becka's fluid. You always get these Becka's fingers.
So this is an instance where you can't go backwards in time and we need to think more creatively about how we solve these. OK. So that's all I really wanted to say about going forwards and backwards in time. What I wanted to end with is something that Alan touched on, and that's the more humanitarian aspect of some of the things that I work on. So some of you might be aware. About 25 years ago now or so, UNICEF launched an initiative
to provide clean, safe water for residents in Bangladesh. And the idea was that they would do a boreholes into the ground so that the Bangladeshis could tap into the ground water and drink this ground water rather than the contaminated surface water. In principle, this was a great idea in practise. It turned out to be a disaster because the groundwater is naturally contaminated with arsenic. This is being described as the largest global mass poisoning affected more than three million people.
So far. But there may be a solution and the solution comes in quite a simple form. It comes in this readily available laterite soil. Laterite is very ayane, which you can see that Mrs. Wedd very wet soil here. Now, iron absorbs arsenic, which means that you can take a filter. I can fill up a cylinder like this full of this laterite soil. I can take some contaminated water in this dustbin at the top here. And this is contaminated to 100 times over the safe limit of arsenic.
And it filters through just through percolation. And what you get out to the bottom here is pure water. So it's incredibly effective. But we ask some questions here, such as how do we know when this photo has expired? It's not like your bitter water filters that you replace every three months or so. But if you don't replace them, it's OK. The water just doesn't taste so good here. If you don't replace this, then you're drinking contaminated
water that could potentially kill you. And you don't know until a year or so later down the line. And how do we upscale for a school or a community? This is a home filter. And this is the filter that's actually in my stocks in India. But if we wanted to upscale, we need to have some mathematical models to understand how much material we might need to use. So I should cheque the volume on this. But you've you've had me talking quite a lot. So I'm going
to give you a rest by me talking. I mean, show you a video. And the video is of me talking. What we're trying to do really is develop mathematical models that can help purify water all over the world. Arsenic contamination and other content. It's been described as the largest global mass poisoning. It's affected more than three million people using a very simple, readily available laterite soil, which is just off hand because it's
readily available. That makes it very cheap. And then you take the contaminated water. Then it passes through this column of laterite soil, which is about a meat torso high. And what you get out of the bottom is almost pure water. If you run this filter for a long time, eventually that filter cannot soak up any more arsenic. It becomes saturated. And so the key question that we're trying to address is how long before
we need to replace this film? You cannot run an experiment for six or seven years to work out how long this this filter would last. So we need to have accelerated tests and winning them on a computer give you that. That's what we wanted to do with our mathematical modelling, is take something that is a very complicated problem and reduce it down to something very simple. But what you want to know ultimately is what is the concentration of arsenic in the water that comes
out? That's when you really want to know. Ideally, we wanted to be able to explain the whole problem in terms of a single parameter. So this number relates to how much mass of soil you're using and the required flow rate and the absorption capacity of that facility itself. The typical exchange rate at the moment is about five or six years. Our predictions say they can last maybe seven
or eight. So it gives you a little bit of extra time for these factors. The key point of our models is that now if you want to make another film for a particular size school or something, then anybody who comes in and says, I want to film with a particular flow, why do we just give them the number and say, this is this is how you would make that film? So what we're trying to make something that that alleviate some of the complications in the end, the problem
for me, innovation is very important. Time is very applied mathematician. And that means that I interact with a lot of different industries. And that's a good way to find out exactly what the key questions are. I think as as mathematician, we can get a bit caught up on some of the fine detail that we find interesting. That might not necessarily be the key
questions. So working with industries, working with experimentalists and the money, the scientific community is a good way to keep us grounded and make sure that we are innovating. So what's the state of play with these now? Well, we have some filters in family homes. We have filters providing schools. That's an order of magnitude of water per day. And another order of magnitude higher is in communities as well. And now we're looking into the removal of flow lines as well.
And reactive dye to most of you are wearing clothing that has been made using reactive dye. And once this has been made, it makes a lot of wastewater, which is then just pumped straight out into the water stream. So trying to remove this dye is pretty complicated and it's carcinogenic
as well. OK. So before my show, my somebody slide, I should acknowledge some people and all of this stuff that I do is in collaboration with lots of different people, as is all of the work that we do here in Oxford. So we that I would like to conclude. And this is kind of a broad picture of the sorts of things that we have looked at here and how you can go backwards in time with all of these. And I hope you don't want to go backwards in time for the last hour,
which you want to go somewhere else. But I hope you've actually learnt something from what I've talked about today. Thank you.