Curl - Grad, Div and Curl (3/3)

both grad and div involve finding fields using partial derivatives we'll look at yet another useful field once again it involves partial derivatives water can flow in many different and often complex ways let's look at a relatively simple case what's the velocity field that describes the flow on the surface of a river we can explore this field by watching floating objects what kind of motion can you see here there appear to be two types of motion downstream and rotation about a vertical axis in fact if we're traveling along with the disc all we see is the rotation this rotation is related to the velocity field but how to answer that question we need to define the velocity field on the surface of the river we'll define the surface velocity vector at any fixed point on the river fixed that is relative to the riverbank as the velocity of any floating object as it passes through that point to model the surface velocity we'll need to make some assumptions both about the shape of the river and about the way that the water flows we'll assume that there are no rocks affecting the flow let's also assume that the river is straight and of uniform width and the flow is steady and streamline it all flows downstream so this is our model River to model the velocity field we first need some axes and a function V of XY that describes the unique velocity vector at each point XY on the surface we'll assume that for this short stretch of the river V of X Y doesn't depend on Y in other words the velocity at any point only depends on its distance X from the left bank this implies a model surface velocity field of the form V of X y equals U of X J but what sort of function is U of X let's go back to the real River for a moment the velocity of the water varies as we go across the river we can assume that at each of the banks the surface speed is zero that is U of 0 and u of D are both 0 where D is the width of the river somewhere in between the surface speed will be a maximum let's assume that there's some symmetry and so that the maximum flow is mid stream what does this imply for U of X well the simplest possibility is parabolic U of x equals CX times D minus X where C is a constant and so our model for the surface velocity field V is this but how can a velocity field cause rotation let's go back to the model can you see what's happening let's subtract the downstream velocity of the center of the disc while the outer edge is being tugged downstream by the flow the nearer edge is being tugged upstream so the disc rotates this indicates the presence of another vector field one that describes the magnitude and direction of the rotation at any point in the river here the rotation is anti-clockwise so using the right hand rule the direction is vertically upwards that is in the direction of K the Cartesian unit vector in the Z direction this new vector field is called the curl of V or just curl V how does this vector curl v-very over the river as we've seen here the rotation is anti-clockwise but here it's clockwise on the left-hand side of the river the velocity difference across the disc causes rotation in the positive k direction while on the right-hand side of the river the velocity difference causes rotation in the negative k direction so what's the rotation midstream well midstream there's no rotation in other words curl V equals zero the vector field showing curl V for the whole river looks like this everywhere curl V is perpendicular to the velocity field V any vector field can have a curl field associated with it and later in this unit you'll see how curl V can be found from partial derivatives for our model River the rotation was localized there was rotation about each point and it's local rotation that curl describes not the bulk rotation of water you see in river bends or in swirling water so what do you think is happening here there's plenty of water movement bulk rotation but what about any local rotation what would curl be let's model the problem by looking at the two-dimensional surface velocity all the waters going round and round in a circle but whilst that near the center is flowing in a vortex further out it gets slower and slower so there are two different kinds of flow to model let's start with the water inside the vortex at the center the velocity is zero was like a hurricane or a tornado where the wind velocity is zero at its center its eye as we move further out the speed increases if you ignore the bulk rotation causing it to flow around in a circular path you can see that the disc is also rotating about its center so with this velocity field there is local rotation but what about the flow outside the vortex clearly there's still bulk rotation of the water but what is the local rotation this time the disc isn't rotating at all about its own center so there's no local rotation in other words this velocity field has a curl of zero but why should anybody want to know about the curl of a vector field in weather patterns you often find concentrations of the vorticity where the curl of the velocity is extremely large these are called cyclones a cyclone is a region where the pressure is lower than normal and so naturally the air tends to flow in towards the center of the cyclone but as it flows in it doesn't just come in radially it acquires a twist in cyclones you get a buildup of the vorticity and when the vorticity gets extremely intense of course you get hurricanes and tornadoes and these can do a great deal of damage so it's very important that we understand how the vorticity is built up and how tornadoes and hurricanes behave so that we can predict them better in future I'm particularly interested in the Sun there are very strong magnetic fields and the curl of the magnetic field is in fact the electric current in most of the Sun the current is zero there's no curl but in small regions where the curl is very large you find dynamic phenomena produced as you go away from the surface of the Sun you might expect going away from a hot body that the temperature would get lower in fact the opposite occurs it actually increases and in the atmosphere of the Sun the temperature is several million degrees five million degrees really hot there are giant tubes of magnetic flux in the atmosphere when these have no curl they just sit there in a quiet state for months or at a time but when the curl builds up when the electric current builds up eventually they can reach a stage where they go unstable and they erupt outwards from the Sun and produce enormous ejections of mass knowing about the curl enables us to understand how and why these ejections occur

Channel: OpenLearn from The Open University

Views: 464,642

Keywords: sin, cos, tangent, ou_MST209, open university, water, velocity field, surface velocity vector, river, downstream, upstream, surface speed, vector field, mathematics, statistics, gradient vector

Id: vvzTEbp9lrc

Length: 10min 28sec (628 seconds)

Published: Tue Jul 26 2011