welcome to lesson 4a the material derivative in this lesson we'll compare the lagrangian to the ellerian description we'll derive and explain the material acceleration and extend it to the material derivative in general and we'll do an example problem there are two primary ways to describe fluid flow first one is the lagrangian description in this description we follow and keep track of individual fluid particles as functions of time let's consider two fluid particles particle a the location of particle a is given by x a where xa is a vector similarly with particle b the velocity of a is v a also a vector and the velocity of b is v b as particle a moves around its position vector is a function of time similarly for particle b if we know these functions we can calculate the velocity of a particle by taking the derivative so v a and v b are also functions of time so the lagrangian description boils down to keeping track of these variables x a and x b as functions of time for example billiard balls banging together in high school physics we did these kinds of experiments where we documented the position vectors of a couple billiard balls we just track each ball and see where they go the physics is relatively simple we just apply conservation of momentum and energy etc directly to each billiard ball the problem in fluid mechanics is we don't really have individual particles rather we have a continuum flow if we were to try to do this in a fluid flow it would be impractical to track billions of fluid particles so the lagrangian description is not often used for fluid flow although i should point out that some sophisticated computer codes do use a lagrangian description but they require super computers for practical problems the second common description is called the eulerian description instead of following individual fluid particles we identify a region of the flow called a control volume and we watch the fluid pass through it for example take some flow along a wall and we're interested in this region or control volume in particular we're interested in the velocity vector for example at some point x y z and if it's unsteady also at some time t at other points in the flow the velocity vector may be different in both magnitude and direction so instead of following fluid particles we describe flow field variables like velocity and pressure as functions of space and time for example the velocity field would be velocity as a function of x y z and t similarly we can define a pressure field p is a function of x y z and t notice here that p is a scalar whereas v is a vector but they're both flow field variables in the eulerian description we don't care about individual fluid particles we just care about the fluid that happens to be in our control volume at a given time the elarian description is usually preferred in fluid mechanics but it can be more difficult to come up with these flow field variables and we have to be careful how we apply the laws of physics this leads us to a discussion of something called the material derivative and the material acceleration i'll derive the material acceleration and then extend it to a general material derivative this is some fluid particle at some time t it has a velocity and an acceleration which is not necessarily in the same direction the goal of this analysis is to transform a particle which is a lagrangian description into a field variable a of x y z and t which is an elarian description fundamental physics is based on this lagrangian description but we want to work in an elarian description fundamental definition of acceleration of a particle is a particle is dv particle dt again this is a lagrangian description but this velocity vector of this particular particle at this particular time is the same as the velocity field at this given location and time mathematically at x particle y particle and z particle and at the given time fluid particle moves with the fluid by definition the particle velocity is a function of time but it's also a function of the location of the particle as a function of time so we have to invoke the chain rule because we have four independent variables these four variables quick review of the chain rule let's do a simple case of f as a function of t and s then df dt is del f del t dt dt plus del f del s d s dt dt dt is just one so we have del f del t plus dou f dou s d s d t let's apply this to a fluid particle the acceleration of a particle is dv of the particle dt and v itself is a function of x y and z of the particle and time so we use our chain rule just like we did up here dv particle dt is del v del t dt dt plus del v del x particle dx particle dt plus similar terms for y and z again dt dt is just one dx particle dt is the x component of velocity which is u similarly dy particle dt is v and dz particle dt is w so the acceleration of a fluid particle is dv dt and this equation reduces to del v del t plus u del v dou x plus v del v del y plus w del v del z this is the expression for the acceleration of a fluid particle now we make an argument at the instant in time under consideration the acceleration field at this location in time given by x y z and t is equal to the acceleration of the particle at that same x y z and t where the right hand side is the acceleration of the fluid particle that happens to be occupying this location at this time since the fluid particle moves with the fluid flow by definition so using our equation above we can write an expression for the acceleration field a is equal to dv dt or del v del t plus u del v del x plus v del v del y plus w del v del z where the velocity field has three components u v and w in cartesian coordinates and u v and w are functions of space and time themselves so we have achieved our goal of writing the acceleration field as a function of x y z and t space and time this is now a field variable in the eulerian description now let's define a material derivative since this is a special derivative we give it a special symbol namely we use capital d instead of small d to indicate the material derivative of the velocity in this case this capital d just emphasizes that this is a total derivative d dt but it's made up of these four parts in the cartesian coordinates so we rewrite our acceleration vector as capital dv dt which is equal to del v del t the first term and you recall from math class that these three terms simplify to v dot del operating on v this del is a vector representing the gradient operator as we've seen in a previous lesson in cartesian coordinates the del operator or the gradient operator is del x del y del z these are the three components of the vector it's an operator because it has to operate on some variable here so continuing our equation we have the dot product between velocity and del and this combined operator is operating on v itself this is thus the material derivative of the velocity which is the acceleration physically this material derivative represents the time derivative formed by following a fluid particle as it moves in the flow although we derived this material derivative for velocity the material derivative itself can operate on any fluid property so in general the material derivative is d dt of something is del del t of that something plus the operator v dot del of that something i'll just put a squiggly cloud to indicate something the fluid property or variable that we're operating on this first term is the local part due to unsteadiness and this second part is the advective or convective part due to movement to a different part of the flow i prefer the word advective instead of convective since convective is often associated with heat transfer when you split it up this way you see that we have an unsteady part and an advective part notice that in a steady flow which means there's no change of anything with time or del del t is zero so this local part is zero but we can still have the advective part but d dt of whatever can be non-zero even in a steady flow this is an important concept and i'll give both a physical example and a mathematical example of this first a physical example consider steady converging duct flow we're talking about a section of duct where the walls of the duct converge let x be to the right and the flow is to the right at the entrance to this duct we have a velocity profile that looks like this notice the no slip condition at the walls but some fluid particle in this duct is definitely accelerating because by the time we get to the exit plane the speed is much higher let's call these speeds u1 and u2 near the center line so u2 is greater than u1 so obviously the fluid particle accelerates from one to two in other words a is not zero even though this flow is steady that's a physical example of this concept that we mentioned here now i'll do a mathematical example suppose we're given a steady two velocity field v of x y with these two components this is a steady velocity field and in two dimensions it has two velocity components u and v so comparing this term to this term u equal 3x and comparing these two terms v equal negative 3y we want to calculate the acceleration field which will also be a function of x and y the solution is to calculate a which is dv dt the uninformed student will look at this velocity field and say v is not a function of time so the acceleration is zero but this is not true as we showed in our physical example a can be non-zero even for a steady flow field the correct solution is to use the material derivative so a is dv dt and let's expand it out as del v del t plus u del v del x plus v del v del y plus w del v del z in this example v is not a function of time so the local or unsteady acceleration is zero since the flow is steady here w and any changes with z are also zero since this is a 2d flow but these two other terms need to be accounted for we plug in our u which is 3x and from here del v del x is 3i where i is the unit vector in the x direction so this term becomes 3x times 3i similarly this term becomes v which is negative three y times negative three j which is del v del y thus we have our result a equal nine x i plus nine y j this is our acceleration field notice that a is not zero even though this flow is steady this is often a hard concept for students to grasp it might help if you think of it this way this acceleration field is the acceleration following a fluid particle as that fluid particle moves around in the flow it has an acceleration since its velocity changes as it moves in the x and y directions so the acceleration is not zero as we also showed in our physical example of the converging duct from the particle's point of view it's accelerating finally this material derivative can be applied to any flow variable or property for example we can write out dp dt the material derivative of pressure this would indicate how the pressure changes as you follow a fluid particle through the flow field for a compressible flow you might want to consider d rho dt this material derivative of density indicates how the density changes as the fluid particle moves around in the flow you can do the same for temperature or any other variable this material derivative will become very important in our later analysis of fluid flows thank you for watching this video please subscribe to my youtube channel for more videos [Music]