The total derivative. Chapter Lagrangian and Eulerian approaches

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1 Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function of time an position But the physics of a system is relate to parcels, which move in space In particular, we are intereste in the change of the properties of a parcel with time In principle we might choose to escribe the flui by means of a set of functions of the various parcels In other wors, we might label each parcel by its coorinates at a given initial time, an then provie their new coorinates as time procees Such an approach is inee possible an is calle the Lagrangian approach, but it is more complicate an less use than the Eulerian approach, where the various quantities are given as a function of the geometrical points of space Hereafter, we will evelop only this latter approach, but we must be aware of the existence of the former because sometimes it is use both in theoretical stuies an in experimental practice When a buoy is release in the ocean, the information it furnishes is relate to its position Since it is swept away by the surrouning water, it will recor the properties of that mass of water, which, to a first approximation, can be thought of as a flui parcel 29

2 30 Franco Mattioli (University of Bologna) In this case the Lagrangian approach becomes the most logical approach The problem arises of how the behavior of the parcels can be relate to the fiels efine in geometrical points In other wors, we must evaluate the time evolution of the properties of the moving parcels as a function of the velocity fiel an of the istribution of these properties in space an time 52 The total erivative For the sake of simplicity, we will first consier the case of a scalar property Let u = u(t, x, y, z) be the velocity fiel an ψ = ψ(t, x, y, z) a scalar property, such as ensity, pressure or temperature The time variation of a quantity, following the motion of a parcel, will be enote by the symbol to istinguish it from the variation of the same quantity in a fixe point of space, which will be enote by The first erivative is calle total erivative, an the secon, partial erivative or local erivative The symbol D Dt is also very common for the total erivative, which is also calle substantial erivative, material erivative or iniviual erivative Let x p (t), y p (t), z p (t) be the coorinates of a parcel moving in space Then the variation of the property ψ of the parcel can be obtaine by applying the rules of the erivative of the function of a function ψ = ψ(t, x p, y p, z p ) = = ψ + ψ x p x + ψ y p y + ψ z p z = ψ + u ψ x + v ψ y + w ψ z

3 Principles of Flui Dynamics (wwwfluiynamicsit) 31 The last step is justifie by the fact that x p / is nothing but the velocity u of the consiere parcel Hence, we can write or, in a more general an symbolic way, ψ = ψ + (u )ψ, (51) = + u δx 2 ψ(t + δt,x + δx) 1 ψ(t, x) Fig 51: While the parcel moves from point 1 to point 2, performing the isplacement δx, the quantity ψ varies from ψ(t, x) to ψ(t + δt, x + δx) We can also see graphically the meaning of the total erivative (Fig 41) During the time interval δt, the parcel passes from point 1 to point 2 In the first point the quantity ψ hols ψ(t, x, y, z), an in the secon point ψ(t + δt, x + δx, y + δy, z + δz) But δx can be evaluate to a first approximation on the basis of the velocity in point 1 an of the time interval δt δx = u δt, δy = v δt, δz = w δt By applying the Taylor series theorem about point 1, we have ψ(t + δt, x + δx, y + δy, z + δz) = ψ(t + δt, x + uδt, y + vδt, z + wδt) = = ψ(t, x, y, z) + ψ ψ ψ ψ δt + u δt + v δt + x y z w δt Subtracting ψ(t, x, y, x) from both members of the equation, iviing by δt, an taking the limit for δt 0, we obtain the above equation (41)

4 32 Franco Mattioli (University of Bologna) 53 The structure of the total erivative Let us further analyze the structure of the total erivative The variation in time of the property ψ of the parcel epens on two factors The former ψ/ represents the variations ue to the fact that at a given point fixe in space the property can increase or ecrease with time The latter (u )ψ, calle the avective term of the total erivative, epens on the fact that the parcel, uring its motion, can pass from a region with a given value of ψ to another with a ifferent, either lower or higher, value of ψ The avective term (u )ψ might also be written as u ψ, or ψ u Both expressions are mathematically correct, but partially hie the unerlying physical meaning, an will be avoie when possible In the case of a stationary flux, ie, a flux inepenent of time, the term ψ/ vanishes everywhere, so that the variation in time of the property is equal to the avective term On the other han, if the flux is uniform in space, then (u )ψ is always zero, an the property can vary only if the fiel simultaneously varies in all points of space In this case, the total an partial erivatives coincie Obviously, what we have sai about a scalar quantity ψ can be extene to any vector quantity v The same expression foun for a scalar quantity must be repeate for each component of the vector In vector notation, the total erivative of a vector takes the form v = v + (u )v (52) Clearly, if a certain quantity associate to a parcel is conserve in time, its total erivative is zero For example, in an incompressible flui the ensity ρ of each parcel is constant in time, so that we have ρ = ρ + (u )ρ = 0 (53) We shoul not confuse a homogeneous flui with an incompressible flui In the former case, the ensity is always the same for all the parcels In the latter, the ensity can vary passing from one parcel to another, but every parcel maintains the same ensity uring its motion A mix of two or more homogeneous fluis is an example of an incompressible flui Strictly speaking, a flui is incompressible when its ensity oes not epen on the pressure Here an in the following we will aopt instea this more restrictive efinition

5 Principles of Flui Dynamics (wwwfluiynamicsit) 33 It shoul be note that, in spite of the complexity of its efinition in a fixe reference system, the total erivative is nevertheless a simple time erivative when referre to the moving parcel Thus, the usual rules for the orinary erivatives hol for the total erivatives as well Problem 51 Show, by applying the Eulerian expression for the total erivative, that if ψ an φ are two scalar fiels variable in time, then 1 (ψφ) = ψ φ + φψ, 2 ψ ψ = 1 ψ 2 2, 3 1 ψ ψ = logψ Problem 52 Show, by applying the Eulerian expression for the total erivative, that x p = u Hint Use the property x p = x 54 The Reynols transport theorem The total erivative allows us to follow the properties of an infinitesimal parcel uring its motion Now, we will exten such an operation to a finite volume of flui Let us consier a quantity given by a volume integral Ψ = ψ, where ψ is a certain scalar property of the flui The time erivative of this integral in a given volume constant in time is simply given by Ψ = ψ ψ = (54) However, if the volume changes with time, then the results is more complicate We can write δψ = Ψ(t + δt) Ψ(t) = ψ(x, t + δt) ψ(x, t) (t+δt) (t)

6 34 Franco Mattioli (University of Bologna) The Taylor expansion theorem in time limite to the linear terms allows us to write ( δψ = ψ + δt ψ ) ψ ψ = δt (t) + ψ (55) δ (t+δt) S(t + δt) S(t) S α n u uδt δ Fig 52: The volume δ between the instants t an t+δt can be compute as the sum of the volumes of the cyliners that we can raw between the surfaces S(t) an S(t + δt) in the irection of the velocity Since the volume of the oblique cyliners is given by the prouct of the area of their basis S by their height uδt cos α, it can be written as S n u δt = u S δt, where n is the normal to the surface S at the center of the basis of the cyliner If the volume changes accoring to the velocity of its parcels, that is, if the volume is always forme by the same matter, the last volume integral can be transforme into a surface integral using the velocity u at which the surface S bouning the volume moves With the help of (Fig 42) we euce that ψ = δt ψ u S = δt (ψu), δ S the last step resulting from an application of the Gauss theorem Then, by using (44), (45) becomes ψ ψ = + (ψu) (56) = ψ + ψu S (57) The two terms at the secon an thir member of the equation correspon, respectively, to the local an avective terms of the total erivative for infinitesimal parcels, which now is represente by the first term of the equation This is known as the Reynols transport theorem S

7 Principles of Flui Dynamics (wwwfluiynamicsit) 35 A particular application of the theorem is obtaine by assuming ψ = 1 In this case we have = = u, (58) where is the volume of Therefore, the volume variation of a mass of flui epens on the ivergence of its velocity fiel By consiering an infinitesimal volume δ over which u can be assume as constant, we can recover the efinition of ivergence alreay seen in (D2) u = 1 δ δ, (59)

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