Chapter 4

The total derivative

4.1 Lagrangian and Eulerian approaches

The representation of a fluid through scalar or vector fields means that each physical quantity under consideration is described as a function of time and position. But the physics of a system is related to parcels, which move in space. In particular, we are interested in the change of the properties of a parcel with time. In principle we might choose to describe the fluid by means of a set of func- tions of the various parcels. In other words, we might label each parcel by its coordinates at a given initial time, and then provide their new coordinates as time proceeds. Such an approach is indeed possible and is called the Lagrangian approach, but it is more complicated and less used than the Eulerian approach, where the various quantities are given as a function of the geometrical points of space. Hereafter, we will develop only this latter approach, but we must be aware of the existence of the former because sometimes it is used both in theoretical studies and in experimental practice. When a buoy is released in the ocean, the information it furnishes is related to its position. Since it is swept away by the surrounding water, it will record the properties of that mass of water, which, to a first approximation, can be thought of as a fluid parcel.

19 20 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 related to the fields defined in geometrical points. In other words, we must evaluate the time evolution of the properties of the moving parcels as a function of the velocity field and of the distribution of these properties in space and time.

4.2 The total derivative

For the sake of simplicity, we will first consider the case of a scalar property. Let u = u(t,x,y,z) be the velocity field and ψ = ψ(t,x,y,z) a scalar property, such as density, pressure or temperature. The time variation of a quantity, following the motion of a parcel, will be denoted by the symbol d dt to distinguish it from the variation of the same quantity in a fixed point of space, which will be denoted by ∂ . ∂t The first derivative is called total derivative, and the second, partial derivative or local derivative. The symbol D Dt is also very common for the total derivative, which is also called substantial derivative, material derivative or individual derivative.

Let xp(t), yp(t), zp(t) be the coordinates of a parcel moving in space. Then the variation of the property ψ of the parcel can be obtained by applying the rules of the derivative of the function of a function dψ dψ(t, xp, yp, zp) = = dt dt

∂ψ ∂ψ dxp ∂ψ dyp ∂ψ dzp ∂ψ ∂ψ ∂ψ ∂ψ = + + + = + u + v + w . ∂t ∂x dt ∂y dt ∂z dt ∂t ∂x ∂y ∂z Principles of Fluid Dynamics (www.fluiddynamics.it) 21

The last step is justified by the fact that dxp/dt is nothing but the velocity u of the considered parcel. Hence, we can write

dψ ∂ψ = +(u ·∇)ψ, (4.1) dt ∂t or, in a more general and symbolic way,

d ∂ = + u ·∇. dt ∂t

...... 2 ...... • ...... ψ(t + δt, x + δx) ...... δx ...... 1 .•...... ψ(t, x) ......

Fig. 4.1: While the parcel moves from point 1 to point 2, performing the displacement 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 derivative (Fig. 4.1). During the time interval δt, the parcel passes from point 1 to point 2. In the first point the quantity ψ holds ψ(t,x,y,z), and in the second point ψ(t + δt,x + δx,y + δy,z + δz). But δx can be evaluated to a first approximation on the basis of the velocity in point 1 and of the time interval δt

δx = uδt, δy = vδt, δz = wδt.

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 ∂ψ ∂ψ ∂ψ = ψ(t,x,y,z) + δt + uδt + vδt + wδt. ∂t ∂x ∂y ∂z

Subtracting ψ(t,x,y,x) from both members of the equation, dividing by δt, and taking the limit for δt → 0, we obtain the above equation (4.1). 22 Franco Mattioli (University of Bologna)

4.3 The structure of the total derivative

Let us further analyze the structure of the total derivative. The variation in time of the property ψ of the parcel depends on two factors. The former ∂ψ/∂t represents the variations due to the fact that at a given point fixed in space the property can increase or decrease with time. The latter (u ·∇)ψ, called the advective term of the total derivative, depends on the fact that the parcel, during its motion, can pass from a region with a given value of ψ to another with a different, either lower or higher, value of ψ. The advective term (u·∇)ψ might also be written as u·∇ψ, or ∇ψ ·u. Both expres- sions are mathematically correct, but partially hide the underlying physical meaning, and will be avoided when possible. In the case of a stationary flux, i.e., a flux independent of time, the term ∂ψ/∂t vanishes everywhere, so that the variation in time of the property is equal to the advective term. On the other hand, if the flux is uniform in space, then (u·∇)ψ is always zero, and the property can vary only if the field simultaneously varies in all points of space. In this case, the total and partial derivatives coincide. Obviously, what we have said about a scalar quantity ψ can be extended to any vector quantity v. The same expression found for a scalar quantity must be repeated for each component of the vector. In vector notation, the total derivative of a vector takes the form dv ∂v = +(u ·∇)v. (4.2) dt ∂t

Clearly, if a certain quantity associated to a parcel is conserved in time, its total derivative is zero. For example, in an incompressible fluid the density ρ of each parcel is constant in time, so that we have

dρ ∂ρ = +(u ·∇)ρ =0. (4.3) dt ∂t

We should not confuse a homogeneous fluid with an incompressible fluid. In the former case, the density is always the same for all the parcels. In the latter, the density can vary passing from one parcel to another, but every parcel maintains the same den- sity during its motion. A mix of two or more homogeneous fluids is an example of an incompressible fluid. Strictly speaking, a fluid is incompressible when its density does not depend on the pressure. Here and in the following we will adopt instead this more restrictive definition. Principles of Fluid Dynamics (www.fluiddynamics.it) 23

It should be noted that, in spite of the complexity of its definition in a fixed reference system, the total derivative is nevertheless a simple time derivative when referred to the moving parcel. Thus, the usual rules for the ordinary derivatives hold for the total derivatives as well.

Problem 4.1 Show, by applying the Eulerian expression for the total derivative, that if ψ and φ are two scalar fields variable in time, then d(ψφ) dφ dψ 1. = ψ + φ , dt dt dt dψ 1 dψ2 2. ψ = , dt 2 dt 1 dψ d log ψ 3. = . ψ dt dt Problem 4.2 Show, by applying the Eulerian expression for the total derivative, that dx p = u. dt

Hint. Use the property xp = x.

4.4 The Reynolds transport theorem

The total derivative allows us to followthe properties of an infinitesimal parcel during its motion. Now, we will extend such an operation to a finite volume of fluid. Let us consider a quantity given by a volume integral

Ψ = ψ dV, V Z where ψ is a certain scalar property of the fluid. The time derivative of this integral in a given volume constant in time is simply given by dΨ d ∂ψ = ψ dV = dV. (4.4) V V dt dt Z Z ∂t However, if the volume changes with time, then the results is more compli- cated. We can write

δΨ=Ψ(t + δt) − Ψ(t) = ψ(x, t + δt) dV − ψ(x, t) dV. V V Z (t+δt) Z (t) 24 Franco Mattioli (University of Bologna)

The Taylor expansion theorem in time limited to the linear terms allows us to write ∂ψ ∂ψ δΨ = ψ + δt dV − ψ dV = δt dV + ψ dV. (4.5) V V V V Z (t+δt) ∂t ! Z (t) Z ∂t Zδ

...... S(t + δt) ...... n ...... S t ...... ( ) ...... α ...... u ...... dS ...... uδt ...... V ...... δ ...

Fig. 4.2: The volume δV between the instants t and t+δt can be computed as the sum of the volumes of the cylinders that we can draw between the surfaces S(t) and S(t + δt) in the direction of the velocity. Since the volume of the oblique cylinders is given by the product of the area of their basis dS by their height uδt cos α, it can be written as dS n · u δt = u · dS δt, where n is the normal to the surface S at the center of the basis of the cylinder.

If the volume changes according to the velocity of its parcels, that is, if the volume is always formed by the same matter, the last volume integral can be transformed into a surface integral using the velocity u at which the surface S bounding the volume moves. With the help of (Fig. 4.2) we deduce that

ψ dV = δtψ u · dS = δt ∇· (ψu) dV, V S V Zδ Z Z the last step resulting from an application of the Gauss’ theorem. Then, by using (4.4), (4.5) becomes d ∂ψ ψ dV = dV + ∇· (ψu) dV (4.6) V V V dt Z Z ∂t Z d = ψ dV + ψu · dS. (4.7) V S dt Z Z The two terms at the second and third member of the equation correspond, respectively, to the local and avective terms of the total derivative for infinitesimal parcels, which now is represented by the first term of the equation. This is known as the Reynolds transport theorem. Principles of Fluid Dynamics (www.fluiddynamics.it) 25

A particular application of the theorem is obtained by assuming ψ = 1. In this case we have d dV dV = = ∇· u dV, (4.8) dt V dt V Z Z where V is the volume of V. Therefore, the volume variation of a mass of fluid depends on the divergence of its velocity field. By considering an infinitesimal volume δV over which ∇· u can be assumed as constant, we can recover the definition of divergence already seen in (D.2) 1 d ∇· u = δV, (4.9) δV dt