1. The Material Derivative in Cartesian Coordinates

Recently, we've had some controversy about some terms in a momentum equation in one of the papers that one of my colleagues is studying. Indeed, for compressible and non constant viscosity fluids, it is not straightforward to come up with the equations of motion as this situation is rarely the case in our daily engineering investigations.

As the problems we are dealing become more complicated (in fact, we add the complication as we solve the problem one step at a time. For example, we start with an inviscid incompressible model, then we add viscosity, then we add compressibility to the inviscid model, then add viscosity to the compressible model etc...).

So anyway, I decided to once and for all settle down the issue and put together a convenient and easy to understand post.

The Material Derivative

First, let us start by going over the material derivative one more time. As you most probably know, there are two reference frames that can be used in studying fluid motion; namely, the Lagrangian and Eulerian frames.

In the Lagrangian description, each particle in the fluid is followed as if it were a "rigid body". This means that each particle has a unique identifier or tag. This description is akin to rigid body dynamics and is a great way of describing a relatively small number of particles. However, when dealing with a fluid, there is a very large number of particles thus rendering a Lagrangian description of a fluid flow problem very tedious and intractable.

In the Eulerian description, flow properties (velocity, pressure, temperature, etc...) are defined as a function of space and time. This means that instead of tagging each particle of the flow, the viewer fixes a volume in space and identifies the flow properties in that region of space. Therefore, it does not matter which particle passes through the volume since that particle will assume the flow property of that point in space.

The natural way of defining the velocity or acceleration is based on a Lagrangian description because it is the easiest (and that's how we've done it in dynamics in high school)! When it comes to fluid flow, the acceleration of a fluid particle as seen in the Eulerian reference is different from the Lagrangian description. This is due to the fact that as the particle moves about, its velocity and position change as well. In Mathematical terms, the particle velocity is

(Eq. 1)

of course, the velocity is a vector

(Eq. 2)

Now the accelerationo is obtained by differentiating the velocity with respect to time. But since the velocity is a function of the position, and the position is a function of time, then we have to use implicit or chain rule differentiation as follows

(Eq. 3)

which yields the following expression for the acceleration of ANY fluid particle as seen by an observer in an Eulerian reference.

(Eq. 4)

If we were in a Lagrangian frame, then the velocity is only a function of time because in that reference, the observer is riding on the particle as it flows through space. [Next Article: The Material Derivative in Cylindrical Coordinates]

Cite as: Saad, T. "1. The Material Derivative in Cartesian Coordinates." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/08/derivation-

of-navier-stokes-equations.html

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Labels: Fluid Mechanics, Mathematics

2. The Material Derivative in Cylindrical Coordinates

This one a little bit more involved than the Cartesian derivation. The reason for this is that the unit vectors in cylindrical coordinates change direction when the particle is moving. In the Lagrangian reference, the velocity is only a function of time. When we switch to the Eulerian reference, the velocity becomes a function of position, which, implicitly, is a function of time as well as viewed from the Eulerian reference. Then

(Eq. 1)

and the material derivative is written as (with the capital D symbol to distinguish it from the total and partial derivatives)

(Eq. 2)

Special attention must be made in evaluating the time derivative in Eq. 2. In dynamics, when differentiating the velocity vector in cylindrical coordinates, the unit vectors must also be differentiated with respect to time. In this case, the partial derivative is computed at a fixed position and therefore, the unit vectors are "fixed" in time and their time derivatives are identically zero. Then, we have

(Eq. 3)

we can now evaluate the remaining terms in Eq. 2 as follows

(Eq. 4)

And

(Eq. 5)

finally

(Eq. 6)

When these are put together, the material derivative in cylindrical coordinates becomes

(Eq. 7)

This was a rather tedious way of deriving the material derivative as one could have used vector technology to obtain an invariant form that works for all coordinates. Nonetheless, it is interesting to see the intricacies of the derivation using chain rule differentiation. Note that if were computing the material derivative for a scalar, the extra terms in Eq. 7 (in the radial and tangential components) would disappear. These are purely reminicsent of the vectorial nature of the velocity field (or any other vector field for that matter). It is very interesting to note the intimate link between the physical nature of the velocity and its mathematical description through vectors. One would pose the following argument: why don't we treat the material derivative of the velocity as that of three scalars, namely, u_r, u_theta, and u_z? Doing this will obviously remove the hassles of dealing with derivatives of unit vectors, but will eventually lead to inconsistent results. So what's the issue here? The problem with that treatment is that in essense, the velocity is one quantity that we describe using vectors: a magnitude and a direction. If we are to use three scalars to describe the velocity we lose an essential ingredient which is the direction. In the end, the material derivative of the velocity can be decomposed into the material derivatives of three scalars (u_r, u_theta, and u_z) plus some correction. This correction stems from the directional nature of the velocity field. In other words, this correction can be thoguht of as being the material derivative of the direction of the velocity field. [Next Article: The Material Derivative in Spherical Coordinates]

Cite as: Saad, T. "2. The Material Derivative in Cylindrical Coordinates." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/08/derivation-of-navier-stokes-

equations_17.html

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3. The Material Derivative in Spherical Coordinates

Spherical coordinates are of course the most intimidating for the untrained eye. For engineers and fluid dynamicists, the farthest we go is usually cylindrical coordinates with rare pop-ups of the spherical problem. Here, I want to derive the material derivative of the velocity field in spherical coordinates. First, let us do that for a scalar. Assume that at point r and time t a fluid particle has a property Q. As this particle moves about, this property will change with time (and space). Again, in the Lagrangian description, Q is only a function of time, i.e.

(Eq. 1)

However, from the Eulerian point of view, any property of the fluid is a function of time and space, which is also a function of time implicitly. Then

(Eq. 2)

Then, the time rate of change of any scalar fluid property is given by the following

(Eq. 3)

where we have used the chain rule to account for the spatial dependence on time. Remembering some of the formulas from dynamics, we have

(Eq. 4)

upon substitution of Eq. 4 into Eq. 3, we finally obtain the material derivative for a scalar

(Eq. 5)

To obtain the material derivative for a vector field, we follow a similar procedure keeping in mind the directional nature of a vector. We illustrate this using the velocity field - keep in mind that this works for any kind of vector field. Again, in a

Lagrangian reference, the velocity is only a function of time. In the Eulerian view, the velocity has the following form

(Eq. 6)

Using the chain rule, the material derivative of the velocity field is written as

(Eq. 7)

Again, noting that the partial derivative with respect to time in Eq. 7 (first term) is evaluated at a fixed position in space, the unit vectors associated with the fluid particle at that point are fixed as viewed from an Eulerian reference, therefore,

(Eq. 8)

To evaluate the remaining terms in Eq. 7, we have to first remember some equations from dynamics or vector calculus about differential changes in unit vectors in spherical coordinates. These are given by

(Eq. 9)

Now we can evaluate the spatial terms in Eq. 7. The radial derivative is

(Eq. 10)

while the tangential derivative takes the following form

(Eq. 11)

Finally, the azimuthal derivative is as follows

(Eq. 12)

Voila!!! Once Eqs. 8 through 12 are put together, one obtains the full expression for the material derivative of a vector field in spherical coordinates. In the next post, I will present an invariant vector form for the material derivative so that we don't have to go through all the hassle of using chain rule differentiation to evaluate the material derivative. But it was worth it to see how it works using good old calculus.

Cite as: Saad, T. "3. The Material Derivative in Spherical Coordinates." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/08/derivation-

of-navier-stokes-equations_18.html

Posted by yNot at 9:19 PM 1 comments Links to this post

Labels: Fluid Mechanics, Mathematics

5. The Reynolds Transport Theorem

[Previous Article: The Material Derivative in Vector Form] For completion purposes, I feel obliged to discuss the Reynolds transport theorem. Although I would like to derive the fluid flow equations from scratch, the Reynolds transport theorem provides an avenue for a simple way to derive them. Henceforth, I decided to use the many ways of deriving the conservation equations, whether in integral or differential form.

I will base the current derivation on the text by James A. Fay Introduction to Fluid Mechanics. I believe it is an excellent text on fluid mechanics that focuses on the essential physics of fluid flows. We first have to distinguish between a material volume and a control volume. A material volume is a volume of fluid that contains the same fluid as it moves and deforms in time.

(Fig. 1)

A control volume is a fixed volume in space where the fluid passes through.

(Fig. 2)

This is tightly linked to the previous discussions on the material derivative and its connection with the Lagrangian and Eulerian views. A material volume is part of a Lagrangian description whereas a control volume is part of the Eulerian description. Now let us consider and "extensive" property B whose "intensive" property is b. For example, mass is an extensive property, whereas the density is the corresponding intensive property. An extensive property describes a specific part of the fluid (e.g. the mass is different for different volumes of the same fluid) while the intensive property is intrinsic (e.g. the density is the same for different volumes of the same fluid). In simple terms, an intensive property is the extensive property per unit mass. The Reynolds transport theorem can be thought of as the integral form of the material derivative. It mainly relates the rate of change of an extensive property of a given material volume to the rate of change of the corresponding intensive property.

The total amount of property B in a given material volume is therefore

(Eq. 1)

As the material volume moves around, the quantity B inside M changes due to external forces or internal reactions for example. Therefore, it is convenient to compute the time rate of change of B

(Eq. 2)

Eq. 2 means that the rate of change of the quantity B in the material volume is equal to the rate of change of B within the fixed control volume plus the net flowrate of the quanity B through the control surface. The RHS of Eq. 2 can be expressed as follows

(Eq. 3)

and

(Eq. 4)

Eq. 4 measures the flux of the quantity B through the control surface. Then, combining the above equations, we get the Reynolds transport theorem

(Eq. 5)

Voila!

There is an alternative way of deriving the Reynolds transport theorem, however, it makes use of the continuity equation which we have not derived yet. So this will be postponed to a later post. [Next Article: How Euler Derived the Continuity Equation]

Cite as: Saad, T. "5. The Reynolds Transport Theorem." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/09/derivation-of-navier-

stokes-equations.html

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Labels: Fluid Mechanics, Mathematics

4. The Material Derivative in Vector Form

[Previous Article: 3. The Material Derivative in Spherical Coordinates] After going over the derivation of the material derivative in Cartesian, cylindrical, and spherical coordinates and seeing all the trouble that we had to go through, it is time to present the material derivative in a vector invariant form.

An invariant vector form is independent of the coordinate system used. The form will be the same for all coordinate. Of course, vector operations (i.e. gradient and curl) are not the same for different coordinate systems [see this post] once they are expanded; but the gradient is always a gradient and the curl is always a curl - only the expansion is different. I will follow Karamcheti's explanation for obtaining the vector form of the material derivative. We start by considering a fluid particle at R measured from the origin of the coordinate system and time t. Consider also a generic scalar fluid property Q such as the temperature, pressure, or density. The scalar restriction will be removed once we obtain the general form for the material derivative. At point R and time t, the property is defined as Q(R, t). At time (t + Δt), the fluid particle moves a distance Ds and the fluid property changes accordingly to Q(R + V Δt, t + Δt)

(Fig. 1)

The total change in Q from t to (t + Δt) is

(Eq. 1)

then, the time derivative is defined as

(Eq. 2)

using Taylor's series for Q, we have

(Eq. 3)

If we substitute Eq. 3 into Eq. 2, we get the following

(Eq. 4)

Note that all high order terms disappear as the limit in the derivative is applied. The second term in Eq. 4 can be cast in vector form because it represents the derivative of Q in the direction of the streamline, tangent to the velocity vector. This means that it can be written as the dot product of the gradient of Q and the unit vector along the streamline, i.e. parallel to the velocity. Mathematically, this can be written as

(Eq. 5)

at the outset, we recover

(Eq. 6)

Voila! This is the expression we are looking for; Eq. 6 represents the time derivative of a transported fluid property as seen from an Eulerian point of view. This also works when Q is a vector field, call it A

(Eq. 7)

However, the form given by Eq. 7 only works for Cartesian coordinates because it not invariant under coordinate transformation. This means that it does not hold true when using curvilinear coordinates such as Cylindrical or Spherical. Fortunately, we can write it using invariant form as follows

(Eq. 8)

Voila! Specifically, when the vector field is the velocity field, then Eq. 8 simplifies quite nicely as

(Eq. 9)

[Next Article: 5. The Reynolds Transport Theorem ]

Cite as: Saad, T. "4. The Material Derivative in Vector Form." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/08/derivation-of-

navier-stokes-equations_20.html

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Labels: Fluid Mechanics, Mathematics

6. How Euler Derived the Continuity Equation

[Previous Article: The Reynolds Transport Theorem]

I thought that it would be interesting to present Euler’s derivation of the continuity equation for incompressible flows. Although d’Alembert, in 1752, had already presented an equivalent equation in his Essai d’une nouvelle théorie de la résistance des fluides (which he had already submitted to the Academy of Sciences of Berlin in 1749), the one proposed by Euler in 1756 (written 1752) is considered to be the most rigorous.

Euler’s contribution to Fluid Mechanics goes beyond what a scientist may imagine, and was mostly due to four manuscripts published between 1752 and 1755. These are

1. Principia Motus Fluidorum (1756) [pdf] 2. Principes généraux de l’état d’équilibre des fluides (1755) [pdf] 3. Principes généraux du mouvement des fluides (1755) [pdf] 4. Continuation des recherches sur la théorie de mouvement des fluides (1755) [pdf]

The final thing I would like to point out is that Euler’s genius lies partly in his ability to synthesize and introduce world class notation. In this way, he was able to supersede all his predecessors.

Without further ado, let’s begin.

Euler starts by saying:

“… I shall posit that the fluid cannot be compressed into a smaller space, and its continuity cannot be interrupted. I stipulate without qualification that, in the course of the motion within the fluid, no empty space is left by the fluid, but it always maintains continuity in this motion…” [Par

Update

subsequently, one can evaluate the position vectors of the remaining vertices Now we have all the required equations to evaluate the area of A’B’C’. For simplicity, we’ll use the cross product. We have Upon a pretty cool evaluation, (which I won’t illustrate here but I still think that you should carry it over because things cancel out very nicely), we obtain Finally, here’s

axes();6</a>, Principia Motus Fluidorum, Translated by Enlin Pan] Hethen argues that if one considers any part of a f luid of this type(i.e. incompressible), then each individual particles fill the sameamount of space as they move around. He then infers that if thishappens for particles, it should happen to the fluid as a w hole (w hicw as his assumption of incompressibility). One is now able to considan arbitrary f luid element and then track its instantaneous changes<em><span style="color: rgb(204, 153, 51);">”to determinethe new portion of space in w hich it w ill be contained after a verysmall time period”</span></em>. The amount of space that is

the punch line. By equating S’ and S, we obtain or Euler then argues that the second term in this last equation is vanishingly small compared to the first term and in the limit of an infinitesimal time increment, we obtain what we call today the continuity equation

Q.E.D I would like to thank the Euler archive for doing a great service for humanity by archiving Euler’s works as well as providing translations for some of his texts. The quoted paragraphs in orange are taken from Enlin Pan's english translation of Euler’s Principia Motus Fluidorum. [Previous Article: The Continuity Equation in Cartesian Coordinates]

Cite as: Saad, T. "6. How Euler Derived the Continuity Equation." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2009/02/derivation-of-

navier-stokes-equations.html

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Labels: Fluid Mechanics, Mathematics

7. Derivation of the Continuity Equation in Cartesian Coordinates

[Previous Article: How Euler Derived the Continuity Equation]

The continuity equation is an expression of a fundamental conservation principle, namely, that of mass conservation. It is a statement that fluid mass is conserved: all fluid particles that flow into any fluid region must flow out. To obtain this equation, we consider a cubical control volume inside a fluid. Mass conservation requires that the the net flow through the control volume is zero. In other words, all fluid that is accumulated inside the control volume (due to compressibility for example) + all fluid that is flowing into the control volume must be equal to the amount of fluid flowing out of the control volume.

Accumulation + Flow In = Flow Out

The mass of the control volume at some time t is

The time rate of change of mass in the control volume is

Now we can compute the net flow through the control volume faces. Starting with the x direction, the net flow is

Similarly, the net flow through the y faces is

while that through the z faces is

Upon adding up the resulting net flow and diving by the volume of the fluid element (i.e. dxdydz), we get the continuity equation in Cartesian coordinates

Voila!

Cite as: Saad, T. "7. Derivation of the Continuity Equation in Cartesian Coordinates." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2009/02/derivation-of-continuity-equation-in.html

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Labels: Fluid Mechanics, Mathematics

8. Derivation of the Continuity Equation in Cylindrical Coordinates

This is one of my favorite derivations. Although it would sound a bit intimidating at first, as none of the standard textbooks carry out the derivation in curvilinear coordinates; it is rather easy to obtain. And guess what? the math is quite rewarding!

So we first have to start by selecting a convenient control volume. The idea here is to pick a volume whose sides are parallel per say to the coordinates. For cylindrical coordinates, one may choose the following control volume

Again, as we did in the previous post, we need to account for all the fluid that is accumulating, and flowing through this control volume, namely:

Rate of Accumulation + Rate of Flow In = Rate of Flow Out

First, let’s get some basics laid out. The velocity field will be described as

I always prefer to use u, v, and w instead of ur, utheta, and uz to save on subscripts, although the latter nomenclature is a bit more descriptive… we’ll get used to it. Now, by construction, the volume of the differential control volume is

while the mass of fluid in the control volume is

The rate of change of mass or accumulation in the control volume is then

For the net flow through the control volume, we deal with it one face at a time. Starting with the r faces, the net inflow is

while the outflow in the r direction is

So that the net flow in the r direction is

Being O(dr^2), the last term in this equation can be dropped so that the net flow on the r faces is

The net flow in the theta direction is slightly easier to compute since the areas of the inflow and outflow faces are the same. At the outset, the net flow in the theta direction is

We now turn our attention to the z direction. This requires a little bit of extra work. The most essential ingredient of computing the flow in the z direction is to compute the face areas. As shown in the figure above, the z faces are essentially trapezoids (in the differential limit) and their area is equal to the average of the bases times the height, in other words

then, the inflow at the lower z face is

while the outflow at the upper z face is

Finally, the net flow in the z direction is

Now we can put things together to obtain the continuity equation

dividing by dV and rearranging the r components of the velocity

Voila!

Cite as: Saad, T. "8. Derivation of the Continuity Equation in Cylindrical Coordinates." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2009/02/8-derivation-of-continuity-equation-in.html

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Labels: Fluid Mechanics, Mathematics