ASPECTS of GEOMETRY for ROCKET PROPULSION Part 1

ASPECTS OF GEOMETRY FOR ROCKET PROPULSION

ERNEST YEUNG

Abstract. I point out aspects of differential geometry related to rocket propulsion. The dynamics of rocket flight is viewed from the point of view of smooth manifolds. General expressions that hold true for the dynamics of rockets in curved spacetime (relativistic or Newtonian) are formulated. Thermo- dynamics is also formulated from the point of view of a manifold of thermodynamics states. This note serves as an appetizer for further work, to motivate applications of topology, symplectic geometry, and numerical relativity to rocket propulsion and related aspects of rocket flight in space.

Part 1. Executive Summary 2 1. Forces 2 2. Energy 3

Part 2. Perspectives and Prospects 3 3. Why? 3 4. Applications 4

Part 3. Some Fluid mechanics 5 5. Mass transport (in Continuum)5 6. Momentum (in Continuum)7 7. Cauchy Stress Tensor9 8. Some Carath´eodory’s Thermodynamics 10 9. Energy Transport 12 References 13

Contents

“Why does TARS have to detach?” “We have to shed the weight to escape the gravity.”

Date: 27 mars 2015. 1991 Mathematics Subject Classification. Fluid Mechanics. Key words and phrases. Aerodynamics, Combustion Theory, Differential Geometry, Folations, Propulsion, Rocket Propulsion, Smooth Manifolds, Symplectic Geometry, Thermodynamics, Topology. Ernest Yeung had been supported by Mr. and Mrs. C.W. Yeung, Prof. Robert A. Rosenstone, Michael Drown, Arvid Kingl, Mr. and Mrs. Valerie Cheng, and the Foundation for Polish Sciences, Warsaw University, during his Masters studies. I am on linkedin: ernestyalumni. I am crowdfunding on Tilt/Open and at Patreon to support basic sciences research: ernestyalumni.tilt.com and ernestyalumni at Patreon. Tilt/Open is an open-source crowdfunding platform that is unique in that it offers open-source tools for building a crowdfunding campaign. Tilt/Open has been used by Microsoft and Dicks Sporting Goods to crowdfund their respective charity causes. Patreon is a subscription crowdfunding service that allows you to directly support the works of artists (and scientists and educators! See the Science and Education section of Patreon), allowing you to be a patron of the arts (and the sciences!). Patreon is run by creators and artists and allows you to be flexible in your support. 1 “Newton’s third law: the only way humans have figured out of getting somewhere is to leave something behind.” -Interstellar (2014)

Part 1. Executive Summary

1. Forces

Let spacetime be a quadruple (M, O, A, g) consisting of a (1 + n)-dimensional smooth manifold M, a choice of topology O, a choice of a smooth atlas A, and a choice of metric g, so that the Newtonian and relativistic cases come into play with the choice of g.

Usually n = 3 for 3 spatial directions.

Consider a total system consisting of a (rigid) rocket with combustion chamber and nozzle, and fuel that will be treated as a ﬂuid. As the rocket is a rigid body, it can be shown that the total momentum of the rocket is such that the rocket acts as if the momentum is at the center of mass and concentrated there like a point particle [1]. Let the mass of the rocket be M0.

Let the 4-acceleration of the rocket be a.

I claim that the full expression for the external forces and dynamics accounted for in rocket propulsion, written in a covariant manner, of a total system consisting of the rocket, with its (rigid) combustion chamber and nozzle, and fuel, on a spacetime that is a 4-manifold M equipped with a metric g, is the following:

(1.1) Z i Z i Z ∂u n i ˙ j ∂u n i n ∂ei M0a + ρ vol ⊗ ei + u dm ⊗ ei + ρu j vol ⊗ ei + ρu vol ⊗ + [u, ei] = B ∂t B ∂x B ∂t X Z = Fext = ∗σ + Fgrav ∂(R+B) where B is a smooth, compact submanifold of the spatial foliation N of spacetime M, with boundary ∂B, enclosing the fuel, ρ is the density of the fuel ρ ∈ C∞(M), u ∈ X(M) is a velocity vector ﬁeld for the fuel, with its spatial coordinate components denoted as u, voln ∈ Ωn(N) is the spatial volume form on N, {ei} is a (coordinate) frame on M, t ∈ R is time, Fext is an external force on the total system, R is a smooth, compact submanifold of the spatial foliation N with boundary ∂R that encloses the rocket, ∗ ∗ σ ∈ Γ(T M ⊗ T M) is the Cauchy-stress tensor on M, and Fgrav is the force due to gravity.

The ﬂuid dynamics of the fuel motivates the form of its dynamics on the left-hand side (LHS) of (1.1) (cf. (6.7)). dm˙ ∈ Ωn(N) is a n-form on N describing the mass transport of an inﬁnitesimal piece of the fuel (cf. (6.6)), which I’ll recap as well here:

˙ ∂ρ n n n dm = vol + diuρvol = L ∂ +uρvol ∂t ∂t

R i ˙ R ˙ In (1.1), if one subtracts from both sides the term B u dm ⊗ ei ≡ B dm ⊗ u, then the so-called thrust, due to the ejection of ﬂuid out of the rocket R, pushing on the rocket (Newton’s third law!), force is obtained as a contribution to the acceleration of the rocket a: Z − dm˙ ⊗ u B 2 R j ∂ui n The term B ρu ∂xj vol ⊗ei in (1.1) also informs us that the geometry of the nozzle can push the rocket forward.

One should also note that the expression for the time derivative of the momentum of the ﬂuid Π˙ at its boundary can also be useful in evaluating rocket performance (cf. (6.7)):

Z i Z Z Z ∂u n ∂ρ n i n i n ∂ei Π˙ = ρvol ⊗ ei + vol ⊗ u + ρu iuvol ⊗ ei + ρu vol ⊗ + [u, ei] B ∂t B ∂t ∂B B ∂t and in particular this term: Z Z ∂ρ n i n vol ⊗ u + ρu iuvol ⊗ ei B ∂t ∂B where the geometry of the surface from which the fluid exits from comes into play in the quantity n iuvol . In (1.1), on the RHS, I claim that the term Z ∗σ ∂(R+B) which is the Hodge star operator applied to the (0, 2)-type symmetric tensor σ, representing the Cauchy stress tensor, is a general, manifestly covariant, expression for the sum of all the forces on a system due to stress and strain at the system’s boundary. What one then does is plug into σ a particular form of the stress tensor. In this note, I apply one example, the perfect fluid with fluid pressure (cf.(7.2)).

2. Energy

Assuming a steady state flow, from calculating the time derivative of the total energy E of a fluid inside a smooth, compact body B with boundary, there is a term that describes the change in energy due to fluid leaving the boundary surface (cf. (9.5)) Z Z 1 2 p n 1 2 p n (2.1) d( u + w − )iuvol = ( u + w − )iuvol B 2 ρ ∂B 2 ρ where Stoke’s law was used and where ρ is the mass density and p the pressure.

It should be noted that no conditions on the thermodynamic process was made as the deﬁnition of internal energy per molar mass w was solely employed to express the time change in energy.

Part 2. Perspectives and Prospects

3. Why?

I became interested in rocket propulsion for two reasons: Elon Musk’s rationale for advancing humanity to become a truly spacefaring civilization so to make (intelligent) life multi-planetary 1 and I wanted enable the observation and creation of physical phenomenon - black holes, worm holes, anti-deSitter (AdS) space - seen in (the best movie ever made) Interstellar (Nolan, 2014) possible.

Apparently, Musk began, after the successful IPO (initial public oﬀering) of PayPal, to have any expertise on rocket propulsion by reading books and talking with experts 2. Reading books was something I could immediately do after my completion of my Master’s thesis in Physics (2014). Books on rocket propulsion he read included Rocket Propulsion Elements by Sutton and Biblarz [2] and Aerothermodynamics of Gas Turbine and Rocket Propulsion by Oates [3].

1A Conversation with Elon Musk https://youtu.be/vDwzmJpI4io 2Richard Feloni “Former SpaceX Exec Explains How Elon Musk Taught Himself Rocket Science”, Business Insider, http://www.businessinsider.com/how-elon-musk-learned-rocket-science-for-spacex-2014-10 3 Also, I did not find any MOOCs (Massive Open Online Courses) for graduate-level rocket propulsion, but I did find one MIT OCW (Open CourseWare) for 16.512 Rocket Propulsion, taught by Prof. Martinez- Sanchez in the Fall 2005 [4],a course suggested by Loretta Treviñoon quora.com 3, which also, in turn, led me to this suggested reading [5].

One small endeavor I would like to help in is open, online education at a high-level, the graduate or professional level, for aerodynamics and propulsion because I want to encourage physicists, particularly theoretical physicists, to interface and help with engineering problems and bring their unique expertise to be applied there. I would point to Dr. Feynman’s role in investigating the Challenger disaster (1986) as an example of how theoretical physicists can help engineers in aerospace 4. So one very small part that I do is to write (a lot of) notes on my readings of these engineering texts and catalogue resources I ﬁnd online, sharing this educational path for (hopefully) others with a physics or pure math background to try the same.

4. Applications

I’ll allow myself, here, to speculate on some possible applications from math and physics to rocket propulsion. What I’ve done, partially, in this note is to write down the dynamics and ﬂuid mechanics in a manifestly covariant manner on a spacetime manifold. At the very least, lowest level, this exercise would reproduce what we already know from a vector space formulation of Newtonian mechanics and classical thermodynamics. But the hope is to try to setup for applications from diﬀerential geometry to gain insight in new phenomenon and new computational techniques in propulsion.

These are other areas that hopefully may see application:

• Topology

• Symplectic Geometry

• Numerical Relativity

4.1. Topology. The flow of a fluid over a smooth, compact spatial manifold B can be seen as part of the group of diffeomorphisms Diff on B [6]. It would be interesting to look at the symmetries of this group, a Lie algebra.

4.2. Symplectic Geometry. In orbital mechanics, we could understand the dynamics and trajectory of rockets in space in terms of Hamiltonian mechanics. The language of Hamiltonian mechanics is naturally symplectic geometry. There has been much advancement in symplectic geometry in the last century, particularly the work of Floer, in this area and orbital mechanics could see application of it there. For thermodynamics, I had tried, in this note, to develop a manifold of thermodynamic states Σ and corresponding 1-forms for heat Q and work W , and internal energy U and volume V as global and local coordinates of Σ, respectively (8). Let T be the temperature and introduce β = (RT )−1 where R is a units conversion factor from temperature to energy, and let p be the pressure. The exterior two form Ω Ω = d(βdU + (βp)dV ) helps to deﬁne a Lagrangian submanifold L such that Ω is zero when restricted on the Lagrangian submanifold, i.e. Ω|L = 0 [7]. It would be interesting to investigate further links between thermodynamics and symplectic geometry.

3http://www.quora.com/What-are-some-resources-for-teaching-myself-Aerospace-Engineering 4Richard P. Feynman “What do You Care What Other People Think?”, W.W. Norton & Company, 1988 4 4.3. Numerical Relavitiy. What I’ve tried to do in this note is to write down the “force law” in a manifestly covariant manner. It’s interesting to note that Newtonian spacetime is a curved spacetime! [8]. In (1.1), I’ve written down the 4-acceleration a for the acceleration of the rocket. It would be interesting to consider and to apply the techniques of numerical relativity to the trajectory of a rocket, as its acceleration could be viewed as given by the covariant derivative of its velocity, in the direction of its velocity:

∇vγ˙ vγ˙ = a where vγ˙ is the tangent vector to the trajectory path γ = γ(τ) that a rocket takes. I’ve also tried to write down the equations for ﬂuid dynamics in a manifestly covariant manner in this note because there are also applications from relativistic hydrodynamics [9], a part of numerical relativity, that one can try for propulsion.

I will end here. What follows are technical derivations (applications of diﬀerential geometry) related to the principles behind rocket propulsion that I’ve found in [2], [3], [4], [5].

Part 3. Some Fluid mechanics

I’ll review some concepts from the ﬂuid mechanics of ideal ﬂuids, but formulated in a manifestly covariant manner on a general spacetime manifold M.

5. Mass transport (in Continuum)

How does one compute the rate of change of an integral when the domain of integration is also changing? -Theodore Frankel

Let ρ = ρ(t, x) ∈ C∞(M) be a C∞ or smooth function on the spacetime manifold M, of dimension dimM = 1 + n. ρ is a function of local coordinates t and x = xi, with i = 1 . . . n, of the manifold M.

ρ represents the mass density.

Denote voln the volume form on a spatial foliation or spatial submanifold of codimension 1 (or dimension n) of M. This n-form voln belongs to the set of all n-forms on M, i.e. voln ∈ Ωn(M).

Note that for 3 dimensions of space, n = 3.

Let B be a compact smooth submanifold of codimension 1 with boundary (i.e. ∂B 6= 0) that lies entirely in a spatial foliation (denote as N) of spacetime M. The B denotes a compact body. In general, B can depend on time t, B = B(t). There is a group of diﬀeomorphisms, parametrized by time t, that act on N, and also B, giving the new spatial conﬁguration of B. Here, I will simply denote this group action with B = B(t).

R n Clearly, B ρvol represents the total mass of a ﬂuid within compact body B. Let m = m(t) denote this total mass of ﬂuid within B: Z m = m(t) = ρvoln B I sought an expression for the instantaneous change in time of m: dm d Z (5.1)m ˙ =m ˙ (t) = = ρvoln dt dt B

Examining Eq. (5.1), there appear to be difficulties in “bringing” the time derivative operation “inside the integral”, as the domain of integration B can change in time as well. Frankel showed ([10, Chapter 4, 5 Sec. 4.3 on Differentiation of Integrals]), by taking the limit of the change of an integral of a differential form, over time, that the operation of the time derivative is equal to applying the Lie derivative of the integrand, in the direction of a 4-velocity for the fluid.

Let u be a vector ﬁeld on spacetime M, i.e. u ∈ X(M), where X(M) is the set of all vector ﬁelds on M.

Let’s stipulate that the velocity vector field is of the following form, locally: ∂ u = u(t, x) = , u ∂t The point is that the time component of a 1 + n = 1 + 3 = 4-velocity should be 1, so that there is only ∂ a ∂t vector component for u. This amounts to wanting time to flow uniformly for our physical object, and to be directed in the future [8]: dt(u) = 1 > 0 u = u(t, x), bold-faced to denote that it only has nonzero components on spatial submanifold N ⊆ M, represents the velocity vector field of the fluid flowing about in M. u can depend on time t.

Then, as developed and shown in [10], Z Z Z d n n n ρvol = Luρvol = L ∂ +uρvol dt B B B ∂t

Applying the properties of the Lie derivative and Cartan’s “Magic Formula” (i.e. LX ω = diX ω + iX dω where ω is a diﬀerential form, and noting that the exterior derivative of the volume form is zero (i.e. dvoln = 0),

Z Z n ∂ρ n n (5.2) Luρvol = vol + diuρvol B B ∂t where bold-faced d denotes an exterior derivative only on the spatial coordinates, i.e.

n X ∂ ∂ d = dxi ∧ ≡ dxi ∧ i = 1 . . . n ∂xi ∂xi i=1 Note that d :Ωp(N) → Ωp+1(N), i.e. d is an antiderivative of degree 1 on the exterior algebra of diﬀerential forms on N, the spatial foliation of spacetime M.

p p−1 iu is the interior product, iu :Ω (M) → Ω (M), an antiderivation of degree −1 on the exterior algebra of diﬀerential forms.

Using Stoke’s law on second term of the right hand side (RHS) of Eq. (5.2), Z Z n n diuρvol = iuρvol B ∂B

Thus, the time derivative of the total mass within a compact body B with boundary can be written in two equivalent manners:

Z ∂ρ n n m˙ = vol + diuρvol = B ∂t (5.3) Z Z ∂ρ n n = vol + iuρvol B ∂t ∂B 6 If we introduce a metric g and equip spacetime manifold M with this metric, then we can write a local n expression for the term diuρvol in Eq. (5.3): √ √ √ g 1 ∂( gρui) 1 ∂(ρui) ∂(ln g) di ρvoln = dρ ui1 dxi2 ∧ · · · ∧ dxin = √ voln = √ + ρui voln u n! i1...in g ∂xi g ∂xi ∂xi

To obtain what would be called a diﬀerential form for the time derivative of m or mass transport, one would argue that the expression should hold for any arbitrary volume and thus the following expression would be obtained: √ ∂ρ 1 ∂(ρui) ∂(ln g) (5.4) + √ + ρui ∂t g ∂xi ∂xi √ So if the usual Cartesian coordinates is employed, g ≡ p|detg| = 1, then the usual expression for mass transport is obtained: ∂ρ ∂(ρui) ∂ρ + = + div(ρu) ∂t ∂xi ∂t R n n Consider the surface integral term ∂B iuρvol of Eq. (5.3). iuρvol is the n − 1 dimensional slice of the volume that is orthogonal to u, the velocity vector, but written in a coordinate-free, manifestly covariant manner. This can be shown by writing it out locally and choosing a Cartesian coordinate system, or in this manifestly covariant manner: n i (∗iuvol )(u) = uiu = 1 if u is a unit vector.

6. Momentum (in Continuum)

For a fluid, the total momentum of a fluid inside a compact, smooth submanifold B, with the fluid having a velocity vector field u ∈ X(M), is given by the integration on B of the following vector-valued 1-form in Ω1(M,TM), Z n i ρvol u ⊗ ei B where voln is the volume form for the spatial manifold N of dimension dimN = n, ρ ∈ C∞(M) is a smooth function on manifold M denoting the mass density of a fluid, and ei is a basis vector in the frame belonging to a frame bundle on M.

One should also note why vector-valued differential forms are needed in the case at hand. One does not integrate vectors over a manifold. It makes no sense to integrate vectors on a manifold. The case where one “sums up” vectors in a “vectorial” manner, adding arrows from head to tail in integration only made some sense in flat Euclidean space. There is a single global chart on a flat Euclidean space and, moreoever, the curvature R for a flat Euclidean space is zero. For a general, curved manifold, it makes no sense to do the same procedure: how does one compare a vector at a point on the manifold to another vector at another point on the manifold, when they are at different points?

What one does is integrate diﬀerential forms [10]. Therefore, in this continuum case, I have sought to formulate the objects we want as diﬀerential forms if we want to integrate over the continuum, represented by a submanifold of spacetime M.

Thus the total momentum of the ﬂuid, inside of B, is Π, with Π ∈ X(M): Z n i (6.1) Π = ρvol u ⊗ ei B

I should note that it’s unclear, right now to me, whether it’s best to treat Π as a vector ﬁeld or as a covector ﬁeld. Note that to use the tangent-cotangent isomorphism or so-called “musical map,” the 7 manifold M must be equipped with a metric g, so that we are considering for spacetime a metric manifold (M, g). This is opposed to not having to consider the additional structure, a metric g.

If one does consider a covector form of Π, note that for the ﬂuid, the integration over B of a covector- valued 1-form in Ω1(M,T ∗M) is needed: Z n i ρvol ui ⊗ e B

So that, with [ denoting the operation of using the metric for the tangent-cotangent isomorphism (this musical map from vectors to covectors): Z [ n i (6.2) Π = ρvol ui ⊗ e B

Going ahead with the case where Π is a vector ﬁeld, take the time derivative of Π to determine the dynamics of this part of the ﬂuid: dΠ d Z = Π˙ = ρvolnui ⊗ e dt dt i Then consider Z Z Z Z d n i n i n i n i (6.3) ρvol u ⊗ ei = L ∂ +u(ρvol u ⊗ ei) = L ∂ +u(ρvol u ) ⊗ ei + ρvol u ⊗ L ∂ +uei dt B B ∂t F ∂t ∂t

Consider this term in the expression for Eq. (6.3):

n i ∂ i n n i L ∂ +u(ρvol u ) = (ρu )vol + Lu(ρvol u ) ∂t ∂t

Consider also this term in Eq. (6.3):

n i i n Lu(ρvol u ) = d(ρu iuvol )

This can be interpreted geometrically as the change of momentum in the ith direction for a small piece of the ﬂuid ﬂowing along the integral curve generated by velocity vector u.

By using Stoke’s theorem, this integral over B can be equated to an integral over ∂B Z Z i n i n d(ρu iuvol ) = ρu iuvol B ∂B

Also, in full generality, consider ∂ L ∂ +uei = ei + [u, ei] ∂t ∂t

The local form for [u, ei] of above is the following: ∂ek ∂uk ∂ (6.4) [u, e ] = uj i − ej i ∂xj i ∂xj ∂xk on a chart (U, x) on M. Right now, to me at least, it appears that [u, ei] in Eq. (6.4) can be interpreted as the rotation of the ﬂuid if the frame of the observer is rotating as well.

Thus, the expression for the time derivative of the total momentum of a piece of the ﬂuid in B, Π can be written as such: Z i Z Z ∂(ρu ) n i n i n ∂ei (6.5) Π˙ = vol ⊗ ei + ρu iuvol ⊗ ei + ρu vol ⊗ + [u, ei] B ∂t ∂B B ∂t

Suppose we want to account for the mass of the fluid leaving B, i.e. mass transport. Recall this equation for mass transport, Eq. (5.3) 8 Z Z Z ∂ρ n n ∂ρ n n m˙ = vol + diuρvol = vol + iuρvol B ∂t B ∂t ∂B Now ∂(ρui) ∂ρ ∂ui ∂ui d(ρuii voln) = ujvoln = ui ujvoln + ρuj voln = uid(i ρvoln) + ρuj voln u ∂xj ∂xj ∂xj u ∂xj Let ˙ ∂ρ n n n (6.6) dm = vol + diuρvol = L ∂ +uρvol ∂t ∂t be a n-form on N, i.e. dm˙ ∈ Ωn(N), describing the mass transport of an infinitesimal piece of the fluid.

Then rewrite Π˙ to be Z ∂ui Z ∂ui Z ∂e (6.7) ˙ n i ˙ j n i n i Π = ρ vol ⊗ ei + u dm ⊗ ei + ρu j vol ⊗ ei + ρu vol ⊗ + [u, ei] B ∂t B ∂x B ∂t

This gives the dynamics of the ﬂuid due to external forces on it and motivates the form of the dynamics on the ﬂuid in the LHS of (1.1).

∂ui In the special case of the fluid not being accelerated, so that ∂t = 0, a uniform flow of fluid, so that ∂ui ∂xj = 0, and a choice of frame on M that doesn’t change at all when moving about the manifold M, ∂ei R i ˙ so that ∂t + [u, ei] = 0, then B u dm ⊗ ei tells us the total force on this part of the fluid. If this fluid, R i ˙ that’s getting expunged, was coming out of something, − B u dm ⊗ ei is the so-called thrust force on that something, due to the expunging of this fluid.

7. Cauchy Stress Tensor

[9] has a description of a stress tensor σ that is manifestly covariant. However, I haven’t seen (please correct me or show me that I’m wrong or what I’m missing [email protected]) this formulation be applied to summing or integrating over an entire surface boundary of a volume the contributions of the stress tensor σ.

i j Recollect that σ is a (0, 2)-type symmetric tensor on M, and so locally it has the form σijdx ⊗ dx . 1 ∗ j This could also be viewed as a covector valued 1-form, i.e. σ ∈ Ω (M,T M), and so let σi = σijdx be 1 the ith component of the 1-form ω, which itself (the ith component) is a 1-form, i.e. σi ∈ Ω (M) I claim that the total force in the ith direction due to all the forces and stress on a body B at its boundary surface ∂B is the following: Z i − ∗σi ⊗ e ∂B with ei being an element in the coframe, a coframe being the dual basis of a frame on a manifold, on a coframe bundle, and the minus sign, −, is for the convention where the orientation on ∂B is ﬁxed such that the normal to the surface boundary of B points outwards from B, so that a minus sign would have the stress forces point inside, towards the interior of body B.

By Stoke’s Theorem, this force can be rewritten as follows: Z Z i i (7.1) − ∗σi ⊗ e = − d ∗ σi ⊗ e ∂B B

As a first example, consider the pressure a fluid experiences for “just being a fluid”: at any (finite volume) place in a fluid, put a body there and it will feel forces, or pressure, all over its body’s surface, a pressure p, that could depend on position x = xi and possibly time t. 9 g is a metric that is an isotropy, invariant under a certain group of diffeomorphisms.

Consider σ = pg, which represents, physically, pressure p in a fluid and hence, the stress across the surface of a body in this fluid, due to the fluid pushing on the body at the surface.

So for (7.2) σ = pg

In plugging in Eq. (7.2) into (7.1), consider the local form of ∗σi in this case of (7.2):

√ g j j i2 in ∗σi = ∗pgijdx = (pgij)dx ∧ · · · ∧ dx n! i2...in Does gij “bring down” the coordinates of the Levi-Civita symbol (which isn’t even a tensor; it’s just a number dependent upon the permutation of coordinate labels) so that one could do the following naive operation: (7.3) g j = ij i2...in ii2...in

If not, then √ 1 ∂( gpgij) d ∗ σ = √ voln i g ∂xj Otherwise, if so, that Eq.(7.3) is true, √ 1 ∂( gp) (7.4) d ∗ σ = √ voln i g ∂xi For the rest of the discussion, I’ll follow the latter case of (7.4).

So Newton’s second law says that the time derivative of the total momentum of a body is equal to the sum of all the external forces on the body: Z Π˙ = − ∗σ] B where the musical map was used to make the stress tensor σ yield a vector.

Thus, from Eq. (6.7), for the time derivative of the total momentum of the ﬂuid inside body B, for the ˙ case of mass conservation, i.e. dm = 0, and for a choice of constant frame {ei} over B, Z i i Z √ ∂u j ∂u n 1 ∂( gp) n ij (7.5) ρ + ρu j vol ⊗ ei = − √ i vol ⊗ g ej B ∂t ∂x B g ∂x

If one supposes the procedure of obtaining a diﬀerential formulation of (7.5), by arguing that it must hold for arbitrary B, arbitrary volume, and comparing terms, then one obtains the following: √ ∂ui ∂ui gij ∂( gp) (7.6) + uj = −√ ∂t ∂xj g ∂xj which yields the celebrated Euler’s equation (1755) if one takes a ﬂat Euclidean metric [11].

8. Some Caratheodory’s´ Thermodynamics

Bamberg and Sternberg has a pedagogically friendly introduction to a formulation of thermodynamics in the language of diﬀerential manifolds [7, Chapter 22]. Frankel [10] also has an introduction. I would like to know of further references 5.

Consider a total thermodynamics system consisting of n number of gas/liquid subsystems in thermal contact, but unable to exchange particles between regions of the subsystem. Let Σ be the manifold

[email protected] 10 of thermodynamic states of this total system. A path in Σ represents a sequence of states. Σ is of dimension dimΣ = n + 1 and on a chart, the local coordinates are v0, v1, . . . , vn with v0 = U, where U is the internal energy of the total system and vi for i = 1 . . . n is the volume of the ith region or subsystem.

Note that v0 = U is stipulated to be a global coordinate for Σ.

Heat Q is a 1-form on Σ i.e. Q ∈ Ω1(Σ) that depends on v0, v1, . . . vn, i.e. Q = Q(v0, v1 . . . vn).

On the other hand, work W is a 1-form on Σ, i.e. W ∈ Ω1(Σ) that depends only on v1 . . . vn and not at all on v0 = U the internal energy, i.e. W = W (v1 . . . vn).

Then, brieﬂy, the ﬁrst law of thermodynamics (energy conservation) is stated as thus:

(8.1) dU = Q − W = Q − pdV and the second law of thermodynamics says, for reversible processes,

(8.2) Q = T dS for temperature T and entropy S ∈ C∞(Σ), so that dS is a 1-form on Σ.

The enthalpy H, a smooth function on Σ i.e. H ∈ C∞(Σ), is

(8.3) H = U + pV

Taking the exterior derivative, dH can be written in two ways dH = dU + d(pV ) = (8.4) = Q + V dp with the second, latter way utilizing the ﬁrst law of thermodynamics.

It is clear what a level curve for an adiabatic or isentropic process is. For an ideal gas or ideal fluid, there is an absence of heat exchange between different parts of the gas or fluid. This adiabatic process is a path a = a(p, V ) or level curve on Σ as the system moves between thermodynamic equilibrium states. An example would be the familiar ideal gas with a level curve defines as such:

pV γ = constant for an adiabatic process.

Consider that this level curve a on Σ is generated by a tangent vector at each state of the system, so, letting a parametrization of this curve to be λ ∈ R on an interval [0, 1], then at λ = τ d a˙(τ) = a(τ) = ξ dλ

The deﬁning property of a level curve a that represents an adiabatic process is this:

(8.5) Q(ξ) = 0

∀ ξ, ∀ λ ∈ [0, 1], i.e. the tangent vector to the level curve is eliminated by the 1-form Q at every point of the curve, or path. ξ is called a null vector.

Also note that as Q = T dS from the second law of thermodynamics, Q(ξ) = 0 implies dS(ξ) = 0 so that the process is isentropic.

By deﬁnition, the volume per unit mass V is related to the density per unit mass ρ as proportional to the inverse of density: NM V = ρ 11 with N being the number of species of the gas molecules and M being the mass of the molecule per mole.

So taking the enthalpy deﬁnition in (8.3), we can express the internal energy U in terms of enthalpy and p and V , and also take its exterior derivative: U = H − pV (8.6) NMp NMp dU = dH − d(pV ) = dH − d( ) = d(H − ) ρ ρ

9. Energy Transport

Let’s write down the total kinetic energy of the ﬂuid, moving with velocity vector ﬁeld u ∈ XM in a smooth, compact body B with surface boundary ∂B: 1 Z (9.1) u2ρvoln 2 B

The time derivative of the total kinetic energy of the ﬂuid inside of B is Z Z Z 2 d 1 2 n 1 2 n 1 ∂u 2 n 2 ˙ (9.2) u ρvol = L ∂ +u(u ρvol ) = ρ + (du )(u) vol + u dm dt 2 B 2 B ∂t 2 B ∂t

Let U, U ∈ C∞(M) be the internal energy per unit mass. The total internal energy of the ﬂuid inside of B, Utot, is Z n (9.3) Utot = Uρvol B

d ˙ The time derivative of the total internal energy, dt U ≡ U, is (9.4) Z Z Z ˙ d n n n ∂U n ˙ Utot = Uρvol = L ∂ +uU ρvol + UL ∂ +uρvol = + dU(u) ρvol + Udm dt B B ∂t ∂t B ∂t

dE ˙ Then the time derivative of the total energy E of the fluid inside B, dt = E is Z 2 Z Z ˙ 1 ∂u ∂U n 1 2 n 1 2 ˙ (9.5) E = + ρvol + (du )i(u) + dUi(u) ρvol + u + U dm B 2 ∂t ∂t B 2 B 2 Consider the second term on the RHS of (9.5) and consider the exterior derivative of U, dU. “Wrap up” ∂ ∂t U + dU into dU if you’d like. Rewrite dU in terms of the internal energy per molar mass w: 1 p 1 p (9.6) (du2)(u) + d(w − )i = d( u2 + w − )i 2 ρ (u) 2 ρ (u) For a steady state condition (time derivatives of quantities is zero!) and a stationary compressible ideal isentropic flow, so that energy is conserved throughout the fluid, and so E˙ = 0, then the differential above is zero for arbitrary volume B and so the celebrated Bernoulli equation is recovered [12]: 1 p u2 + w − = constant 2 ρ

I want to point out that in writing dU in terms of enthalpy, only the definitions were used. The condition that we are on an adiabatic level curve with null vector ξ would have implied (??) so that only the work done on the fluid by pressure at its boundary would’ve been sufficient to describe the change in internal energy. (9.6) is a general expression. a 12 References

[1] L.D. Landau and E.M. Lifshitz, Mechanics, Volume 1 of Course of Theoretical Physics, Elsevier Butterworth- Heinemann, Third Edition, Reprinted, 2000. ISBN 0 7506 2896 0 [2] George P. Sutton, Oscar Biblarz, Rocket Propulsion Elements, John Wiley & Sons, Inc., Seventh Edition, 2001. ISBN 0-471-32642-9 [3] Gordon C. Oates, Aerothermodynamics of Gas Turbine and Rocket Propulsion, American Institute of Aero- nautics and Astronautics, Inc., Reston, Virginia, Third Edition, 1998. ISBN 1-56347-241-4 [4] Manuel Martinez-Sanchez. 16.512 Rocket Propulsion, Fall 2005. (Massachusetts Institute of Technology: MIT Open- CourseWare), http://ocw.mit.edu (Accessed 6 Apr, 2015). License: Creative Commons BY-NC-SA [5] Philip G. Hill, Mechanics and Thermodynamics of Propulsion, Addison-Wesley Publishing Company, Second Edition, 1992. ISBN 0-201-14659-2 [6] Vladimir I. Arnold, Boris A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag New York, 1998. ISBN 0-387-94947-X [7] Paul Bamberg and Shlomo Sternberg, A Course in Mathematics for students of physics: 2, Cambridge Univer- sity Press, 1990. [8] F. Schuller, Lecture 9: Newtonian spacetime is curved! (International Winter School on Gravity and Light 2015), The WE-Heraeus International Winter School on Gravity and Light, Feb. 12, 2015, https://youtu.be/ IBlCu1zgD4Y [9] E. Gourgoulhon, An Introduction to Relativistic Hydrodynamics, Stellar Fluid Dynamics and Numerical Simu- lations: From the Sun to Neutron Stars, EAS Publications Series, 21 (2006) 43-79. [10] T. Frankel, The Geometry of Physics, Cambridge University Press, Second Edition, 2004. [11] L. D. Landau and E.M. Lifshitz, Fluid Mechanics, Volume 6 of Course of Theoretical Physics, Elsevier Science, Second English Edition, Revised, 2004. ISBN 0 7506 2767 0 [12] R. Abraham, J.E. Marsden, T.Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, 2003.

There is a Third Edition of T. Frankel’s The Geometry of Physics [10], but I don’t have the funds to purchase the book (about $ 71 US dollars, with sales tax). It would be nice to have the hardcopy text to see new updates and to use for research, as the second edition allowed me to formulate ﬂuid mechanics and elasticity in a covariant manner. Please help me out and donate at ernestyalumni.tilt.com or at subscription based Patreon, patreon.com/ernestyalumni.

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