Ideal fluid mechanics

Transcription

1 Lecture notes Ieal flui mechanics Simon J.A. Malham Simon J.A. Malham (6th Feb 2010) Maxwell Institute for Mathematical Sciences an School of Mathematical an Computer Sciences Heriot-Watt University, Einburgh EH14 4AS, UK Tel.: Fax:

2 2 Simon J.A. Malham 1 Introuction The erivation of the equations of motion for an ieal flui by Euler in 1755, an then for a viscous flui by Navier (1822) an Stokes (1845) were a tour-e-force of 18th an 19th century mathematics. These equations have been use to escribe an explain so many physical phenomena aroun us in nature, that currently billions of ollars of research grants in mathematics, science an engineering now revolve aroun them. They can be use to moel the couple atmospheric an ocean flow use by the meteorological office for weather preiction own to any application in chemical engineering you can think of, say to evelopment of the thrusters on NASA s Apollo programme rockets. The incompressible Navier Stokes equations are given by u t + u u = ν 2 u p + f, u = 0, where u = u(x, t) is a three imensional incompressible flui velocity (inicate by the last equation), p = p(x, t) is the pressure an f is an external force fiel. The frictional force ue to stickiness of a flui is represente by the term ν 2 u. Here we will consier ieal fluis only, an this correspons to the case ν = 0, when the equations above are known as the Euler equations for a homogeneous incompressible ieal flui. We will erive the Euler equations an in the process learn about the subtleties of flui mechanics an along the way see lots of interesting applications. 2 Flui flow 2.1 Flow A material exhibits flow if shear forces, however small, lea to a eformation which is unboune we coul use this as efinition of a flui. A soli has a fixe shape, or at least a strong limitation on its eformation when force is applie to it. With the category of fluis, we inclue liquis an gases. The main istinguishing feature between these two fluis is the notion of compressibility. Gases are usually compressible as we know from everyay aerosols an air canisters. Liquis are generally incompressible a feature essential to all moern car braking mechanisms. Fluis can be further subcatergorize. Here we will only consier ieal fluis. This means that the only internal force present is pressure which acts so that flui flows from a region of high pressure to one of low pressure. However fluis can exhibit internal frictional forces which moel a stickiness property of the flui which involves energy loss such fluis are known as viscous fluis we will not consier them here. Some fluis/material known as non-newtonian or complex fluis exhibit even stranger behaviour, their reaction to eformation may epen on: (i) past history (earlier eformations), for example some paints; (ii) temperature, for example some polymers or glass; (iii) the size of the eformation, for example some plastics or silly putty. The notion of an ieal flui however, can be use to escribe an preict a lot of phenomena observe in common working fluis, such as air, water, bloo, an so forth, an have been applie to wing an aircraft esign (as a limit of high Reynols number flow).

3 Ieal flui mechanics Continuum hypothesis For any real flui there are three natural length scales: 1. L molecular, the molecular scale characterize by the mean free path istance of molecules between collisions; 2. L flui, the meium scale of a flui parcel, the flui roplet in the pipe or ocean flow; 3. L macro, the macro-scale which is the scale of the flui geometry, the scale of the container the flui is in, whether a beaker or an ocean. An, of course we have the asymptotic inequalities: L molecular L flui L macro. We will assume that the properties of an elementary volume/parcel of flui, however small, are the same as for the flui as a whole i.e. we suppose that the properties of the flui at scale L flui propagate all the way own an through the molecular scale L molecular. This is the continuum assumption. For everyay flui mechanics engineering, this assumption is extremely accurate (Chorin an Marsen [3, Page 2]). 2.3 Conservation principles Our erivation of the basic equations unerlying the ynamics of ieal fluis is base on three basic principles (see Chorin an Marsen [3, Page 2]): 1. Conservation of mass, mass is neither create or estroye; 2. Newton s 2n law/balance of momentum, for a parcel of flui the rate of change of momentum equals the force applie to it; 3. Conservation of energy, energy is neither create nor estroye. In turn these principles generate the: 1. Continuity equation which governs how the ensity of the flui evolves locally an thus inicates compressibility properties of the flui; 2. Euler equations of motion for a flui which inicates how the flui moves aroun from regions of high pressure to those of low pressure; 3. Equation of state which inicates the mechanism of energy exchange within the flui. 3 Trajectories an streamlines Suppose that our ieal flui is containe with a region/omain D R where = 2 or 3, an x = (x, y, z) D is a position/point in D. Imagine a small flui particle or a speck of ust moving in a flui flow fiel prescribe by the velocity fiel u(x, t) = (u, v, w). Suppose the position of the particle at time t is recore by the variables `x(t), y(t), z(t). The velocity of the particle at time t at position `x(t), y(t), z(t) is ẋ(t) = u`x(t), y(t), z(t), t, ẏ(t) = v`x(t), y(t), z(t), t, ż(t) = w`x(t), y(t), z(t), t.

4 4 Simon J.A. Malham In shorter vector notation this is x(t) = u(x(t), t). t The trajectory or particle path of a flui particle is the curve trace out by the particle as time progresses. It is the solution to the ifferential equation above (with suitable initial conitions). Suppose now for a given flui flow u(x, t) we fix time t. A streamline is an integral curve of u(x, t) for t fixe, i.e. it is a curve x = x(s) parameterize by the variable s, that satisfies the system of equations x(s) = u(x(s), t), s with t hel constant. If the velocity fiel u is time-inepenent, i.e. u = u(x) only, or equivalently t u = 0, then trajectories an streamlines coincie. Flows for which t u = 0 are sai to be stationary. Example. Suppose a velocity fiel u(x, t) = (u, v, w) is given for t > 1 by u = x 1 + t, v = y t an w = z. To fin the particle paths or trajectories, we must solve the system of equations x t = u, y t = v an z t = w, an then eliminate the time variable t between them. Hence for the particle paths we have x t = x 1 + t, y t = y z t an t = z. Using the metho of separation of variables an integrating in time from t 0 to t, in each of the three equations, we get «««x 1 + t y «ln = ln, ln = 2ln 2 t x t 0 y t 0 an «z ln = t, z 0 t 0 where we have assume that at time t 0 the particle is at position (x 0, y 0, z 0 ). Exponentiating the first two equations an solving the last one for t, we get x x 0 = 1 + t 1 + t 0, y y 0 = ( t)2 ( t 0) 2 an t = t 0 ln(z/z 0 ). We can use the last equation to eliminate t so the particle path/trajectory through (x 0, y 0, z 0 ) is the curve in three imensional space given by x = x 0 `1 + t0 ln(z/z 0 ) (1 + t 0 ), an y = y 0 ` t 0 ln(z/z 0 ) 2 ( t 0) 2. To fin the streamlines, we fix time t. We must then solve the system of equations x s = u, y s = v an z s = w,

5 Ieal flui mechanics 5 with t fixe, an then eliminate s between them. Hence for streamlines we have x s = x 1 + t, y s = y t an z s = z. Assuming that we are intereste in the streamline that passes through the point (x 0, y 0, z 0 ), we again use the metho of separation of variables an integrate with respect to s from s 0 to s, for each of the three equations. This gives «««x y z ln x 0 = s s t, ln y 0 = s s t an ln z 0 = s s 0. Using the last equation, we can substitute for s s 0 into the first equations. If we then multiply the first equation by 1 + t an the secon by t, an use the usual log law ln a b = b ln a, then exponentiation reveals that «1+t x y = « t = z z 0, x 0 y 0 which are the equations for the streamline through (x 0, y 0, z 0 ). 4 Conservation of mass 4.1 Continuity equation Recall, we suppose our ieal flui is containe with a region/omain D R (here we will assume = 3, but everything we say is true for the collapse two imensional case = 2). Hence x = (x, y, z) D is a position/point in D. At each time t we will suppose that the flui has a well efine mass ensity ρ(x, t) at the point x. Further, each flui particle traces out a well efine path in the flui, an its motion along that path is governe by the velocity fiel u(x, t) at position x at time t. Consier an arbitrary subregion Ω D. The total mass of flui containe insie the region Ω at time t is ρ(x, t) V. Ω where V is the volume element in R. Let us now consier the rate of change of mass insie Ω. By the principle of conservation of mass, the rate of increase of the mass in Ω is given by the mass of flui entering/leaving the bounary Ω of Ω. To compute the total mass of flui entering/leaving the bounary Ω, we consier a small area patch S on the bounary of Ω, which has unit outwar normal n. The total mass of flui flowing out of Ω through the area patch S per unit time is mass ensity flui volume leaving per unit time = ρ(x, t) u(x, t) n(x) S, where x is at the center of the area patch S on Ω. Note that to estimate the flui volume leaving per unit time we have ecompose the flui velocity at x Ω, time t, into velocity components normal (u n) an tangent to the surface Ω at that point. The velocity component tangent to the surface pushes flui across the surface no flui enters or leaves Ω via this component. Hence we only retain the normal component see Fig. 2.

6 6 Simon J.A. Malham Ω D Fig. 1 The flui of mass ensity ρ(x, t) swirls aroun insie the container D, while Ω is an imaginary subregion. u.n n u S Fig. 2 The total mass of flui moving through the patch S on the surface Ω per unit time, is given by the mass ensity ρ(x, t) times the volume of the cyliner shown which is u ns. Returning to the principle of conservation of mass, this is now equivalent to the integral form of the law of conservation of mass: ρ(x, t)v = ρu n S. t Ω Ω The ivergence theorem an that the rate of change of the total mass insie Ω equals the total rate of change of mass ensity insie Ω imply, respectively, (ρu)v = (ρu) n S Ω Ω an ρ ρ V = t Ω Ω t V. Using these two relations, the law of conservation of mass is equivalent to Ω ρ + (ρu) V = 0. t Now we use that Ω is arbitrary to euce the ifferential form of the law of conservation of mass or continuity equation that applies pointwise: ρ + (ρu) = 0. t This is the first of our three conservation laws.

7 Ieal flui mechanics Incompressible flow Having establishe the continuity equation we can now efine a subclass of flows which are incompressible. The classic examples are water, an the brake flui in your car whose incompressibility properties are vital to the effective transmission of peal pressure to brakepa pressure. Definition 1 (Incompressibility) A flui with the property u = 0 is incompressible. The continuity equation an the ientity, (ρ u) = ρ u + ρ u, imply ρ + u ρ + ρ u = 0. t Hence since ρ > 0, a flow is incompressible if an only if ρ + u ρ = 0. t If the flui is homogeneous so that ρ is constant in space, then the flow is incompressible if an only if ρ is constant in time. 4.3 Stream functions A stream function exists for a given flow u = (u, v, w) if the velocity fiel u is solenoial, i.e. u = 0, an we have an aitional symmetry that allows us to eliminate one coorinate. For example, a two imensional incompressible flui flow u = u(x, y, t) is solenoial since u = 0, an has the symmetry that it is uniform with respect to z. For such a flow we see that u = 0 u x + v y = 0. This equation is satisfie if an only if there exists a function ψ(x, y, t) such that ψ y ψ = u(x, y, t) an = v(x, y, t). x The function ψ is calle Lagrange s stream function. A stream function is always only efine up to any arbitrary aitive constant. Further note that for t fixe, streamlines are given by constant contour lines of ψ. Note that if we use plane polar coorinates so u = u(r, θ, t) an the velocity components are u = (u r, u θ ) then u = 0 1 r r (r ur) + 1 r u θ r = 0. This is satisfie if an only if there exists a function ψ(r, θ, t) such that 1 ψ ψ = ur(r, θ, t) an r θ r = u θ(r, θ, t). Example Suppose that in Cartesian coorinates we have the two imensional flow u = (u, v) given by (u, v) = (k x, k y),

8 8 Simon J.A. Malham for some constant k. Note that u = 0 so there exists a stream function satisfying ψ y ψ = k x an = k y. x Consier the first partial ifferential equation. Integrating with respect to y we get ψ = k xy + C(x) where C(x) is an arbitrary function of x. However we know that ψ must simultaneously satisfy the secon partial ifferential equation above. Hence we substitute this last relation into the secon partial ifferential equation above to get ψ x = k y k y + C (x) = k y. We euce C (x) = 0 an therefore C is an arbitrary constant. Since a stream function is only efine up to an arbitrary constant we take C = 0 for simplicity an the stream function is given by ψ = k xy. Now suppose we use plane polar coorinates instea. The corresponing flow u = (u r, u θ ) is given by (u r, u θ ) = (k r cos 2θ, k r sin 2θ). First note that u = 0 using the polar coorinate form for u inicate above. Hence there exists a stream function ψ = ψ(r, θ) satisfying 1 ψ ψ = k r cos 2θ an = k r sin 2θ. r θ r As above, consier the first partial ifferential equation shown, an integrate with respect to θ to get ψ = 1 2 k r2 sin 2θ + C(r). Substituting this into the secon equation above reveals that C (r) = 0 so that C is a constant. We can for convenience set C = 0 so that ψ = 1 2 k r2 sin 2θ. Comparing this form with its Cartesian equivalent above, reveals they are the same. 5 Balance of momentum 5.1 Differentiation following the flui Recall our image of a small flui particle moving in a flui flow fiel prescribe by the velocity fiel u(x, t). The velocity of the particle at time t at position x(t) is x(t) = u(x(t), t). t As the particle moves in the velocity fiel u(x, t), say from position x(t) to a nearby position an instant in time later, two ynamical contributions change: (i) a small instant

9 Ieal flui mechanics 9 in time has elapse an the velocity fiel u(x, t), which epens on time, will have change a little; (ii) the position of the particle has change in that short time as it move slightly, an the velocity fiel u(x, t), which epens on position, will be slightly ifferent at the new position. Let us compute the acceleration of the particle to explicitly observe these two contributions. By using the chain rule we see that 2 t 2 x(t) = t u`x(t), t = u x x t + u y x = t x + y t = u u + u t. y t + u z z y + z t t + u t z «u + u t Inee for any function F(x, y, z, t), scalar or vector value, the chain rule implies t F `x(t), y(t), z(t), t = F t + u F. Definition 2 (Material erivative) If the velocity fiel components are u = (u, v, w) an u u x + v y + w z, then we efine the material erivative following the flui to be D Dt := t + u. 5.2 Internal flui forces Let us consier the forces that act on a small parcel of flui in a flui flow. There are two types: 1. external or boy forces, these may be ue to gravity or external electromagnetic fiels. They exert a force per unit volume on the continuum. 2. surface or stress forces, these are forces, molecular in origin, that are applie by the neighbouring flui across the surface of the flui parcel. The force per unit area exerte across a surface (imaginary in the flui) is calle the stress. Let S be a small imaginary surface in the flui centere on the point x see Fig. 3. The force F on sie (2) by sie (1) of S in the flui/material is given by F = Σ(n)S. Here Σ is the stress at the point x. It is a function of the normal irection n to the surface S, in fact it is given by: Σ(n) = σ(x) n. Note σ = [σ ij ] is a 3 3 matrix known as the stress tensor. The iagonal components of σ ij, with i = j, generate normal stresses, while the off-iagonal components, with

10 10 Simon J.A. Malham F (2) x n (1) S Fig. 3 The force F on sie (2) by sie (1) of S is given by Σ(n) S. i j, generate tangential or shear stresses. Here we will only consier normal stresses, in fact those for which there exists a scalar function p such that (here I is the 3 3 ientity matrix) σ(x) = p I Σ(n) = p n. Definition 3 (Ieal flui) For any motion of an ieal flui there is a function p(x, t) calle the pressure such that the stress across a surface S, which has unit normal n at the point x S at time t, is given by p(x, t)n, i.e. the stress across S at x = p(x, t) n. Remark 1 Hence there are no tangential forces in an ieal flui. This has some important consequences, quoting from Chorin an Marsen [3, Page 5]:...there is no way for rotation to start in a flui, nor, if there is any at the beginning, to stop even here we can etect trouble for ieal fluis because of the abunance of rotation in real fluis (near the oars of a rowboat, in tornaoes, etc. ). 5.3 Equation of motion (Euler 1755) Consier again an arbitrary imaginary subregion Ω of D as in Fig. 1. At any instant in time t, the total force exerte on the flui insie Ω through the stresses exerte across its bounary Ω is given by p n S p V. Ω Ω If f(x, t) is a boy force (external force) per unit mass then the total boy force on the flui insie Ω is ρ f V. Ω Thus on any parcel of flui, the force per unit volume acting on it is p + ρ f. Hence using Newton s 2n law (force = mass acceleration) we can euce the following relation the ifferential form of the balance of momentum: ρ Du Dt = p + ρ f. This is Euler s equation of motion for an ieal flui. It represents the secon of our three conservation laws.

11 Ieal flui mechanics 11 6 Transport theorem Suppose that the region within which the flui is moving is D. Suppose Ω is a subregion of D ientifie at time t = 0. As the flui flow evolves the flui particles that originally mae up Ω will subsequently fill out a volume Ω t at time t. We think of Ω t as the volume moving with the flui. Theorem 1 (Transport theorem) For any function F = F(x, t) an ensity function ρ = ρ(x, t) satisfying the continuity equation, we have ρ F V = ρ DF t Ω t Ω t Dt V. The transport theorem is useful because it allows us to euce Euler s equation of motion from the primitive integral form of the balance of momentum. From Newton s secon law this is ρ u V = p + ρ f V. t Ω t Ω t Hence, using the transport theorem with F u an that Ω an thus Ω t are arbitrary, we see that Euler s equation of motion must hol pointwise at each x D. Proof There are four steps; see Chorin an Marsen [3, Pages 6 11]. Step 1: Flui flow map. For any x D let ϕ(x, t) enote the trajectory followe by the particle that starts at position x at time t = 0. Let ϕ t enote the map x ϕ(x, t), i.e. ϕ t is the map that avances each particle at position x at time t = 0 to its position at time t later; it is the flui flow-map. Hence, for example ϕ t (Ω) = Ω t. We assume ϕ t is sufficiently smooth an invertible for all our subsequent manipulations. Step 2: Change of variables. For any two functions ρ = ρ(x, t) an F = F(x, t) we can perform the change of variables with J(x, t) as the Jacobian of ϕ t : t ρ F V = (ρ F)(ϕ(x, t), t) J(x, t) V Ω t t Ω = `(ρ F)(ϕ(x, t), t) J(x, t) V Ω t = Ω t (ρ F)(ϕ(x, t), t) J(x, t) + (ρ F)(ϕ(x, t), t) J(x, t) rv t «D = (ρ F) (ϕ(x, t), t) J(x, t) + (ρ F)(ϕ(x, t), t) J(x, t) V. Ω Dt t Step 3: Evolution of the Jacobian. By irect computation (see Appenix A) t J(x, t) = ` u(ϕ(x, t), t) J(x, t). Step 4: Conservation of mass. We see that we thus have «D ρ F V = t Ω t Ω Dt (ρ F) + (ρ F)` u (ϕ(x, t), t) J(x, t) V D = Ω t Dt (ρ F) + `ρ «u F V = ρ DF Ω t Dt V, where in the last step we have use the conservation of mass equation.

12 12 Simon J.A. Malham 7 Conservation of energy We have erive two conservation laws thusfar, first, conservation of mass, ρ + (ρu) = 0, t an secon, balance of momentum, ρ Du = p + ρ f. Dt If we assume we are in three imensional space so = 3, we have four equations, but five unknowns namely u, p an ρ. We cannot specify the flui motion completely without specifying one more conition. Definition 4 (Kinetic energy) The kinetic energy of the flui in the region Ω D is E := 2 1 ρ u 2 V. Ω The rate of change of the kinetic energy, using the transport theorem, is given by E t = «t 12 ρ u 2 V Ω t = 2 Ωt 1 ρ D u 2 V Dt = 2 1 ρ D (u u) V Ω t Dt = ρu Du Ω t Dt V = u ρ Du «V. Ω t Dt Here we assume that all the energy is kinetic. The principal of conservation of energy states (from Chorin an Marsen, page 13): the rate of change of kinetic energy in a portion of flui equals the rate at which the pressure an boy forces o work. In other wors we have E t = p u n S + ρ u f V. Ω t Ω t We compare this with our expression above for the rate of change of the kinetic energy. Equating the two expressions, using Euler s equation of motion, an noticing that the boy force term immeiately cancels, we get p u n S = u p V Ω t Ω t (up) V = u p V Ω t Ω t (up) V = (up) ( u) p V Ω t Ω t ( u) p V = 0. Ω t

13 Ieal flui mechanics 13 Since Ω an therefore Ω t is arbitrary we see that the assumption that all the energy is kinetic implies u = 0. Hence our thir conservation law, conservation of energy has lea to the equation of state, u = 0, i.e. that an ieal flui is incompressible. Hence the Euler equations for a homogeneous incompressible flow in D are u t + u u = 1 p + f, ρ u = 0, together with the bounary conition on D which is u n = 0. Remark 2 Often Euler s equation is quote with ensity ρ not present. Since ρ is constant ( p)/ρ (p/ρ), an we coul re-label the term p/ρ to be p. Equivalently, we coul simply normalize the uniform constant ensity ρ to unity. Example (sink or bath rain) As the water (of uniform ensity ρ) flows out through a hole at the bottom of a bath the resiual rotation is confine to a core of raius a, so that the water particles may be taken to move on horizontal circles with ( Ωr, r a, u θ = Ωa 2 r, r > a. As we have all observe when water runs out of a bath or sink, the free surface of the water irectly over the rain hole has a epression in it see Fig. 4. The question is, what is the form/shape of this free surface epression? p 0 z a r Fig. 4 Water raining from a bath. We know that the pressure at the free surface is uniform, it is atmospheric pressure, say P 0. We nee the Euler equations for a homogeneous incompressible flui in cylinrical coorinates (r, θ, z) with the velocity fiel u = (u r, u θ, u z). These are u r t + (u )ur u2 θ r = 1 p ρ r + fr, u θ t + (u )u θ + uru θ = 1 p r ρr θ + f θ, u z t + (u )uz = 1 p ρ z + fz,

14 14 Simon J.A. Malham where p = p(r, θ, z, t) is the pressure, ρ is the uniform constant ensity an f = (f r, f θ, f z) is the boy force per unit mass. Here we also have u = u r r + u θ r θ + uz z. Further the incompressibility conition u = 0 is given in cylinrical coorinates by 1 (ru r) + 1 u θ r r r θ + uz z = 0. Now we look at the setting we are presente with for this problem. Note the flow is steay an u r = u z = 0, f r = f θ = 0. The force ue to gravity implies f z = g. The whole problem is also symmetric with respect to θ, so that all partial erivatives with respect to θ shoul be zero. Combining all these facts reuces Euler s equations above to u2 θ r = 1 p ρ r, 0 = 1 ρr p θ an 0 = 1 p ρ z g. The incompressibility conition is satisfie trivially. The secon equation above tells us the pressure p is inepenent of θ, as we might have alreay suspecte. Hence we assume p = p(r, z) an focus on the first an thir equation above. Assume r a. Using that u θ = Ωr in the first equation we see that p r = ρω2 r p(r, z) = 1 2 ρω2 r 2 + C(z), where C(z) is an arbitrary function of z. If we then substitute this into the thir equation above we see that 1 p ρ z = g C (z) = ρg, an hence C(z) = ρgz + C 0 where C 0 is an arbitrary constant. Thus we now euce that the pressure function is given by p(r, z) = 1 2 ρω2 r 2 ρgz + C 0. At the free surface of the water, the pressure is constant atmospheric pressure P 0 an so if we substitute this into this expression for the pressure we see that P 0 = 1 2 ρω2 r 2 ρgz + C 0 z = (Ω 2 /2g) r 2 (C 0 P 0 )/ρg. Hence the epression in the free surface for r a is a parabolic surface of revolution. Note that pressure is only ever globally efine up to an aitive constant so we are at liberty to take C 0 = 0 or C 0 = P 0 if we like. For r > a a completely analogous argument using u θ = Ωa 2 /r shows that p(r, z) = ρω2 a 4 2 r 2 ρgz + K 0, where K 0 is an arbitrary constant. Since the pressure must be continuous at r = a, we substitute r = a into the expression for the pressure here for r > a an the expression for the pressure for r a, an equate the two. This gives 1 2 ρω2 a 2 ρgz + K 0 = 1 2 ρω2 a 2 ρgz K 0 = ρω 2 a 2.

15 Ieal flui mechanics 15 Hence the pressure for r > a is given by p(r, z) = ρω2 a 4 2 r 2 ρgz + ρω 2 a 2. Using that the pressure at the free surface is p(r, z) = P 0, we see that for r > a the free surface is given by z = Ω2 a 4 g r 2 + Ω2 a 2. g 8 Bernoulli s Theorem Theorem 2 (Bernoulli s Theorem) Suppose we have a homogeneous incompressible stationary flow with a conservative boy force f = φ, where φ is the potential function. Then the quantity H := 1 2 u 2 + p ρ + φ is constant along streamlines. Proof We nee the following ientity that can be foun in Appenix B: 1 2 ` u 2 = u u + u ( u). Since the flow is stationary, Euler s equation of motion imply p u u = φ. ρ Using the ientity above we see that 1 2 ` u 2 p u ( u) = φ ρ 12 u 2 + p ρ + φ = u ( u) H = u ( u), using the efinition for H given in the theorem. Now let x(s) be a streamline that satisfies x (s) = u`x(s). By the funamental theorem of calculus, for any s 1 an s 2, s2 H`x(s 2 ) H`x(s 1 ) = H`x(s) s 1 x(s2) = H x (s) s x(s 1) x(s2) = `u ( u) u`x(s) s x(s 1) = 0, where we use that (u a) u 0 for any vector a (since u a is orthogonal to u). Since s 1 an s 2 are arbitrary we euce that H oes change along streamlines.

16 16 Simon J.A. Malham z=0 P = air pressure 0 h z= h P0 U Typical streamline Fig. 5 Torricelli problem: the pressure at the top surface an outsie the puncture hole is atmospheric pressure P 0. Suppose the height of water above the puncture is h. The goal is to etermine how the velocity of water U out of the puncture hole varies with h. Example (Torricelli 1643). Consier the problem of an oil rum full of water that has a small hole puncture into it near the bottom. The problem is to etermine the velocity of the flui jetting out of the hole at the bottom an how that varies with the amount of water left in the tank the setup is shown in Fig 5. We shall assume the hole has a small cross-sectional area α. Suppose that the cross-sectional area of the rum, an therefore of the free surface (water surface) at z = 0, is A. We naturally assume A α. Since the rate at which the amount of water is ropping insie the rum must equal the rate at which water is leaving the rum through the puncture hole, we have h «A = U α. t We observe that if U is of orer one, an A α so that α/a 1, then h t = U (α/a) 1, an hence the velocity of the free surface is asymptotically small. Since the flow is quasi-stationary, incompressible as it s water, an there is conservative boy force ue to gravity, we apply Bernoulli s Theorem for one of the typical streamlines shown in Fig. 5. This implies that the quantity H is the same at the free surface an at the puncture hole outlet, hence «2 1 h 2 + P 0 t ρ = 1 2 U2 + P 0 ρ gh. If we ignore the term (h/t) 2, which is small, an cancel the P 0 /ρ terms then we can euce that U = p 2gh. Remark 3 Note the pressure insie the container at the puncture hole level is P 0 +ρgh. The ifference between this an the atmospheric pressure P 0 outsie, accelerates the water through the puncture hole.

17 Ieal flui mechanics 17 9 Vorticity The vorticity fiel of a flow with velocity fiel u is efine as ω = u. It encoes the magnitue of, an irection of the axis about which, the flui rotates, locally. Theorem 3 Suppose x R 3 an x + h R 3 is a nearby point. Then for any function u = u(x) we can always write u(x + h) = u(x) + D(x) h ω h + O(h2 ), where D(x) is a 3 3 symmetric matrix. Proof By Taylor expansion we have u(x + h) = u(x) + ` u(x) h + O(h 2 ), where ( u) h is simply matrix multiplication of the 3 3 matrix u by the column vector h. We can always write Set u = 1 2 `( u) + ( u) T `( u) ( u) T. D := 1 2 `( u) + ( u) T, R := 1 2 `( u) ( u) T. Note that D is symmetric. Direct computation reveals that ω 3 ω 2 R = 2 ω 3 0 ω 1 A ω 2 ω 1 0 where ω 1, ω 2 an ω 3 are the three components of ω = u, i.e. ω = (ω 1, ω 2, ω 3 ). Hence again by irect computation we see that R h = 1 2 ω h. The symmetric matrix D is the eformation tensor. Since it is symmetric, there is an orthonormal basis ê 1, ê 2, ê 3 in which D is iagonal, i.e. if X = [ê 1, ê 2, ê 3 ] then X 1 DX A

18 18 Simon J.A. Malham Now consier the motion of a flui particle labelle by x + h where x is fixe an h is small (for example suppose that only a short time has elapse). Then the position of the particle is given by (x + h) = u(x + h) t h = u(x + h) t h t u(x) + D(x) h ω(x) h. Let us consier in turn each of the effects on the right shown. The term u(x) is simply uniform translational velocity (the particle being pushe by the ambient flow surrouning it). Now consier the secon term D(x) h. If we ignore the other terms then, approximately, we have h = D(x) h. t Making a local change of coorinates so that h = Xĥ we get t 0 1 B ĥ ĥ 1 ĥ 2 A A B ĥ 2 A. ĥ ĥ 3 We see that we have pure expansion or contraction (epening on whether i is positive or negative, respectively) in each of the characteristic irections ĥi, i = 1, 2, 3. Inee the small linearize volume element ĥ1ĥ2ĥ3 satisfies t (ĥ1ĥ2ĥ3) = ( )(ĥ1ĥ2ĥ3). Note that = Tr(D) = u. Let us now consier the effect of the thir term 2 1 ω h. Ignoring the other two terms we have h t = 1 2ω(x) h. Direct computation shows that h(t) = Φ(t, ω(x))h(0), where Φ(t, ω(x)) is the matrix that represents the rotation through an angle t about the axis ω(x). Note also that `ω(x) h = 0. Definition 5 (Circulation) Let C be a simple close contour in the flui at time t = 0. Suppose that C is carrie along by the flow to the close contour C t at time t, i.e. C t = ϕ t (C). The circulation aroun C t is efine to be the line integral I K = u r. C t Theorem 4 (Kelvin s circulation theorem) For incompressible flow without external forces, the circulation K for any close contour C t is constant in time.

19 Ieal flui mechanics Exercises Exercise (trajectories an streamlines) For a given velocity flow fiel u = u(x, t) prescribe at position x an time t, the particle trajectories are given by the solutions to the system of orinary ifferential equations x t = u(x(t), t). Streamlines are given by the solutions to the system of orinary ifferential equations (where t is fixe) x s = u`x(s), t. (a) Explain what particle trajectories an streamlines are, an their ifference. (b) For the two-imensional flow in Cartesian coorinates, (u, v) = `u 0, v 0 cos(kx αt), where u 0, v 0, k an α are constants, fin the general equation for a streamline. Show that the streamline passing through (x, y) = (0, 0) at t = 0 is y = v 0 ku 0 sin(kx). Fin the equation for the path of the particle which is at (x, y) = (0, 0) at t = 0. Comment briefly on the contrast between the above streamline an particle path in the two separate limiting cases: first α 0; secon k 0. Exercise (channel flow) Consier the two-imensional channel flow (with U a given constant) «! u = 0, U 1 x2 a 2, 0, between the two walls x = ±a. Show that there is a stream function an fin it. (Hint: a stream function ψ exists for a velocity fiel u = (u, v, w) when u = 0 an we have an aitional symmetry. Here the aitional symmetry is uniformity with respect to z. You thus nee to verify that if u = ψ y an v = ψ x, then u = 0 an then solve this system of equations to fin ψ.) Show that approximately 91% of the volume flux across y = y 0 for some constant y 0 flows through the central part of the channel x 3 4 a. Exercise (Couette flow) Let Ω be the region between two concentric cyliners of raii R 1 an R 2, where R 1 < R 2. Suppose the velocity fiel u = (u r, u θ, u z) of the flui flow insie Ω, in cylinrical coorinates, is given by u r = 0, u z = 0, an u θ = A r + Br, where Show that the: A = R2 1R 2 2(ω 2 ω 1 ) R 2 2 R2 1 an B = R2 1ω 1 R2ω 2 2 R2 2. R2 1

20 20 Simon J.A. Malham (a) velocity fiel u = (u r, u θ, u z) is a stationary solution of Euler s equations of motion for an ieal flui with ensity ρ 1 (hint: you nee to fin a pressure fiel p that is consistent with the velocity fiel given); (b) angular velocity of the flow on the two cyliners is ω 1 an ω 2. Exercise (Venturi tube) Consier the Venturi tube shown in Fig. 6. Assume that the ieal flui flow through the construction is homogeneous, incompressible an steay. The flow in the wier section of cross-sectional area A 1, has velocity u 1 an pressure p 1, while that in the narrower section of cross-sectional area A 2, has velocity u 2 an pressure p 2. Separately within the uniform wie an narrow sections, we assume the velocity an pressure are uniform themselves. (a) Why oes the relation A 1 u 1 = A 2 u 2 hol? Why is the flow faster in the narrower region of the tube compare to the wier region of the tube? (b) Use Bernoulli s theorem to show that 1 2 u p 1 ρ 0 = 1 2 u2 2 + p 2 ρ 0, where ρ 0 is the constant uniform ensity of the flui. (c) Using the results in parts (a) an (b), compare the pressure in the narrow an wie regions of the tube. () Give a practical application where the principles of the Venturi tube is use or might be useful. A, u, p A, u, p A, u, p streamline Fig. 6 Venturi tube: the flow in the wier section of cross-sectional area A 1 has velocity u 1 an pressure p 1, while that in the narrower section of cross-sectional area A 2 has velocity u 2 an pressure p 2. Separately within the uniform wie an narrow sections, we assume the velocity an pressure are uniform themselves. Exercise (hurricane) We evise a simple moel for a hurricane. (a) Using the Euler equations for an ieal incompressible flow in cylinrical coorinates (see the bath or sink rain problem in the main text) show that at position (r, θ, z), for a flow which is inepenent of θ with u r = u z = 0, we have u 2 θ r = 1 ρ 0 p r, 0 = 1 ρ 0 p z + g,

21 Ieal flui mechanics 21 where p = p(r, z) is the pressure an g is the acceleration ue to gravity (assume this to be the boy force per unit mass). Verify that any such flow is inee incompressible. (b) In a simple moel for a hurricane the air is taken to have uniform constant ensity ρ 0 an each flui particle traverses a horizontal circle whose centre is on the fixe vertical z-axis. The (angular) spee u θ at a istance r from the axis is u θ = where Ω an a are known constants. ( Ωr, for 0 r a, Ω a3/2 r 1/2, for r > a, (i) Now consier the flow given above in the inner region 0 r a. Using the equations in part (a) above, show that the pressure in this region is given by p = P ρ 0Ω 2 r 2 gρ 0 z, where P 0 is a constant. A free surface of the flui is one for which the pressure is constant. Show that the shape of a free surface for 0 r a is a paraboloi of revolution, i.e. it has the form z = Ar 2 + B, for some constants A an B. Specify the exact form of A an B. (ii) Now consier the flow given above in the outer region r > a. Again using the equations in part (a) above, an that the pressure must be continuous at r = a, show that the pressure in this region is given by p = P 0 ρ 0 r Ω2 a 3 gρ 0 z ρ 0Ω 2 a 2, where P 0 is the same constant (reference pressure) as that in part (i) above. Exercise (Clepsyra or water clock) A clepsyra has the form of a surface of revolution containing water an the level of the free surface of the water falls at a constant rate as the water flows out through a small hole in the base. The basic setup is shown in Fig. 7. Show that the container must have the form z = C r 4 in cylinrical polars, where C is a constant. z 0 r Fig. 7 Clepsyra (water clock).

22 22 Simon J.A. Malham Exercise (rigi boy rotation) An ieal flui of constant uniform ensity ρ 0 is containe within a fixe right-circular cyliner (with symmetry axis the z-axis). The flui moves uner the influence of a boy force fiel f = (αx + βy, γx + δy, 0) per unit mass, where α, β, γ an δ are inepenent of the space coorinates. Use Euler s equations of motion to show that a rigi boy rotation of the flui about the z-axis, with angular velocity ω(t) given by ω = 2 1 (γ β) is a possible solution of the equation an bounary conitions. Show that the pressure is given by p = p ρ 0`(ω 2 + α)x 2 + (β + γ)xy + (ω 2 + δ)y 2, where p 0 is the pressure at the origin. Exercise (evolution of vorticity) By taking the curl of Euler s equation for an ieal, incompressible, homogeneous flui, subject to a conservative boy force, show that the vorticity ω satisfies ω + u ω = ω u. t 11 Notes 11.1 Streaklines A streakline is the locus of all the flui elements which at some time have past through a particular point, say (x 0, y 0, z 0 ). We can obtain the equation for a streakline through (x 0, y 0, z 0 ) by solving the equations x(t) = u(x(t), t), t assuming that at t = t 0 we have `x(t 0 ), y(t 0 ), z(t 0 ) = (x 0, y 0, z 0 ). Eliminating t 0 between the equations generates the streakline corresponing to (x 0, y 0, z 0 ). For example, ink ye injecte at the point (x 0, y 0, z 0 ) in the flow will trace out a streakline Isentropic flows A compressible flow is isentropic if there is a function π, calle the enthalpy, such that π = 1 ρ p. The Euler equations for an isentropic flow are thus u + u u = π + f t ρ + (ρu) = 0, t in D, an on D, u n = 0 (or matching normal velocities if the bounary is moving).

23 Ieal flui mechanics 23 For compressible ieal gas flow, the pressure is often proportional to ρ γ, for some constant γ 1, i.e. p = C ρ γ, for some constant C. This is a special case of an isentropic flow, an is an example of an equation of state. In fact we can actually compute π = ρ p (z) z z = γ C ργ γ 1, an the internal energy (see Chorin an Marsen, pages 14 an 15) ǫ = π (p/ρ) = C ργ γ 1. A Evolution of the flow-map Jacobian Our goal is to establish the following result for the Jacobian J(x, t) of the flow-map ϕ t: t J(x, t) = ` u(ϕ(x, t), t) J(x, t). Recall, for a fixe position x D we enote by ξ(x, t) = (ξ, η, ζ) the position of the particle at time t, which at time t = 0 was at x. We use ϕ t to enote the map x ξ(x, t), it is calle the flow-map. Hence by efinition J(x, t) = et` ξ(x, t). We know that a particle at position ξ(x, t) = `ξ(x, t), η(x, t), ζ(x, t), which starte at x at time t = 0, evolves accoring to Since x is fixe this is equivalent to t ξ(x, t) = u`ξ(x, t), t. t ξ(x, t) = u`ξ(x, t), t. Taking the graient with respect to x of this relation, an swapping over the graient an / t operations on the left, we see that Using the chain rule we have t ξ(x, t) = u`ξ(x, t), t. xu`ξ(x, t), t = ξ u`ξ(x, t), t ` xξ(x, t). Combining the last two relations we see that t ξ = ( ξu) ξ. Abel s Theorem then tells us that J = et ξ evolves accoring to t et ξ = `Tr( ξ u) et ξ. Since Tr( ξ u) u we have establishe the require result.

24 24 Simon J.A. Malham B Multivariable calculus ientities We provie here some useful multivariable calculus ientities. Here φ an ψ are generic scalars, an u an v are generic vectors u = i j k 1 0 w y / x / y / za B v 1 z u z w C x A. u v w v x u y 2. ( φ) = 2 φ = 2 φ x φ y φ z ( φ) ( u) ( u) = ( u) 2 u. 6. (φψ) = φ ψ + ψ φ. 7. (u v) = (u )v + (v )u + u ( v) + v ( u). 8. (φu) = φ( u) + u φ. 9. (u v) = v ( u) u ( v). 10. (φu) = φ u + φ u. 11. (u v) = u( v) v( u) + (v )u (u )v. Acknowlegements These lecture notes have to a large extent grown out of a merging, of lectures on Ieal Flui Mechanics given by Dr. Frank Berkshire [2] in the Spring of 1989 at Imperial College, an the style an content of the excellent text by Chorin an Marsen [3]. They have also benefitte from lecture notes by Prof. Trevor Stuart [13] on Viscous Flui Mechanics an Prof. Frank Leppington [8] on Electromagnetism. References 1. Batchelor, G.K An introuction to flui mechanics, CUP. 2. Berkshire, F Lecture notes on ieal flui ynamics, Imperial College Mathematics Department. 3. Chorin, A.J. an Marsen, J.E A mathematical introucton to flui mechanics, Thir eition, Springer Verlag, New York. 4. Evans, L.C Partial ifferential equations Grauate Stuies in Mathematics, Volume 19, American Mathematical Society. 5. Keener, J.P Principles of applie mathematics: transformation an approximation, Perseus Books. 6. Krantz, S.G How to teach mathematics, Secon Eition, American Mathematical Society. 7. Lamb, H Hyroynamics, 6th Eition, CUP. 8. Leppington, F Electromagnetism, Imperial College Mathematics Department. 9. McCallum, W.G. et. al Multivariable calculus, Wiley. 10. Maja, A.J. an Bertozzi, A.L Vorticity an incompressible flow, Cambrige Texts in Applie Mathematics, Cambrige University Press. 11. Marsen, J.E. an Ratiu, T.S Introuction to mechanics an symmetry, Secon eition, Springer. 12. Saffman, P.G Vortex ynamics, Cambrige Monographs in Mechanics an Applie Mathematics, Cambrige University Press. 13. Stuart, J.T Lecture notes on Viscous Flui Mechanics, Imperial College Mathematics Department.