Lai Ch6 Meta

Lai Chapter 6 Meta Notes: Newtonian Viscous Fluids PhL 6.28.12 This is Lai's offering on the subject of fluids. Here is a little hierarchy fluid mechanics fluid statics fluid dynamics (before 1900 this whole field was called hydrodynamics, and this name lingers) aerodynamics (air and other gases in motion) hydrodynamics (water and other liquids in motion) Wiki gives a list of 172 "popular" books on the subject of fluid mechanics!!! Amazing. I have attached that list at the end of this meta notes. Chapter is 57 pages, raw notes are 34 pages, these meta notes are 9 pages. 6.1 Fluids. (353) 2 6.2. Incompressible subclass of fluids (354). 2 6.3 Hydrostatics (355). 2 6.4 Newtonian Fluids (357). 2 6.5 Interpretation of λ and μ (358). 2 6.6 Incompressible Newtonian Flow (359). 3 6.7 Navier-Stokes for incompressible fluids (360). 3 6.8 Navier-Stokes for incompressible fluids in cylindricals and sphericals (364). 4 6.9 Boundary Conditions (365). 4 6.10 Streamline vs Pathline, Steady vs Unsteady, Laminar vs Turbulent (365). 4 6.11 Plane Couette Flow (368). 4 6.12 Plane Poiseuille Flow (368). 5 6.13 Hagen-Poiseuille Flow (371). 5 6.14 Plane Couette Flow with two layers, gravity on, no tilt (372). 5 6.15 Couette Flow (374). 5 6.16 Trying to drive a transverse wave into a NS fluid (375). 5 6.17 Dissipation functions for Newtonian fluids (376). 6 6.19 The Vorticity Vector (379). 6 6.20 Irrotational Flow (381). 6 6.21 Irrotational Flow for inviscid incompressible (381). 6 6.22 Irrotational Flow and Navier-Stokes (384). 7 6.23 The Vorticity Transport Equation (385). 7 6.24 Concept of a Boundary Layer (388). 7 6.25 7 Equations for a Compressible Newtonian fluid (389). 7 6.26. Energy equation in terms of enthalpy (390). 8 6.27. Waves in a compressible inviscid fluid (392). 8 6.28. Irrotational inviscid + (barotropic compressible) flow (395). 8 6.29. One dimensional compressible flow (398). 9 6.30. Compressible fluid through a nozzle at bottom of a big tank (399). 9 6.31. Steady laminar flow in an elastic tube (blood vessel) (401). 9 Wiki book list on fluid dynamics: 9 6.1 Fluids. (353) If there is no on-going deformation (Dij = 0) near a fluid cube, then Tij(x) = -p(x) δij. There are only normal forces, no shear forces. 6.2. Incompressible subclass of fluids (354). Incompressible means from continuity that div v = 0 and ρ = constant. Later we see that this means Δ = tr(D) = 0 as well. 6.3 Hydrostatics (355). General Cauchy EOM says ∂jTij + ρBi = ρai, so when Tij = -pδij get -∂ip+ ρBi = ρai. Consideration of the static situation leads to p = ρgh + constant (incompressible). If compressible, you need some p = p(ρ) type connection (such as Boyle or adiabatic) later called barotropic. Cylinder floating on bottom-tether in a pool leads to Archimedes. Glass accelerating to the right has tipped surface. Notion of "static" situations like this where motion is "rigid" and Dij = 0. 6.4 Newtonian Fluids (357). Newtonian fluid assumes first isotropic, second Tij = -pδij + T'ij where T'ij = QijabDab, and then isotropic tensor theory tells you that T'ij = λΔδij+ 2μDij, so the Newtonian model is this Tij = -pδij + λΔδij+ 2μDij Δ = tr(D) // = 0 when incompressible Δ = the rate of dilatation Recall Dij = (v)Sij (rate of deformation tensor) and Wij = (v)Aij (spin tensor). I show why this second object does not appear in Tij. Note that λ and μ are unrelated to elastic solid parameters of the same names. 6.5 Interpretation of λ and μ (358). In a river where 1 is the flow direction and 2 across, find T12 = μ ∂2v1 shear force. There is a rate of deformation here D12 = ∂2v1/2, so there is a shear force. The meaning of μ is that large μ means large shear force for a given D12 and we interpret this as viscosity. As for the meaning of λ, we learn that (average of normal T'ii elements) = (λ+2μ/3) Δ = k Δ k = bulk viscosity So consider <T'ii> = kΔ and then Δ = <T'ii>/k. If you have some set force <T'ii> crushing in or pulling out, then large k means the rate of dilatation is small, it is "gooey" in a bulk sense. So the name bulk viscosity is reasonable. We don't have a separate interpretation for λ outside of this combination. The case k = 0 does occur, meaning the material is extremely squashable so any <T'ii> gives a huge Δ rate. This case is called the Stokes Assumption. 6.6 Incompressible Newtonian Flow (359). If your Newtonian fluid is incompressible, then Δ = 0 and then Tij = -pδij + 2μDij is the simplified constitutive equation (this is Hooke's law in the elastic theory). Since div v = 0 as well, it is easy to show that (divT) = -p + μ2v where (divT)i = ∂jTij. So these two terms are going to appear in the equation of motion for this kind of fluid. 6.7 Navier-Stokes for incompressible fluids (360). Put the above in the Cauchy EOM and out comes Navier-Stokes for an incompressible Newtonian fluid Navier-Stokes incompressible ρ [ ∂tv + (v)v ] = ρB -p + μ2v and div v = 0 More generally, the last term should be written μ (divT) where as usual (divT)i = ∂jTij. In the general case we have Tij = -pδij + λΔδij + 2μDij and then ∂jTij = -∂ip +λ∂iΔ + 2μ∂jDij. But in this last term we have 2∂jDij = ∂j(∂ivj + ∂jvi) = 2vi + ∂i (div v) and then NS says ρ [ ∂tv + (v)v ] = ρB -p + μ(divT) ρ [ ∂tv + (v)v ] = ρB -p + μ2v + μ (div v) + λ [ tr(D)] But we also know that tr(D) = Dii = ∂ivi = div v, so we can combine the last two terms ρ [ ∂tv + (v)v ] = ρB -p + μ2v + (μ + λ) (div v) and further if we write k = λ + 2μ/3 then (μ + λ) = μ + (k -2μ/3) = μ/3 + k, and then ρ [ ∂tv + (v)v ] = ρB -p + μ2v + (μ/3 + k) (div v) so the above is NS for a compressible fluid, and this is the way you see things in wiki http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations#Derivation_and_description So we now have to use the term "inviscid" to mean that μ=0 and k = 0, which means λ = 0, and then even when div v ≠0, we still recover our old NS form ρ [ ∂tv + (v)v ] = ρB -p + μ2v Even the simpler compressible NS equation shown just above is non-linear due to the (v)v term. It seems to have "heat conduction equation" aspects to it since ∂t and 2 appear, but the variable is not scalar temperature, it is vector v. You sort of have diffusion of v. Since no ∂t2, it seems distant from the wave equation, but somehow waves will appear below. The longitudinal waves require ρ ≠ constant I think, and transverse ones die out quickly. First example is for uniform flow where have only v1(x2,x3) and v1 constant along flow axis. Dij= 0 and T'ij = 0 and nothing but Tij= -pδij. Second example has a river going downhill and defines the so called piezometric head h = p/(ρg) + z , but this had no real meaning for me. Need to see more. Third example seems to justify the intuitive ρgh idea you would apply at any point in any body of water that is not deforming. 6.8 Navier-Stokes for incompressible fluids in cylindricals and sphericals (364). Lai writes NS and div v = 0 in Cylindricals and Sphericals. I am totally happy with this. 6.9 Boundary Conditions (365). We learn about the non-slip condition that seems valid for almost all kinds of fluids. All three components of v = 0 at a hard boundary. This is the BC set for a problem, seems Dirichlet like. 6.10 Streamline vs Pathline, Steady vs Unsteady, Laminar vs Turbulent (365). We learn here the meaning of a streamline as opposed to a pathline, the first being a strobe photo thing, the second you put in a little boat. Flow can be steady or unsteady, which must means steady or changing in time. The final distinction is between laminar flow and turbulent flow. Laminar means adjacent layers don't mix at all. We may later learn that things are laminar when the Reynolds number is less than 2100, but for now we don't even know what this number even means. 6.11 Plane Couette Flow (368). Plane Couette Flow. Think of an infinitely deep river with vertical walls. One wall is at rest, the other moves at v0, wall spacing is d. This gives the simplest of all possibly cases of varying v -- linear variation p 368 picture. No gravity. My notes talk about this as a vertical situation instead where the top of the river (infinitely wide) has a plate dragging it along. There is shear force between the layers, you get D12 = (v0/2d) and T12 = 2μD12 = μ (v0/d). The two walls exert shear force on the river touching them. This very prototype of cases shows Lai's general method. We make an ansatz about a few things (in this case that p = constant along the flow, for example) and then we show that we have a self-consistent situation which solves both NS and continuity. This then generates the prototype, and luckily we have nice names to glue to the prototypes. 6.12 Plane Poiseuille Flow (368). Now both walls are at rest, separation 2b, we no longer get a linear variation for v, it is in fact a simple quadratic curve shown p 369. So think of this as an infinitely deep river no gravity and both side walls are at rest (both dragging on the water). One ansatz here is a constant driving force ∂1p to make the flow go! So in this river, p drops as you go downstream, but at any point the flow Q is the same and he computes it. He then tips the river at an angle and deduces some facts concerning pressure. // pwa-zoy'-ee 6.13 Hagen-Poiseuille Flow (371). Whereas plane is the river situation with duct-like bed, this HP thing is flow in a rigid tube and again you expect some quadratic v profile and that is what you get. Q is again computed. I do a calculation here regarding water in my PVC pipe sprinkler system and get a fairly reasonable answer. We are using incompressible Newtonian N-S all the time in these examples. Flow rate Q is proportional to ∂zp and d2 for a pipe. If you have the same end pressures and double the length of a pipe, ∂zp drops in half and Q drops in half, so hose flow drops as hose gets long! The dependence on diameter is quadratic as you would expect (area). I did not study the nozzle effect, maybe in the problems. I had to get actual μ for water. 6.14 Plane Couette Flow with two layers, gravity on, no tilt (372). Lai solves this, using T12 continuous at the boundary and p as well and v as well (no slip between layers!). A bit of fiddling was needed here. 6.15 Couette Flow (374). This is a rather strange apparatus and you don't just get a linear velocity ramp as in the plane case. You have two cylinders rotating at different rates with the fluid between them! Lai computes everything. Note on naming flows: Lai wiki plane Couette Couette Couette Taylor-Couette. Couette was 1890 or so, Taylor 1923 actually did those cylinders. There was a 1999 paper in fact accounting for effects of the ends of finite cylinders Not surprisingly, Bessel functions appear. 6.16 Trying to drive a transverse wave into a NS fluid (375). But there is no wave equation, it is a diffusion equation as shown p 375 6.16.2. If you shear-wiggle a plane boundary, the effect just diffuses into the fluid and quickly dies away. An interesting situation to consider I think. 6.17 Dissipation functions for Newtonian fluids (376). Lai shows that for an incompressible fluid we find that the stress power (per unit volume) is Ps = 2μ tr(D2) ≡ Φinc. This is the rate at which power is converted to heat due to the viscosity. For a compressible fluid, there is an extra term here, Φ = Φinc + λ (trD)2, and then Ps = Φ - p tr(D) for a second extra term. Somehow this means that work goes not just into heat but also into pressure storage (last term). An example is given and I conclude that for plane Couette action a cubic meter of water would only dissipate 1 mW. This is where I concluded that μ for water is "low". 6.18. Energy Equation for Newtonian fluids (378). We did this earlier in the general continuum case and Tij appears in the energy conservation equation (6.18.1). Also appearing are energy density u and flow rate of heat div q and also possible heat sources. If you assume only Fourier heat conduction flow, get 6.18.2. Then if you can say that u = u(Θ) = cΘ, you end up with a PDE in temperature Θ which is 6.18.3. Note that earlier Ps = Tij∂jvi = 2μ tr(D2) and this same term appears in the Θ equation and that is why you see Φinc sitting in there. If you then neglect the viscous energy loss, you end up with a simple diffusion equation for heat flow in a fluid where then the constant is a combination of the Fourier κ and ρ and c. So here is a derivation of the heat equation in one circumstance, an equation Stakgold studied in amazingly great detail. In an example, Lai then finds the Θ distribution in plane Couette flow where your two planes are held at certain fixed temperatures. This example is interesting for several reasons: (1) it is a combination of the NS problem for velocity, and the energy equation for Θ; (2) if you ignore μ losses, the plot is linear; (3) this is justified if a lot of heat is flowing through the fluid from the plates. 6.19 The Vorticity Vector (379). We first learn that if you go into the principal axes system, then the three normal vectors in that system rotate together (George Shearing) according to spin tensor W. In this system, the sugar cube does not shear, though it can scale its edges and rotate as just stated. Next, we learn that if you call ω the vector associated with the spin tensor W (going way back on this), then in fact you find that 2ω = curl v. For irrotational flow, ω = 0 and W = 0 and a cube in the principle axis system neither rotates nor shears, but only stretches along its axes! The axes n are fixed. The vorticity vector is defined as ζ = 2ω = curl v. Lai computes curl v in our various prototype situations. 6.20 Irrotational Flow (381). If you want curl v = 0 and div v = 0 (incompressible), you can say v = -φ and you automatically satisfy the curl thing and the div thing becomes Laplace. So here is the origin of "potential theory" in fluid dynamics. 6.21 Irrotational Flow for inviscid incompressible (381). For Lai, the term inviscid means both μ = 0 and incompressible (Δ=0), and so for Lai, inviscid Tij = -pδij. Our NS equation was only stated by Lai for incompressible, so we just set μ = 0 in the NS and the simpler resulting equations are called the Euler Equations of Motion 6.21.2. Now, if you also have B = -Ω , this Euler equation becomes 6.21.6. Finally, if in addition to the above the flow is irrotational, Euler then becomes Bernoulli's equations of motion: -∂tφ + v2/2 + p/ρ + Ω = f(t) v = -φ v2/2 + p/ρ + Ω = constant // for steady flow Ω = gy for gravity. So again, Bernoulli is for an inviscid ( incompressible) Newtonian fluid which is doing irrotational flow. The last equation relates p,ρ,v at different points in a steady flow fluid. Lai notes that when μ = 0, you can violate the no-slip condition. Two examples are given, the first academic, the second is the hole in can problem. Here you end up with the famous Torricelli's Formula v2 = 2gh, as if the water were dropped from h. I find a table that shows that μ for water is quite low compared to kerosene or oil, etc, so treating it as inviscid is probably OK in many circumstances. 6.22 Irrotational Flow and Navier-Stokes (384). When μ ≠ 0, for irrotational flow 2v = 0 as is trivially shown raw notes. So you still get the Euler equations when μ ≠ 0! But: the problem is that in this case, the non-slip condition is in force and you then get both Dirichlet and Neumann BC's on φ for every problem, so there is really no solution except in contrived cases. The example here shows the interesting fact that in Couette flow, there is a setting for the two rotation rates that causes curl v = 0 in the fluid, so you can have irrotational in this special case. This has nothing to do with the observation above. 6.23 The Vorticity Transport Equation (385). If you just define ζ = curl v , Lai and I separately show that the NS and div v = 0 can be written in terms of ζ as Dtζ = (v)ζ + (μ/ρ)2ζ p 386 in the case of incompressible. B can be present if in its potential form Ω, and as you see, Ω does not even show in this vorticity equation. In the example of 2D rotational flow, Lai shows that ζ ≡ ζ3 solves the diffusion equation, so you can think of "vorticity" as something that diffuses away from a source, like heat. The source is usually a boundary where the no-slip condition creates vorticity and then this created substance diffuses away from the boundary. If it stays close to the boundary only, you get a boundary layer. 6.24 Concept of a Boundary Layer (388). See previous section. You tend to get the layer if the flow is fast on a wing, so things are fairly laminar except right at the surface of the wing where v = 0. Lift is not covered by Lai! 6.25 7 Equations for a Compressible Newtonian fluid (389). For general analysis, there are 7 equations and 7 variables, v,p,ρ,u,Θ . The 7 equations are (1,2,3) NS, which counts as three; (4) continuity; (5) energy conservation; (6) equation of state relating p and ρ; (7) equation relating stored energy u to temperature Θ. So you have a tough problem to solve in the fully general case where everything moves. 6.26. Energy equation in terms of enthalpy (390). Enthalpy is the thermodynamic "energy" thing (potential) which is conserved in a constant S and constant p process (H = U + pV, dH = TdS - Vdp). In the fluid context, this thing is written h = u + p/ρ and if you add in KE v2/2 you get h0, the stagnation enthalpy. All this Lai stuff is per unit mass so think δm = 1 and so ρ = 1/δV. You can then write the energy equation from the above equation list as 6.26.3, in which energy u is replaced by energy h0 and we find this h0 is then driven by T'ij. If you have a steady inviscid fluid situation (T'=0), this becomes ho = constant, and you can solve problems that way. If you also have ideal gas type equations, you get 6.26.9 for h0 which is constant. But no example problems are solved using this result. 6.27. Waves in a compressible inviscid fluid (392). Lai obtains the wave equation for longitudinal sound waves in an inviscid compressible fluid μ=0,k=0,λ=0. He starts with NS-for-inviscid (which recall works for compressible or incompressible inviscid fluids) and throws out quadratic terms in smallness, so waves are valid when they have small amplitude. The wave speed comes out being c02 = (dp/dρ)0 . In the usual gas case of adiabatic (aka isentropic) expansion, p = βργ and then you get c2 = γp/ρ. [ Ie, this is the equation of state describing p(ρ).] What about sound in water? On page 218 we did longitudinal waves in an elastic solid and we found there that cL2 ~ k/ρ0 where k is the solid's bulk modulus. A similar equation holds for hard-to-compress fluids, but Lai never derives it. This site http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html claims that this is in fact the formula for water, and k is a large number so we get cL = 1500 m/sec. Speed of sound in air is about 350 m/sec, and in steel 4500 m/sec. In first example, we learn that small amplitude rule means ε << ρ0co2 = γpo. The second example does head on reflections and shows that ρcL is the equivalent of the index of refraction (fluid impedance). 6.28. Irrotational inviscid + (barotropic compressible) flow (395). Here we derive Bernoulli's equations for the compressible case (previous shot was incompressible only). We end up with p/ρ replaced by∫dp/ρ as in 6.28.7,8. If you have p = p(ρ), that is called barotropic, and adiabatic gas law is an example where p(ρ) = βργ, the isentropic deal. In this case we get 6.28.10 which relates p, ρ and v (and γ). In the example we compare two points in a gas system. At the p0 point we have zero velocity v0 = 0 and ρ0. At the other point we have p, v, ρ. We can then use 6.28.10 being equal at these two locations to solve for (p0/p) as in 6.28.11, no big deal. What is interesting is that the ratio v/c appears for the first time, where c is that adiabatic sound speed c2 = γp/ρ at the p,v,ρ location. This really does remind me of special relativity. This ratio is the Mach number v/c! In the second exercise, we get a simplified version of 6.28.11 (namely, 6.28.12) by assuming M << 1. 6.29. One dimensional compressible flow (398). Imagine flow through a duct whose cross section A varies in some manner. We assume that our v field is uniform across this cross section (perhaps ignore boundary layers at the duct surfaces). By using the fancier Bernoulli equation and mass conservation (ρAv = constant), we find (dA/A) = (dv/v)(M2-1) known as the Hugoniot equation. Normally you would think dA>0 would give dv < 0 as is the case for compressible and which is just intuitive. This is in fact true below Mach One! Above the opposite is true! If you increase A, v increases as well, very strange! 6.30. Compressible fluid through a nozzle at bottom of a big tank (399). The first section discusses a "convergent nozzle". We apply our new Bernoulli tool and solve for v2, the speed at the tank section 2 region 6.30.4. We then compute dm/dt at that same location as 6.30.6. We next show that when (p2/p1) = some critical value, v2 = c, the speed of sound. Now if we apply Hugoniot to the convergent shape of the nozzle, we conclude that since dA < 0 in (dA/A) = (dv2/v2)(M2-1). If we could lower p2, trying to make dv2 > 0 from our v2 formula, we get v2 > c or M > 1. This is a contradiction, so we have to conclude that v2 ≤ c at any point like section 2 in such a nozzle. The flow cannot go supersonic and we cannot have (p2/p1) < f(γ), there is a floor on the nozzle interior pressure. In effect we have two conditions for the nozzle (the first in order to get any flow at all! ) p2 ≥ pR => p2 ≥ max(pR, pc) pR = ambient pressure outside tank p2 ≥ pc Case 1: if max = pR the dm/dt is given by 6.30.9 Case 2: if max = pc the dm/dt is given by 6.30.10 which result is independent of pR Lai next considers a convergent nozzle followed by a divergent one as in the p 401 picture. This thing has a throat. We know that to the left of this throat, we are subsonic. If sonic exactly at the throat, "choked flow". Lai gives a qualitative description of what might happen as you vary pR in such a way that you hit this choke point and then "go beyond". You end up with a pressure discontinuity which is a shock front, but we have never really studied such things, so he is just commenting. Thinking flight 477 Nova and the shocks coming off the wings when stalling. 6.31. Steady laminar flow in an elastic tube (blood vessel) (401). In a completely systematic manner, Lai obtains an expression for the flow Q of fluid through an elastic "artery" tube as a function of the pressures at the two ends. Along the way he finds r(p), the radius at some point along the tube, as in 6.31.7. If EY for the tubing is large enough, you can approximate this as in 6.31.10 and then the flow rate Q simplifies to 6.31.12. This problem combines elastic solid theory and compressible flow fluid theory! Wiki book list on fluid dynamics: