Additional file 1: Figure S1. Particle size distributions of milled wheat straw. (a) [log scale for particle size] and (b) [linear scale for particle size]: 3 mm straw freeze milled for 1 (green circle), 2 (blue square) and 3 (red triangle) minutes at 8 cycles per second. (c) [log scale for particle size] and (d) [linear scale for particle size]: < 1 mm straw (blue circle) and after freeze milling for 1 min at 8 cycles per second (green triangle). (a) [log scale for particle size] and (b) [linear scale for particle size]: 3.6 g samples of < 0.25 mm straw before and after being pre-frozen and then subject to 5 × 3 min freeze milling cycles with an impact rate of 8 cycles per second

Andrea L. Harper (4486165)

David Wilson (57684)

Graham Moates (3466112)

Ian Bancroft (55072)

Keith Waldron (3417029)

Samuel Collins (3466118)

FigShare

MOESM1 of Variation across a wheat genetic diversity panel for saccharification of hydrothermally pretreated straw

BACKGROUND: Wheat straw forms an important, reliable source of lignocellulosic biomass for use in second-generation ethanol production. However, there is limited understanding of the variation in quality of straw from current breeding cultivars, and studies on such variation have generally employed suboptimal pretreatments. There is also a degree of confusion regarding phenotypic characteristics relevant to optimising the enzymatic saccharification of cellulose after suitable pretreatments for biorefining compared with those which determine good ruminant digestibility. The aim of this study has been to (a) evaluate and compare the levels of glucose enzymatically released from straw obtained from 89 cultivars of winter wheat after optimised hydrothermal pretreatments and (b) identify the underlying phenotypic characteristics relevant to enhanced glucose production with special reference to the ratios of constituent tissue types. RESULTS: Optimised pretreatment involved hydrothermal extraction at 210 °C for 10 min. Using excess cellulases, quantitative saccharification was achieved within 24 h. The amount of glucose released ranged from 192 to 275 mg/g. The extent of glucose release was correlated with (a) the level of internode tissue (R = 0.498; p = 6.84 × 10-7), (b) stem height (R = 0.491; p = 1.03 × 10-6), and (c) chemical characteristics particular to stem tissues including higher levels of cellulose (R = 0.552; p = 2.06 × 10-8) and higher levels of lignin R = 0.494; p = 8.67 × 10-7. CONCLUSIONS: In order to achieve maximum yields of cellulosic glucose for second-generation ethanol production, a predisposition for wheat to produce cellulose-enriched internode stem tissue, particularly of longer length, would be beneficial. This contrasts with the ideotype for ruminant nutrition, in which an increased proportion of leaf tissue is preferable

Collins, Samuel R A

Wilson, David R.

Moates, Graham K

Harper, Andrea L

Bancroft, Ian

Waldron, Keith W.

White Rose Research Online

Variation across a wheat genetic diversity panel for saccharification of hydrothermally pretreated straw

Samuel R. A. Collins

David R. Wilson

Graham K. Moates

Andrea L. Harper

Ian Bancroft

Keith W. Waldron

Crossref

Biotechnology for Biofuels

Abstract Background Wheat straw forms an important, reliable source of lignocellulosic biomass for use in second-generation ethanol production. However, there is limited understanding of the variation in quality of straw from current breeding cultivars, and studies on such variation have generally employed suboptimal pretreatments. There is also a degree of confusion regarding phenotypic characteristics relevant to optimising the enzymatic saccharification of cellulose after suitable pretreatments for biorefining compared with those which determine good ruminant digestibility. The aim of this study has been to (a) evaluate and compare the levels of glucose enzymatically released from straw obtained from 89 cultivars of winter wheat after optimised hydrothermal pretreatments and (b) identify the underlying phenotypic characteristics relevant to enhanced glucose production with special reference to the ratios of constituent tissue types. Results Optimised pretreatment involved hydrothermal extraction at 210 °C for 10 min. Using excess cellulases, quantitative saccharification was achieved within 24 h. The amount of glucose released ranged from 192 to 275 mg/g. The extent of glucose release was correlated with (a) the level of internode tissue (R = 0.498; p = 6.84 × 10−7), (b) stem height (R = 0.491; p = 1.03 × 10−6), and (c) chemical characteristics particular to stem tissues including higher levels of cellulose (R = 0.552; p = 2.06 × 10−8) and higher levels of lignin R = 0.494; p = 8.67 × 10−7. Conclusions In order to achieve maximum yields of cellulosic glucose for second-generation ethanol production, a predisposition for wheat to produce cellulose-enriched internode stem tissue, particularly of longer length, would be beneficial. This contrasts with the ideotype for ruminant nutrition, in which an increased proportion of leaf tissue is preferable

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Nakaki, Tatsuyuki

Kyoto University Research Information Repository

TitleNumerical Computation to the Advection in the Field of SomePoint Vortices (Numerical Solution of Partial DifferentialEquations and Related Topics)Author(s)Nakaki, TatsuyukiCitation数理解析研究所講究録 (2000), 1145: 145-152Issue Date2000-04URL http://hdl.handle.net/2433/63939RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityNumerical Computation to the Advectionin the Field of Some Point VorticesTatsuyuki Nakaki ( ) $*$Graduate School of MathematicsKyushu University, Fukuoka 812-8581 Japan1 IntroductionThe analysis of the motion of point vortices is a classical problem, which is treated insome detail in many textbooks on fluid mechanics. The point vortex is defined on the twodimensional perfect fluid occupying the whole of a plane. When several vortices are inthe fluid. an interaction occurs: every vortex drifts away with the flow due to the othervortices. Such a phenomenon is described by the ordinary differential equation$\frac{d}{dt}\overline{\sim_{j}\vee(t)}=k\sum_{j\neq}.\frac{\Gamma_{k}}{2r_{1}i}\frac{1}{\approx_{j}(t)-z_{k}(t)}$ , $j=1.2,$ $\ldots,$ $n$ , (1)where $n$ is a number of vortices. $\approx_{j}(t)$ is the complex position of the $j\mathrm{t}\mathrm{h}$ point vortex ofstrength $\Gamma_{j}$ . and $i=\sqrt{-1}$. Since the strength of vortex in the perfect fluid is constant intime $t$ . we treat $\Gamma_{j}\mathrm{a}_{\iota}\mathrm{s}$ a given constant.We summarize some results on the solution of (1). When $n=2$ . it is easy to deter-mine the motion of two point vortices; see textbooks on fluid mechanics (Marchioro andPulvirenti [5]. for example). A qualitative analysis of the motion of three point vortices isdone by Aref [1]. When $n\geq 4$ . (1) is not solved yet in genera] cases. However some specia1cases are investigated. For examples. Morikawa and Swenson [6] analyze the case whereidentical point vortices are on the vertices of a regular polygon. Chaotic motion of fourpoint vortices is investigated by Aref and Pomphrey [2]. Nakaki [7] shows that five pointvortices. which satisfy$(z_{j}(0).\Gamma_{j})=\{$$( \exp\frac{j-1}{2}.\pi i.1)$ . $j=1,2.3.4$ ;(0.-1.5). $j=5$ . (2)exhibit relaxation oscillation (see Section 2).*email: $\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{k}\mathrm{i}\Phi \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{k}\mathrm{y}\mathrm{u}\mathrm{s}\mathrm{h}\mathrm{u}-\mathrm{u}.\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}$1145 2000 145-152 145In this paper, taking (2) into considerations, we treat five point vortices such that$z_{1}(0)= \frac{\alpha+i}{2},$ $z_{2}(0)= \frac{-\alpha+i}{2}.,$ $z_{3}(0)= \frac{-\alpha-i}{2},$ $z_{4}(0)= \frac{-\alpha-i}{2}$ . $z_{5}(0)=0$ (3)and$\Gamma_{j}=\{$1, $j=1.2.3,4$ ;$-\kappa$ . $j=5$ , (4)where $\alpha(\geq 1)$ and $\kappa\in R$ are parameters. When $\alpha=1$ and $\kappa=1.5$ . (3)$-(4)$ is consistentwith (2). Aref and $\mathrm{P}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{h}\mathrm{r}\mathrm{e}}}\mathrm{y}\mathrm{s}$ results correspond to $\alpha=1.75$ or $\alpha=2$ and $\kappa=0$ . forexample.Our goal in this paper is as follows: We show the existence of periodic or quasi-periodicmotion of the five vortices. Some periodic motions are demonstrated by numerical sim-ulations. Following Aref [3], Babiano et al. [4] and Neufeld and T\’el [8]. we analyze thedynamics of passively advected particles in the field of the perio.dic point vortices fromnumerical points of view.2 Relaxation oscillationIn this section, we describe the motion of five vortices when the parameters in (3)$-(4)$ are$\alpha=1$ and $\kappa\in R$ . When $\kappa=1.5$ . this problem is analyzed by [7].In this problem. it is easily found that$z_{j}(t)=z_{j}(0)\exp(i\Omega t)$ , $t\in R$ (5)is a trivial solution of (1), where$\Omega=\frac{\kappa-1.5}{2\pi}$ (6)is the angular velocity of the trivial solution. By the standard linearized analysis. thetrivial solution is unstable for $\kappa>0.5,\cdot$ and the real part of linearized eigenvalues are allzero for $\kappa<0.5$ . For the latter case. we haveTheorem 1. Let $\kappa<0.5$ . Then the trivial solution (5) is stable against the $p$erturba-tion $Satisf_{\mathrm{J}^{\gamma ing}}$$z_{1}(t)=-z_{3}(t)$ . $z_{2}(t)=-z_{4}(t)$ and $z_{5}(t)=0$ $(t\in R)$ . (7)By our numerical simulations. the solution (5) with $\kappa<0.5$ seems to be stable against theperturbation which does not satisfy (7).In the rest of this section. we consider the case of $\kappa>0.5$ . that is. the trivial solutionis unstable in the linearized sense. First of all, we demonstrate the numerical simulation.Figure 1 shows the numerical solution when $\kappa=1.5$ . In this case. we have $\Omega=0$ .which means that the trivial solution is steady state in time. The numerical solution does146$\bullet(\mathrm{a})\nearrow_{\bullet}^{\bullet}\backslash _{\bullet}$$\backslash _{\bullet}/$$\bullet\bullet\underline{(\mathrm{b})}$$\bullet-\bullet|\bullet|$$\bullet(\mathrm{C})/_{\bullet}^{\bullet}\backslash _{\bullet}$$\lambda_{\bullet}/$Figure 1: The numerical solution when $\alpha=1$ and $\kappa=1.5$ . The relaxation osciuationoccurs. $(\mathrm{a}):0\leq t\leq 30$ . $(\mathrm{b}):53\leq t\leq 100$ , $(\mathrm{c}):123\leq t\leq 160$ .Figure 2: The time evolution of vortices between (a) and (b) in Figure 1.not move for a while, however. it suddenly begin to move toward another configuration(Figure 1: $(\mathrm{a})arrow(\mathrm{b})$ ). After taking some rest, the solution resumes moving $((\mathrm{b})arrow(\mathrm{c}))$ . Themotion of vortices from Figure 1 (a) to (b) is shown in Figure 2. The reason why such arelaxation oscillation occurs can be explained by the existence of a heteroclinic orbit whichconnects two steady state solutions. If the heteroclinic orbit exists, the solution of (1), theinitial configuration of which is close to a steady state solution, travels near the orbits.The motion of solution is slow when the solution stays near the steady state solution, andbecomes fast when it goes away from the steady state solution. By the influence of theheteroclinic orbit. the solution approaches to another steady state solution and the motionbecomes slow down. As a result, we observe the relaxation osciUation. For the $\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{S}\mathrm{t}.\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$of the heteroclinic orbits, we obtainTheorem 2. Let $\kappa>0.5$ , and define $w_{j}(t)=z(t)\exp(-i\Omega t)(j=1,2, \ldots , 5)$ . Thenthere exist a solution $\{z_{j}(t)\}$ of (1) and constants $\eta_{1}^{+},$ $\eta_{2}^{+}\in C$ which satisfy$w_{j}(\pm\infty)=\eta_{j}^{\pm}$ , $j=1,2,$ $\ldots,$ $5$ , (8)where$\eta_{1}^{-}=1$ , $\eta_{2}^{-}=i$ , $\eta_{j+2}^{\pm}=-\eta_{j}(\pm j=1,2)$ $\eta_{5}^{\pm}=0$ . (9)We note that $\{w_{j}(t)\}$ in Theorem 2 indicates the solution $\{z_{j}(t)\}$ observed in the corotatingcoordinate with angular velocity $\Omega$ . When $\kappa=1.5$ , we find that $\eta_{1}^{+}=\frac{1+i}{\sqrt{2}}$ and $\eta_{2}^{+}=\frac{1-i}{\sqrt{2}}$ ,147Figure 3: The numerical values of $\eta_{1}^{+}$ and $\eta_{2}^{+}$ in Theorem 2. Using the value of $\theta_{1}$ in thisfigure, we have $\eta_{1}^{+}=\exp i\theta_{1}$ and $\eta_{2}^{+}=\eta_{1}^{+}\exp(-\pi/2)$ .and the steady state solutions $\{\eta_{j}^{-}\}$ and $\{\eta_{j}^{+}\}$ correspond to (a) and (b) in Figure 1.respectively. For another value of $\kappa_{t}$. we do not know the values of $\eta_{1}^{+}$ and $\eta_{2}^{+}$ . however. wecan compute approximate values by numerical simulations (see Figure 3).3 Periodic and quasi-periodic motionLet us treat the case of $\alpha>1$ . In this case, Aref and Pomphrey [2] shows that the motionof vortices is chaotic for $\alpha=2$ and $\kappa=0$ . They also demonstrate the motion for $\alpha=1.75$and $\kappa=0$ (see Figure 4). The chaotic motion occurs because the (quasi-)periodic motionFigure 4: Time trajectory of the point vortex $z_{1}(t)$ with $\kappa=0$ . Aref and Pomphrey [2]demonstrate the cases of $\alpha=2$ (left) and $\alpha=1.75$ (right). In the former case. chaoticmotion occurs. A regular precession motion appears in the latter case.becomes unstable. In fact, the motion of numerical solution of (1) under the condition$z_{1}(t)=-z_{3}(t)$ . $z_{2}(t)=-z_{4}(t)$ and $z_{5}(t)=0$ $(t\in R)$ (10)is not chaotic as shown in Figure 5. The existence of periodic or quasi-periodic motion isshown by the following.Theorem 3. Let $\alpha>1$ and $\kappa\in R.$ Then there exists a periodic or $q\mathrm{u}$asi-periodicsolution of (1), (3) and (4).148Figure 5: Numerical solution with $\alpha=2$ and $\kappa=0$ under the condition (10). The motionis not chaotic anymore.(a) (b) (c) (d)(e) (f) (g) (h)$0$Figure 6: The time evolution of the point vortex $z_{1}(t)$ with $\alpha=2$ . $(\mathrm{a}):\kappa=0.3506\cdots$ ,$(\mathrm{b}):t\iota,$ $=0.4051\cdots$ , $(\mathrm{c}):\kappa=0.5$ . $(\mathrm{d}):\kappa=0.5817\cdot\cdot,1$ $(\mathrm{e}):\kappa=1.036\cdots$ , $(\mathrm{f}):\kappa=$$1.571\cdots$ . $(\mathrm{g}):\kappa=1.705\cdots.$, $(\mathrm{h}):\kappa’=1.923\cdots$ .The stability of the solution is not proved yet. The numerical simulations in Figures 4 and 5suggest that the solution with $\alpha=2$ and $\kappa=0$ is unstable. however, is stable against theperturbation satisfying (10).Next let us show numerical simulations with $\kappa\neq 0$ . The aspect ratio $\alpha$ is taken to $\alpha=2$ ,for simplicity. $\mathrm{F}\mathrm{i}\mathrm{g}_{11\mathrm{r}}\mathrm{e}6$ displays the numerical solution with $\alpha=2$ . For special value of$\kappa$ , periodic solutions appears. Mathematically we haveTheorem 4. Let $\alpha=2$ and $\kappa=0.5$ . Then the solution $\{z_{j}(t)\}$ of (1), (3) and (4) is aperiodic $sol\mathrm{u}$ tion.Theorem 4 shows that the solution (c) in Figure 6 is periodic. Until now, we do not knowwhether or not other solutions in Figure 6 is peri$o\mathrm{d}\mathrm{i}\mathrm{c}$ .We also make numerical simulations for another value of $\alpha$ . The solutions with a similarshape of (c) in Figure 6 are shown in Figure 7. We can observe that the vortices move onthe wide range and the vallle of $\kappa$ is small when $\alpha$ becomes large.On numerical methods. it is not easy to specify the value of two parameters under which aperiodic solution appears. We try to compute such a value on a narrow region of parameters(see Figure 8). We remark that they are not all value of parameters under which a periodic149ta) (b)$\ovalbox{\tt\small REJECT}$(d)Figure 7: Trajectory of numerical solution $z_{1}(t)$ . $(\mathrm{a}):\alpha=1.1$ and $\kappa=0.9917\cdots$ . $(\mathrm{b}):\alpha=$$1.5$ and $\kappa=0.6984\cdots$ , $(\mathrm{c}):\alpha=2.5$ and $\kappa=0.3095\cdots$ , $(\mathrm{d}):\alpha=3$ and $\kappa=0.08712\cdots$ .3$\theta$2 8’2. 6 $\theta$$\theta$$\mathrm{o}$ a. $\triangleleft$$.\underline{\overline{\overline{\approx}}}\mathrm{a}$ . a $\mathrm{c}$$\mathrm{R}’$ ’ $\theta$ $c$ $\delta$ $c$ ’ $\backslash$ $\theta$$\overline{\circ}$2$\theta"\cdots \mathrm{C}<C\wedge$ $\phi\circ$ $\backslash \cdot. c\mathrm{c}\theta^{\vee}\vee\prime c..\cdot\wedge\sigma o\mathrm{o}\mathrm{o},$,$\theta$$\mathrm{Q}_{4}\Phi \mathrm{t}\hslash 1$8 $c$[$\mathrm{d}1$ 6 $\theta$1. 4 $0$1. 2 $\nu$$o$ $\kappa$1$\emptyset$ $\emptyset.5$ 1 1. 5 aa.5 3Figure 8: The value of parameters under which a periodic solution appears.solution appears. We numerically check the periodic solution with $0<T\leq 100$ , where$T$ is the period of the solution on some parameter region. To find the law between suchparameters is one of our further problems.4 Motion of passively advected particlesIn this section, folowing Aref [3]. Babiano et al. [4] and Neufeld and T\’el [8]. we analyzethe dynamics of passively advected particles in the field of the periodic point vortices fromnumerical points of view. We use the periodic point vortices in Theorem 4. Let $x(t)$ bethe complex position of the particle. Then we have$\frac{d}{dt}\overline{x(t)}=\sum_{=j1}^{5}.\frac{\Gamma_{j}}{2\pi i}\frac{1}{x(t)-zj(t)}$ , (11)where $\{z_{j}(t)\}$ is the periodic solution in Theorem 4. We numerically solve (11) under theinitial condition$x(0)=X_{0}$ . (12)where $x_{0}\in C$ is the initial position of particle.150$\Leftrightarrow$Figure 9: The position of particle at $t=1000T$, 10001 $T,$ $..‘’ 10000T$, where $T$ is the periodof periodic solution in Theorem 4. The initial position of particle is $x_{0}=3$ (left) and$x_{0}=2i$ (right).If the initial particle is located far away from vortices, it stays there and rotates aroundthe vortices. To see this, we draw some positions of particle on the Poincar\’e section, thatis.,$P=\{x(n\tau);n=1000, 1001, ..., 10000\}$ , (13)where $T$ is the period of periodic solution $\{z_{j}(t)\}$ (see Figure 9). As shown in this figure,there are two types of trajectory $P$ of the particle.If the particle is some position at the initial time, chaotic behavior of the particle appearsas shown in Figure 10. Until now, we do not succeeded in proving that the motion inFigure 10 is chaos, however. the figure suggests that it is chaotic.References[1] H. Aref. Motion of three vortices, Phys. Fluids, 22 (1979), 393-400.[2] H. Aref and N. Pomphrey, Integrable and chaotic motions of four vortices I. The caseof identical vortices, Proc. Roy. Soc. London Ser. A, 380 (1982), 359-387.[3] H. Aref. Stirring by chaotic advection, J. Fluid Mech., 143 (1983), 1-21.Figure 10: The position of particle at t $=1000T$, 10001T, \ldots , 10000T with $x_{0}=2.2$ .151[4] A. Babiano, G. Boffetta, A. Provenzale and A. Vulpiani, Chaotic advection in pointvortex models and two-dimensional turbulence, Phys. Fluids, 6 (1994), 2465-2474.[5] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible NonviscousFluids, Springer-Verlag, 1994.[6] G. K. Morikawa and E. V. Swenson, Interacting motion of rectilinear geostrophicvortices, Phys, Fluids, 14 (1971), 1058-1073.[7] T. Nakaki, Behavior of point vortices in a plane and existence of heteroclinic orbits,Dynam. Contin. Discrete Impuls. Systems, 5 (1999), 159-169.[8] Z. Neufeld and T. T\’el, The vortex dynamics analogue of the restricted three-bodyproblem: advection in the field of three identical point vortices, J. Phys. A: $\wedge \mathrm{W}\prime \mathrm{a}th$.Gen., 30 (1997), 2263-2280.152