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Alan N. Jones\aff1 \corresp [email protected]    H.-C. Smith\aff1    J.Q. Long\aff2 \aff1STM Journals, Cambridge University Press, The Printing House, Shaftesbury Road, Cambridge CB2 8BS, UK \aff2DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Abstract

This file contains information for authors planning to submit a paper to the Journal of Fluid Mechanics. The document was generated in using the JFM class file and supporting files provided on the JFM website here, and the source files can be used as a template for submissions (please note that this is mandatory for JFM Rapids). Full author instructions can be found on the JFM website. The present paragraph appears in the abstract environment. All papers should feature a single-paragraph abstract of no more than 250 words which must not spill onto the seond page of the manuscript. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract. Dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract dummy text of abstract.

keywords:
Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see Keyword PDF for the full list). Other classifications will be added at the same time.

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1 First-order heading

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1.1 Second-order Heading

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1.1.1 Third-order Heading

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2 Figures and Tables

Refer to caption

Figure 1: Trapped-mode wavenumbers, k​dkd, plotted against a/da/d for three ellipses:
——, b/a=1b/a=1; ⋯\cdots ⋯\cdots, b/a=1.5b/a=1.5.

2.1 Figures

Each figure should be accompanied by a single caption, to appear beneath, and must be cited in the text. Figures should appear in the order in which they are first mentioned in the text. For example see figures 1 and 2.

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Refer to caption

Figure 2: The features of the four possible modes corresponding to (a) periodic
and (b) half-periodic solutions.

2.2 Tables

Tables, however small, must be numbered sequentially in the order in which they are mentioned in the text. Words table 1, table 2 should be lower case throughout. See table 1 for an example.

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a/da/d M=4M=4 M=8M=8 Callan et al.
0.1 1.56905 001.560 1.56904
0.3 1.50484 001.504 1.50484
0.55 1.39128 001.391 1.39131
0.7 1.32281 010.322 1.32288
0.913 1.34479 100.351 1.35185
Table 1: Values of k​dkd at which trapped modes occur when ρ​(θ)=a\rho(\theta)=a.

3 Notation and style

Generally any queries concerning notation and journal style can be answered by viewing recent pages in the Journal. However, the following guide provides the key points to note. It is expected that Journal style and mathematical notation will be followed, and authors should take care to define all variables or entities upon first use. Also note that footnotes are not normally accepted. Abbreviations must be defined at first use, glossaries or lists/tables of abbreviations are not permitted.

3.1 Mathematical notation

3.1.1 Setting variables, functions, vectors, matrices etc

  • •

    Italic font should be used for denoting variables, with multiple-letter symbols avoided except in the case of dimensionless numbers such as \Rey\Rey, \Pran\Pran and \Pen\Pen (Reynolds, Prandtl, and Péclet numbers respectively, which are defined as \Rey, \Pran and \Pen in the template).

  • •

    Upright Roman font (or upright Greek where appropriate) should be used for:

    1. 1.

      (vI) label, e.g. T. t (transpose)

    2. 2.

      Fixed operators: sin, log, d, Δ\Delta, exp etc.

    3. 3.

      Constants: i (−1\sqrt{-1}), \upi\upi (defined as \upi),e etc.

    4. 4.

      Special Functions: \Ai\Ai, \Bi\Bi (Airy functions, defined as \Ai and \Bi), \Real\Real (real part, defined as \Real), \Imag\Imag (imaginary part, defined as \Imag), etc.

    5. 5.

      Physical units: cm, s, etc.

    6. 6.

      Abbreviations: c.c. (complex conjugate), h.o.t. (higher-order terms), DNS, etc.

  • •

    Bold italic font (or bold sloping Greek) should be used for vectors (with the centred dot for a scalar product also in bold): 𝒊⋅𝒋\boldsymbol{i\cdot j}

  • •

    Bold sloping sans serif font, defined by the \mathsfbi macro, should be used for tensors and matrices: \mathsfbi​D\mathsfbi{D}

  • •

    Calligraphic font (for example 𝒢\mathcal{G}, ℛ\mathcal{R}) can be used as an alternative to italic when the same letter denotes a different quantity use \mathcal in )

3.1.2 Other symbols

Large numbers that are not scientific powers should not include commas, but should use a non-breaking space, and use the form 1600 or 16 000 or 160 000. Use O to denote ‘of the order of’, not the  𝒪\mathcal{O}.

The product symbol (×\times) should only be used to denote multiplication where an equation is broken over more than one line, to denote a cross product, or between numbers . The ⋅\boldsymbol{\cdot} symbol should not be used, except to denote a scalar product of vectors specifically.

3.1.3 Example Equations

This section contains sample equations in the JFM style. Please refer to the source file for examples of how to display such equations in your manuscript.

(∇2+k2)​Gs=(∇2+k2)​Ga=0(\nabla^{2}+k^{2})G_{s}=(\nabla^{2}+k^{2})G_{a}=0 (1)
\bnabla​\bcdot​𝒗=0,∇2P=\bnabla​\bcdot​(𝒗×𝒘).\bnabla\bcdot\boldsymbol{v}=0,\quad\nabla^{2}P=\bnabla\bcdot(\boldsymbol{v}\times\boldsymbol{w}). (2)
Gs,Ga∼1/(2​\upi)​ln⁡ras r≡|P−Q|→0,G_{s},G_{a}\sim 1/(2\upi)\ln r\quad\mbox{as\ }\quad r\equiv|P-Q|\rightarrow 0, (3)
\p​Gs\p​y=0on y=0,Ga=0on y=0,}\left.\begin{array}[]{ll}\displaystyle\frac{\p G_{s}}{\p y}=0\quad\mbox{on\ }\quad y=0,\\[8.0pt] \displaystyle G_{a}=0\quad\mbox{on\ }\quad y=0,\end{array}\right\} (4)
−12​\upi​∫0∞γ−1​[e​x​p​(−k​γ​|y−η|)+e​x​p​(−k​γ​(2​d−y−η))]​cos⁡k​(x−ξ)​t​dt,0<y,η<d,-\frac{1}{2\upi}\int_{0}^{\infty}\gamma^{-1}[\mathrm{e}xp(-k\gamma|y-\eta|)+\mathrm{e}xp(-k\gamma(2d-y-\eta))]\cos k(x-\xi)t\>\mathrm{d}t,\qquad 0<y,\quad\eta<d, (5)
γ​(t)={−i​(1−t2)1/2,t≤1(t2−1)1/2,t>1.\gamma(t)=\left\{\begin{array}[]{ll}-\mathrm{i}(1-t^{2})^{1/2},&t\leq 1\\[2.0pt] (t^{2}-1)^{1/2},&t>1.\end{array}\right. (6)
−12​\upi​\pvi​B​(t)​cosh⁡k​γ​(d−y)γ​sinh⁡k​γ​d​cos⁡k​(x−ξ)​t​d​t-\frac{1}{2\upi}\pvi B(t)\frac{\cosh k\gamma(d-y)}{\gamma\sinh k\gamma d}\cos k(x-\xi)t\>\mathrm{d}t
G=−14​i​(H0​(k​r)+H0​(k​r1))−1\upi​\pvi​e−\kgdγ​sinh⁡\kgd​cosh⁡k​γ​(d−y)​cosh⁡k​γ​(d−η)G=-\frac{1}{4}\mathrm{i}(H_{0}(kr)+H_{0}(kr_{1}))-\frac{1}{\upi}\pvi\frac{\mathrm{e}^{-\kgd}}{\gamma\sinh\kgd}\cosh k\gamma(d-y)\cosh k\gamma(d-\eta) (7)

Note that when equations are included in definitions, it may be suitable to render them in line, rather than in the equation environment: 𝒏q=(−y′​(θ),x′​(θ))/w​(θ)\boldsymbol{n}_{q}=(-y^{\prime}(\theta),x^{\prime}(\theta))/w(\theta). Now Ga=\squart​Y0​(k​r)+\GatG_{a}=\squart Y_{0}(kr)+\Gat where r={[x​(θ)−x​(ψ)]2+[y​(θ)−y​(ψ)]2}1/2r=\{[x(\theta)-x(\psi)]^{2}+[y(\theta)-y(\psi)]^{2}\}^{1/2} and \Gat\Gat is regular as k​r​\ttzkr\ttz. However, any fractions displayed like this, other than \thalf\thalf or \squart\squart, must be written on the line, and not stacked (ie 1/3).

\ndq​(14​Y0​(k​r))\displaystyle\ndq\left(\frac{1}{4}Y_{0}(kr)\right) ∼\displaystyle\sim 14​\upi​w3​(θ)​[x′′​(θ)​y′​(θ)−y′′​(θ)​x′​(θ)]\displaystyle\frac{1}{4\upi w^{3}(\theta)}[x^{\prime\prime}(\theta)y^{\prime}(\theta)-y^{\prime\prime}(\theta)x^{\prime}(\theta)] (8)
=\displaystyle= 14​\upi​w3​(θ)​[ρ′​(θ)​ρ′′​(θ)−ρ2​(θ)−2​ρ′⁣2​(θ)]as k​r​\ttz.\displaystyle\frac{1}{4\upi w^{3}(\theta)}[\rho^{\prime}(\theta)\rho^{\prime\prime}(\theta)-\rho^{2}(\theta)-2\rho^{\prime 2}(\theta)]\quad\mbox{as\ }\quad kr\ttz.
12​ϕi=\upiM​\sumjm​ϕj​Ki​ja​wj,i=1,…,M,\frac{1}{2}\phi_{i}=\frac{\upi}{M}\sumjm\phi_{j}K_{ij}^{a}w_{j},\qquad i=1,\,\ldots,\,M, (9)

where

Ki​ja={\p​Ga​(θi,θj)/\p​nq,i≠j\p​\Gat​(θi,θi)/\p​nq+[ρi′​ρi′′−ρi2−2​ρi′⁣2]/4​\upi​wi3,i=j.K_{ij}^{a}=\left\{\begin{array}[]{ll}\p G_{a}(\theta_{i},\theta_{j})/\p n_{q},&i\neq j\\[2.0pt] \p\Gat(\theta_{i},\theta_{i})/\p n_{q}+[\rho_{i}^{\prime}\rho_{i}^{\prime\prime}-\rho_{i}^{2}-2\rho_{i}^{\prime 2}]/4\upi w_{i}^{3},&i=j.\end{array}\right. (10)
ρl=limζ→Zl−​(x)ρ​(x,ζ),ρu=limζ→Zu+​(x)ρ​(x,ζ)\rho_{l}=\lim_{\zeta\rightarrow Z^{-}_{l}(x)}\rho(x,\zeta),\quad\rho_{u}=\lim_{\zeta\rightarrow Z^{+}_{u}(x)}\rho(x,\zeta) (11​a,b)
(ρ​(x,ζ),ϕζ​ζ​(x,ζ))=(ρ0,N0)forZl​(x)<ζ<Zu​(x).(\rho(x,\zeta),\phi_{\zeta\zeta}(x,\zeta))=(\rho_{0},N_{0})\quad\mbox{for}\quad Z_{l}(x)<\zeta<Z_{u}(x). (12)
{subeqnarray}

τ_ij & = (¯¯u_i ¯u_j - ¯u_i¯u_j) + (¯¯u_iu^SGS_j + u^SGS_i¯u_j) + ¯u^SGS_iu^SGS_j,
τ^θ_j = (¯¯u_j¯θ - ¯u_j ¯θ) + (¯¯u_jθ^SGS + u^SGS_j ¯θ) + ¯u^SGS_jθ^SGS.

\slsQC=[−ω−2​Vw′−(αt​ω)−1000βα​ω2​Vw′000i​ω−1i​ω−10000i​Rδ−1​(αt+ω−1​Vw′′)0−(i​αt​Rδ)−100i​βα​ω​Rδ−1​Vw′′0000(i​αt)−1​Vw′(3​Rδ−1+ct​(i​αt)−1)0−(αt)−2​Rδ−10].\slsQ_{C}=\left[\begin{array}[]{ccccc}-\omega^{-2}V^{\prime}_{w}&-(\alpha^{t}\omega)^{-1}&0&0&0\\ \displaystyle\frac{\beta}{\alpha\omega^{2}}V^{\prime}_{w}&0&0&0&\mathrm{i}\omega^{-1}\\ \mathrm{i}\omega^{-1}&0&0&0&0\\ \displaystyle\mathrm{i}R^{-1}_{\delta}(\alpha^{t}+\omega^{-1}V^{\prime\prime}_{w})&0&-(\mathrm{i}\alpha^{t}R_{\delta})^{-1}&0&0\\ \displaystyle\frac{\mathrm{i}\beta}{\alpha\omega}R^{-1}_{\delta}V^{\prime\prime}_{w}&0&0&0&0\\ (\mathrm{i}\alpha^{t})^{-1}V^{\prime}_{w}&(3R^{-1}_{\delta}+c^{t}(\mathrm{i}\alpha^{t})^{-1})&0&-(\alpha^{t})^{-2}R^{-1}_{\delta}&0\\ \end{array}\right]. (13)
\etbt=\etb^​expt⁡[i​(αt​x1t−ω​t)],\etb^{t}={\hat{\etb\mkern 2.0mu}\mkern-2.0mu}{}^{t}\exp[\mathrm{i}(\alpha^{t}x^{t}_{1}-\omega t)], (14)

where \etb^=t𝒃exp(iγx3t){\hat{\etb\mkern 2.0mu}\mkern-2.0mu}{}^{t}=\boldsymbol{b}\exp(\mathrm{i}\gamma x^{t}_{3}).

Det​[ρ​ω2​δp​s−Cp​q​r​st​kqt​krt]=0,\mbox{Det}[\rho\omega^{2}\delta_{ps}-C^{t}_{pqrs}k^{t}_{q}k^{t}_{r}]=0, (15)
⟨k1t,k2t,k3t⟩=⟨αt,0,γ⟩\langle k^{t}_{1},k^{t}_{2},k^{t}_{3}\rangle=\langle\alpha^{t},0,\gamma\rangle (16)
𝒇​(θ,ψ)=(g​(ψ)​cos⁡θ,g​(ψ)​sin⁡θ,f​(ψ)).\boldsymbol{f}(\theta,\psi)=(g(\psi)\cos\theta,g(\psi)\sin\theta,f(\psi)). (17)
f​(ψ1)=3​b\upi​[2​(a+b​cos⁡ψ1)]3/2​∫02​\upi(sin⁡ψ1−sin⁡ψ)​(a+b​cos⁡ψ)1/2[1−cos⁡(ψ1−ψ)]​(2+α)1/2​dx,\displaystyle f(\psi_{1})=\frac{3b}{\upi[2(a+b\cos\psi_{1})]^{{3}/{2}}}\int^{2\upi}_{0}\frac{(\sin\psi_{1}-\sin\psi)(a+b\cos\psi)^{1/2}}{[1-\cos(\psi_{1}-\psi)](2+\alpha)^{1/2}}\mathrm{d}x, (18)
g​(ψ1)\displaystyle g(\psi_{1}) =\displaystyle= 3\upi​[2​(a+b​cos⁡ψ1)]3/2∫02​\upi(a+b​cos⁡ψ2+α)1/2{\astrutf(ψ)[(cosψ1−bβ1)S+β1P]\displaystyle\frac{3}{\upi[2(a+b\cos\psi_{1})]^{{3}/{2}}}\int^{2\upi}_{0}\left(\frac{a+b\cos\psi}{2+\alpha}\right)^{1/2}\left\{\astrut f(\psi)[(\cos\psi_{1}-b\beta_{1})S+\beta_{1}P]\right. (19)
×sin⁡ψ1−sin⁡ψ1−cos⁡(ψ1−ψ)+g(ψ)[(2+α−(sin⁡ψ1−sin⁡ψ)21−cos⁡(ψ−ψ1)−b2γ)S\displaystyle\mbox{}\times\frac{\sin\psi_{1}-\sin\psi}{1-\cos(\psi_{1}-\psi)}+g(\psi)\left[\left(2+\alpha-\frac{(\sin\psi_{1}-\sin\psi)^{2}}{1-\cos(\psi-\psi_{1})}-b^{2}\gamma\right)S\right.
+(b2cosψ1γ−abα)F(12\upi,δ)−(2+α)cosψ1E(12\upi,δ)]\astrut}dψ,\displaystyle\left.\left.\mbox{}+\left(b^{2}\cos\psi_{1}\gamma-\frac{a}{b}\alpha\right)F(\frac{1}{2}\upi,\delta)-(2+\alpha)\cos\psi_{1}E(\frac{1}{2}\upi,\delta)\right]\astrut\right\}\mathrm{d}\psi,
α=α​(ψ,ψ1)=b2​[1−cos⁡(ψ−ψ1)](a+b​cos⁡ψ)​(a+b​cos⁡ψ1),β−β​(ψ,ψ1)=1−cos⁡(ψ−ψ1)a+b​cos⁡ψ.\alpha=\alpha(\psi,\psi_{1})=\frac{b^{2}[1-\cos(\psi-\psi_{1})]}{(a+b\cos\psi)(a+b\cos\psi_{1})},\quad\beta-\beta(\psi,\psi_{1})=\frac{1-\cos(\psi-\psi_{1})}{a+b\cos\psi}. (20)
H(0)=ϵ​C¯vv~T1/2​(1−β),H′(0)=−1+ϵ2/3C¯u+ϵC^;u′H′′​(0)=ϵ​u∗2v~T1/2​uP2,H′​(∞)=0.}\left.\begin{array}[]{l}\displaystyle H(0)=\frac{\epsilon\overline{C}_{v}}{\tilde{v}^{{1}/{2}}_{T}(1-\beta)},\quad H^{\prime}(0)=-1+\epsilon^{{2}/{3}}\overline{C}_{u}+\epsilon{\hat{C\mkern 5.0mu}\mkern-5.0mu}{}_{u}^{\prime};\\[16.0pt] \displaystyle H^{\prime\prime}(0)=\frac{\epsilon u^{2}_{\ast}}{\tilde{v}^{{1}/{2}}_{T}u^{2}_{P}},\quad H^{\prime}(\infty)=0.\end{array}\right\} (21)
Lemma 1

Let f​(z)f(z) be a trial Batchelor (1971, pp. 231–232) function defined on [0,1][0,1]. Let Λ1\varLambda_{1} denote the ground-state eigenvalue for −d2​g/d​z2=Λ​g-\mathrm{d}^{2}g/\mathrm{d}z^{2}=\varLambda g, where gg must satisfy ±d​g/d​z+α​g=0\pm\mathrm{d}g/\mathrm{d}z+\alpha g=0 at z=0,1z=0,1 for some non-negative constant α\alpha. Then for any ff that is not identically zero we have

α​(f2​(0)+f2​(1))+∫01(d​fd​z)2​dz∫01f2​dz≥Λ1≥(−α+(α2+8​\upi2​α)1/24​\upi)2.\frac{\displaystyle\alpha(f^{2}(0)+f^{2}(1))+\int_{0}^{1}\left(\frac{\mathrm{d}f}{\mathrm{d}z}\right)^{2}\mathrm{d}z}{\displaystyle\int_{0}^{1}f^{2}\mathrm{d}z}\geq\varLambda_{1}\geq\left(\frac{-\alpha+(\alpha^{2}+8\upi^{2}\alpha)^{1/2}}{4\upi}\right)^{2}. (22)
Corollary 1

Any non-zero trial function ff which satisfies the boundary condition f​(0)=f​(1)=0f(0)=f(1)=0 always satisfies

∫01(d​fd​z)2​dz.\int_{0}^{1}\left(\frac{\mathrm{d}f}{\mathrm{d}z}\right)^{2}\mathrm{d}z. (23)

4 Citations and references

All papers included in the References section must be cited in the article, and vice versa. Citations should be included as, for example “It has been shown (Rogallo, 1981) that…” (using the \citep command, part of the natbib package) “recent work by Dennis (1985)…” (using \citet). The natbib package can be used to generate citation variations, as shown below.
\citet[pp. 2-4]{Hwang70}:
Hwang et al (1970, pp. 2-4)
\citep[p. 6]{Worster92}:
(Worster, 1992, p. 6)
\citep[see][]{Koch83, Lee71, Linton92}:
(see Koch, 1983; Lee, 1971; Linton and Evans, 1992)
\citep[see][p. 18]{Martin80}:
(see Martin, 1980, p. 18)
\citep{Brownell04,Brownell07,Ursell50,Wijngaarden68,Miller91}:
(Brownell, 2004; Brownell and Su, 2007; Ursell, 1950; Wijngaarden, 1968; Miller, 1991)
(Briukhanovetal et al, 1967)
Bouguet (2008)
(Josep and Saut, 1990)

The References section can either be built from individual \bibitem commands, or can be built using BibTex. The BibTex files used to generate the references in this document can be found in the JFM template files folder provided on the website here.

Where there are up to ten authors, all authors’ names should be given in the reference list. Where there are more than ten authors, only the first name should appear, followed by et al.

\backsection

[Supplementary data]Supplementary material and movies are available at
https://doi.org/10.1017/jfm.2019…

\backsection

[Acknowledgements]Acknowledgements may be included at the end of the paper, before the References section or any appendices. Several anonymous individuals are thanked for contributions to these instructions.

\backsection

[Funding]Please provide details of the sources of financial support for all authors, including grant numbers. Where no specific funding has been provided for research, please provide the following statement: ”This research received no specific grant from any funding agency, commercial or not-for-profit sectors.”

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Appendix A

In order not to disrupt the narrative flow, purely technical material may be included in the appendices. This material should corroborate or add to the main result and be essential for the understanding of the paper. It should be a small proportion of the paper and must not be longer than the paper itself.

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