Navier-Stokes

Paper Preparation

Springer provides you with a complete integrated LaTeX document class (llncs.cls) for multi-author books such as those in the LNCS series. Papers not complying with the LNCS style will be reformatted. This can lead to an increase in the overall number of pages. We would therefore urge you not to squash your paper.

Please always cancel any superfluous definitions that are not actually used in your text. If you do not, these may conflict with the definitions of the macro package, causing changes in the structure of the text and leading to numerous mistakes in the proofs.

If you wonder what LaTeX is and where it can be obtained, see the “LaTeX project site” (http://www.latex-project.org) and especially the webpage “How to get it” (http://www.latex-project.org/ftp.html) respectively.

When you use LaTeX together with our document class file, your text is typeset automatically in Computer Modern Roman (CM) fonts. Please do than the preset fonts, kindly submit these with your files.

Please use the commands \label and \ref for cross-references and the commands \bibitem and \cite for references to the bibliography, to enable us to create hyperlinks at these places.

For preparing your figures electronically and integrating them into your source file we recommend using the standard LaTeX graphics or graphicx package. These provide the \includegraphics command. In general, please refrain from using the \special command.

Remember to submit any further style files and fonts you have used together with your source files.

Headings.

Headings should be capitalized (i.e., nouns, verbs, and all other words except articles, prepositions, and conjunctions should be set with an initial capital) and should, with the exception of the title, be aligned to the left. Words joined by a hyphen are subject to a special rule. If the first word can stand alone, the second word should be capitalized.

Here are some examples of headings: “Criteria to Disprove Context-Freeness of Collage Language", “On Correcting the Intrusion of Tracing Non-deterministic Programs by Software", “A User-Friendly and Extendable Data Distribution System", “Multi-flip Networks: Parallelizing GenSAT", “Self-determinations of Man".

Lemmas, Propositions, and Theorems.

The numbers accorded to lemmas, propositions, and theorems, etc. should appear in consecutive order, starting with Lemma 1, and not, for example, with Lemma 11.

Figures

For LaTeX users, we recommend using the graphics or graphicx package and the \includegraphics command.

Please check that the lines in line drawings are not interrupted and are of a constant width. Grids and details within the figures must be clearly legible and may not be written one on top of the other. Line drawings should have a resolution of at least 800 dpi (preferably 1200 dpi). The lettering in figures should have a height of 2 mm (10-point type). Figures should be numbered and should have a caption which should always be positioned under the figures, in contrast to the caption belonging to a table, which should always appear your source.

Please center the figures or your tabular material by using the \centering declaration. Short captions are centered by default between the margins and typeset in 9-point type (Fig. \ref{fig:example} shows an example). The distance between text and figure is preset to be about 8 mm, the distance between figure and caption about 6 mm.

To ensure that the reproduction of your illustrations is of a reasonable quality, we advise against the use of shading. The contrast should be as pronounced as possible.

If screenshots are necessary, please make sure that you are happy with the print quality before you send the files.

Please define figures (and tables) as floating objects. Please avoid using optional location parameters like “[h] " for “here".

In the printed volumes, illustrations are generally black and white (halftones), and only in exceptional cases, and if the author is prepared to cover the extra cost for color reproduction, are colored pictures accepted. Colored pictures are welcome in the electronic version free of charge. If you send colored figures that are to be printed in black and white, please make sure that they really are legible in black and white. Some colors as well as the contrast of converted colors show up very poorly when printed in black and white.

\label{fig:example} One kernel at \(x_s\) (dotted kernel) or two kernels at \(x_i\) and \(x_j\) (left and right) lead to the same summed estimate at \(x_s\). This shows a figure consisting of different types of lines. Elements of the figure described in the caption should be set in italics, in parentheses, as shown in this sample caption.

Formulas

Displayed equations or formulas are centered and set on a separate line (with an extra line or halfline space above and below). Displayed expressions should be numbered for reference. The numbers should be consecutive within each section or within the contribution, with numbers enclosed in parentheses and set on the right margin – which is the default if you use the equation environment, e.g.,

\[\psi (u) = \int_{o}^{T} \left[\frac{1}{2} \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt \; .\]

Equations should be punctuated in the same way as ordinary text but with a small space before the end punctuation mark.

Footnotes

The superscript numeral used to refer to a footnote appears in the text either directly after the word to be discussed or – in relation to a phrase or a sentence – following the punctuation sign (comma, semicolon, or period). Footnotes should appear at the bottom of the normal text area, with a line of about 2 cm set immediately above them.1

Program Code

Program listings or program commands in the text are normally set in typewriter font, e.g., CMTT10 or Courier.

Example of a Computer Program program Inflation (Output) {Assuming annual inflation rates of 7%, 8%, and 10%,... years}; const MaxYears = 10; var Year: 0..MaxYears; Factor1, Factor2, Factor3: Real; begin Year := 0; Factor1 := 1.0; Factor2 := 1.0; Factor3 := 1.0; WriteLn('Year 7% 8% 10%'); WriteLn; repeat Year := Year + 1; Factor1 := Factor1 * 1.07; Factor2 := Factor2 * 1.08; Factor3 := Factor3 * 1.10; WriteLn(Year:5,Factor1:7:3,Factor2:7:3,Factor3:7:3) until Year = MaxYears end. (Example from Jensen K., Wirth N. (1991) Pascal user manual and report. Springer, New York)

Citations

For citations in the text please use LaTeX’s \cite mechanism.

Page Numbering and Running Heads

There is no need to include page numbers. If your paper title is too long to serve as a running head, it will be shortened. Your suggestion as to how to shorten it would be most welcome.


  1. The footnote numeral is set flush left and the text follows with the usual word spacing.