formulasearchengine - User contributions [en]

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2014-10-20T19:21:21Z

182.64.157.100:

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2014-08-20T14:52:15Z

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2014-01-13T08:50:03Z

182.64.206.26:

{{redirect|Black letter|the legal concept|Black letter law}}
{{redirect|Gothic letter||Gothic alphabet}}
{{Infobox Writing system
| name = Latin script (Blackletter hand)
| type = [[Alphabet]]
| time = 12th century – 1946
| languages = [[European languages]]
| fam1 = [[Latin alphabet|Latin script]]
| fam2 = [[Carolingian minuscule]]
| children = [[Fraktur (typeface sub-classification)|Fraktur]]¹, [[Kurrent]]schrift, including [[Sütterlin]]
| iso15924 = Latf
| unicode = <code>1D504</code>–<code>1D537</code>²
| footnotes =
# [[Fraktur (script)|Fraktur]] and black letter are sometimes used interchangeably.
# With some exceptions; [[#Unicode|see below]]
| sample = Calligraphy.malmesbury.bible.arp.jpg|200px
}}
'''Blackletter''', also known as '''Gothic script''', '''Gothic minuscule''', or '''Textura''', was a [[script (styles of handwriting)|script]] used throughout [[Western Europe]] from approximately 1150 to well into the 17th century. It continued to be used for the [[German language]] until the 20th century. [[Fraktur (script)|Fraktur]] is a notable script of this type, and sometimes the entire group of faces is known as Fraktur. Blackletter is sometimes called '''Old English''', but it is not to be confused with the [[Old English|Old English language]], despite the popular, though mistaken, belief that the language was written with blackletter. The Old English (or Anglo-Saxon) language predates black letter by many centuries, and was itself written in the [[insular script]], or [[Futhorc]] runes before that.

==Origins==
[[File:AberdeenBestiaryFol56rPhoenix.jpg|thumb|left|Folio 56r of the [[Aberdeen Bestiary]], an early example of black letter from the 12th century]]
{{stack begin}}
[[File:Piers plowman drolleries.gif|right|thumb|Page from a 14th-century [[psalter]] ([[Latin Psalters#Enumeration|Vulgate]] Ps 93:16-21), with black letter "''sine pedibus''" text]]
[[File:Rudolf Koch gebrochene Schriften.png|right|thumb|German black-letter typefaces]]
[[File:Gebrochene Schriften.png|thumb|black-letter typefaces]]
[[File:Old English typeface.svg|right|thumb|Modern interpretation of blackletter script in the form of the font "Old English" which includes several [[anachronistic]] glyphs, such as [[Arabic numerals]], ampersand (instead of [[Tironian et]]) and several punctuation marks, but lacks letter alternatives like long s and r rotunda, scribal abbreviations and ligatures and contains several modernised letters, such as x]]
{{stack end}}

[[Carolingian minuscule]] was the direct ancestor of blackletter. Blackletter developed from Carolingian as an increasingly literate 12th-century Europe required new books in many different subjects. New [[Medieval university|universities]] were founded, each producing books for [[business]], [[law]], [[grammar]], [[history]], and other pursuits, not solely religious works for which earlier [[script (styles of handwriting)|scripts]] typically had been used.

These books needed to be produced quickly to keep up with demand. Carolingian, though legible, was time-consuming and labour-intensive to produce. Its large size consumed a lot of [[manuscript]] space in a time when writing materials were very costly. As early as the 11th century, different forms of Carolingian were already being used, and by the mid-12th century, a clearly distinguishable form, able to be written more quickly to meet the demand for new books, was being used in northeastern [[France]] and the [[Low Countries]].

==The name ''Gothic'' script==
[[File:Incunabulum.JPG|thumb|left|Page of a rare black-letter Bible, 1497, printed in Strasbourg by [[Johann Grüninger]]. The coloured chapter initials were handwritten after printing]]
The term ''Gothic'' was first used to describe this script in 15th-century [[Italy]], in the midst of the [[Renaissance]], because [[Renaissance humanism|Renaissance Humanists]] believed it was barbaric. ''Gothic'' was a synonym for ''barbaric''. [[Flavio Biondo]], in ''Italia Illustrata'' (1531) thought it was invented by the [[Lombards]] after their invasion of Italy in the 6th century.

Not only were black-letter forms called ''Gothic script'', but any other seemingly barbarian script, such as [[Visigothic script|Visigothic]], [[Beneventan script|Beneventan]], and [[Merovingian script|Merovingian]], were also labeled "Gothic", in contrast to [[Carolingian minuscule]], a highly legible script which the Humanists called ''[[humanist minuscule|littera antiqua]]'', "the ancient letter", wrongly believing that it was the script used by the [[ancient Rome|Romans]]. It was invented in the reign of [[Charlemagne]], although only used significantly after that era.

The black letter must not be confused either with the ancient [[Gothic alphabet|alphabet of the Gothic language]], nor with the [[sans-serif]] [[typefaces]] that are also sometimes called ''Gothic''.

==Forms of black letter==

===Textualis===
''Textualis'', also known as ''textura'' or ''Gothic bookhand'', was the most [[calligraphy|calligraphic]] form of black letter, and today is the form most associated with "Gothic". [[Johannes Gutenberg]] carved a textualis typeface – including a large number of [[ligature (typography)|ligatures]] and common abbreviations – when he printed his [[Gutenberg Bible|42-line Bible]]. However, the textualis was rarely used for typefaces afterwards.

According to Dutch scholar Gerard Lieftinck, the pinnacle of black-letter use occurred in the 14th and 15th centuries. For Lieftinck, the highest form of ''textualis'' was ''littera textualis formata'', used for ''de luxe'' manuscripts. The usual form, simply ''littera textualis'', was used for literary works and university texts. Lieftinck's third form, ''littera textualis currens'', was the [[cursive]] form of black letter, extremely difficult to read and used for textual [[Gloss (annotation)|glosses]], and less important books.

''Textualis'' was most widely used in France, the Low Countries, [[England]], and [[Germany]]. Some characteristics of the script are:
*tall, narrow letters, as compared to their Carolingian counterparts.
*letters formed by sharp, straight, angular lines, unlike the typically round Carolingian; as a result, there is a high degree of "breaking", i.e. lines that do not necessarily connect with each other, especially in curved letters.
*[[Ascender (typography)|ascender]]s (in letters such as b, d, h) are vertical and often end in sharp [[Stroke ending (typography)|finial]]s
*when a letter with a bow (in b, d, p, q) is followed by another letter with a bow (such as "be" or "po"), the bows overlap and the letters are joined by a straight line (this is known as "biting").
*a related characteristic is the [[half r]], the shape of ''r'' when attached to other letters with bows; only the bow and tail were written, connected to the bow of the previous letter. In other scripts, this only occurred in a [[ligature (typography)|ligature]] with the letter o.
*similarly related is the form of the letter d when followed by a letter with a bow; its ascender is then curved to the left, like the [[uncial]] d. Otherwise the ascender is vertical.
*the letters g, j, p, q, y, and the hook of h have descenders, but no other letters are written below the line.
*the letter a has a straight back stroke, and the top loop eventually became closed, somewhat resembling the number 8. The letter s often has a diagonal line connecting its two bows, also somewhat resembling an 8, but the [[long s]] is frequently used in the middle of words.
*[[minim (palaeography)|minim]]s, especially in the later period of the script, do not connect with each other. This makes it very difficult to distinguish i, u, m, and n. A 14th-century example of the difficulty minims produced is, ''mimi numinum niuium minimi munium nimium uini muniminum imminui uiui minimum uolunt'' ("the smallest mimes of the gods of snow do not wish at all in their life that the great duty of the defences of the wine be diminished"). In black letter this would look like a series of single strokes. Dotted i and the letter j developed because of this{{Citation needed|date=June 2008}}. Minims may also have finials of their own.
*the script has many more [[scribal abbreviation]]s than Carolingian, adding to the speed in which it could be written.

===Schwabacher===
{{main|Schwabacher}}
'''Schwabacher''' was a black-letter form that was much used in early German print typefaces. It continued to be used occasionally until the 20th century. Characteristics of Schwabacher are:

*The small letter o is rounded on both sides, though at the top and at the bottom, the two strokes join in an angle. Other small letters have analogous forms.
*The small letter g has a horizontal stroke at its top that forms crosses with the two downward strokes.
*The capital letter H has a peculiar form somewhat reminiscent of the small letter h.

===Fraktur===
{{main|Fraktur}}
'''Fraktur''' is a form of black letter that became the most common German black-letter typeface by the mid 16th century. Its use was so common that often any black-letter form is called ''Fraktur'' in Germany. Characteristics of Fraktur are:

*The left side of the small letter o is formed by an angular stroke, the right side by a rounded stroke. At the top and at the bottom, both strokes join in an angle. Other small letters have analogous forms.
*The capital letters are compound of rounded c-shaped or s-shaped strokes.

Here is the entire alphabet in Fraktur (minus the [[long s]] and the [[ß|sharp s (ß)]]), using the [[TeX]] \mathfrak font (see [[Help: Displaying a formula]]):

<math>\mathfrak{A} \mathfrak{B} \mathfrak{C} \mathfrak{D} \mathfrak{E} \mathfrak{F} \mathfrak{G} \mathfrak{H} \mathfrak{I} \mathfrak{J} \mathfrak{K} \mathfrak{L} \mathfrak{M} \mathfrak{N} \mathfrak{O} \mathfrak{P} \mathfrak{Q} \mathfrak{R} \mathfrak{S} \mathfrak{T} \mathfrak{U} \mathfrak{V} \mathfrak{W} \mathfrak{X} \mathfrak{Y} \mathfrak{Z} </math>

<math>\mathfrak{a} \mathfrak{b} \mathfrak{c} \mathfrak{d} \mathfrak{e} \mathfrak{f} \mathfrak{g} \mathfrak{h} \mathfrak{i} \mathfrak{j} \mathfrak{k} \mathfrak{l} \mathfrak{m} \mathfrak{n} \mathfrak{o} \mathfrak{p} \mathfrak{q} \mathfrak{r} \mathfrak{s} \mathfrak{t} \mathfrak{u} \mathfrak{v} \mathfrak{w} \mathfrak{x} \mathfrak{y} \mathfrak{z} </math>

===Cursiva===
''[[Cursiva]]'' refers to a very large variety of forms of black letter; as with modern [[cursive writing]], there is no real standard form. It developed in the 14th century as a simplified form of ''textualis'', with influence from the form of ''textualis'' as used for writing [[charter]]s. ''Cursiva'' developed partly because of the introduction of [[paper]], which was smoother than [[parchment]]. It was therefore, easier to write quickly on paper in a [[cursive|cursive script]].

In ''cursiva'', descenders are more frequent, especially in the letters f and s, and ascenders are curved and looped rather than vertical (seen especially in the letter d). The letters a, g, and s (at the end of a word) are very similar to their Carolingian forms. However, not all of these features are found in every example of ''cursiva'', which makes it difficult to determine whether or not a script may be called ''cursiva'' at all.

Lieftinck also divided ''cursiva'' into three styles: ''littera cursiva formata'' was the most legible and calligraphic style. ''Littera cursiva textualis'' (or ''libraria'') was the usual form, used for writing standard books, and it generally was written with a larger pen, leading to larger letters. ''Littera cursiva currens'' was used for textbooks and other unimportant books and it had very little standardization in forms.

===Hybrida===
''[[Hybrida]]'' is also called ''[[bastarda]]'' (especially in France), and as its name suggests, refers to a hybrid form of the script. It is a mixture of ''textualis'' and ''cursiva'', developed in the early 15th century. From ''textualis'', it borrowed vertical ascenders, while from ''cursiva'', it borrowed long f and [[long s|ſ]], single-looped a, and g with an open descender (similar to Carolingian forms).

===Donatus-Kalender===
The ''[[Donatus-Kalender]]'' (also known as Donatus-und-Kalender or D-K) is the name for the metal type design that [[Johannes Gutenberg|Gutenberg]] used in his earliest surviving printed works, dating from the early 1450s. The name is taken from two works: the [[Ars grammatica]] of [[Aelius Donatus]], a Latin grammar, and the Kalender (calendar).<ref>John Man, How One Man Remade the World with Words</ref> It is a form of textura.

==Black-letter typesetting==
While an [[antiqua script|antiqua]] typeface is usually compound of [[roman type]]s and [[italic type]]s since the 16th-century French typographers, the black-letter typefaces never developed a similar distinction. Instead, they use [[Emphasis (typography)#Letter spacing|letterspacing]] (German ''sperren'') for emphasis. When using that method, black-letter ligatures like ''ch'', ''ck'', ''tz'' or ''ſt'' remain together without additional letterspacing (''ſt'' is dissolved, though). The use of bold text for emphasis is also alien to black-letter typefaces.

Words from other languages, especially from Romance languages including Latin, are usually typeset in antiqua instead of black letter. Like that, single antiqua words or phrases may occur within a black-letter text. This does not apply, however, to loanwords that have been incorporated into the language.

==National forms==

===England===

====''Textualis''====
[[File:calligraphy.malmesbury.bible.arp.jpg|left|thumb|right|Black letter in a Latin Bible of AD 1407, on display in [[Malmesbury Abbey]], Wiltshire, England]]
[[English language|English]] black letter developed from the form of Caroline minuscule used there after the [[Norman Conquest]], sometimes called "Romanesque minuscule." ''Textualis'' forms developed after 1190 and were used most often until approximately 1300, afterward being used mainly for ''de luxe'' manuscripts. English forms of black letter have been studied extensively and may be divided into many categories. ''Textualis formata'' ("Old English" or "Black Letter"), ''textualis prescissa'' (or ''textualis sine pedibus'', as it generally lacks feet on its minims), ''textualis quadrata'' (or ''psalterialis'') and ''semi-quadrata'', and ''textualis rotunda'' are various forms of high-grade ''formata'' styles of black letter.

The [[University of Oxford]] borrowed the ''littera parisiensis'' in the 13th century and early 14th century, and the ''littera oxoniensis'' form is almost indistinguishable from its Parisian counterpart; however, there are a few differences, such as the round final "s" forms, resembling the number 8, rather than the long "s" used in the final position in the Paris script.

[[Chaucer]] was originally printed in blackletter, but most presses were switched over to [[Roman type]] around 1590, following the trend of the [[Renaissance]].<ref>''Pica Roman Type in Elizabethan England'' (1989)</ref> [[Horace Walpole]] wrote in 1781 that "I am too, though a Goth, so modern a Goth that I hate the black letter, and I love Chaucer better in Dryden and Baskerville than in his own language and dress".<ref>Caroline F.E. Spurgeon, ''Five Hundred Years of Chaucer Criticsm and Allusion (1357-1900)'', “Introduction” (London: Chaucer Society, 1923) xliv - xx.</ref> The final uses of blackletter in the 17th century were for printing ballads, romance novels, and jokebooks.<ref>"[http://www.jstor.org/discover/10.2307/459873 Black Letter as a Social Discriminant in the Seventeenth Century]" ''Proceedings of the Modern Language Association of America'' 68.3</ref>

====''Cursiva''====
English ''cursiva'' began to be used in the 13th century, and soon replaced ''littera oxoniensis'' as the standard university script. The earliest cursive black-letter form is ''Anglicana'', a very round and looped script, which also had a squarer and angular counterpart, ''Anglicana formata''. The ''formata'' form was used until the 15th century and also was used to write vernacular texts. An ''Anglicana bastarda'' form developed from a mixture of ''Anglicana'' and ''textualis'', but by the 16th century the principal cursive black letter used in England was the Secretary script, which originated in [[Italy]] and came to England by way of France. Secretary script has a somewhat haphazard appearance, and its forms of the letters a, g, r, and s are unique, unlike any forms in any other English script.

===France===

====''Textualis''====
[[French language|French]] ''textualis'' was tall and narrow compared to other national forms, and was most fully developed in the late 13th century in Paris. In the 13th century there also was an extremely small version of [[textualis]] used to write miniature Bibles, known as "pearl script." Another form of French textualis in this century was the script developed at the [[University of Paris]], ''littera parisiensis'', which also is small in size and designed to be written quickly, not calligraphically.

====''Cursiva''====
{{anchor|French cursiva}}French ''cursiva'' was used from the 13th to the 16th century, when it became highly looped, messy, and slanted. ''Bastarda'', the "hybrid" mixture of ''cursiva'' and ''textualis'', developed in the 15th century and was used for vernacular texts as well as Latin. A more angular form of ''bastarda'' was used in [[Duchy of Burgundy|Burgundy]], the ''lettre de forme'' or ''lettre bourgouignonne'', for [[book of hours|books of hours]] such as the [[Très Riches Heures]] of [[John, Duke of Berry]].

===Germany===
[[File:Fraktur alte schwabacher.png|left|frame|[[Schwabacher]] lettering]]
Despite the frequent association of black letter with [[German language|German]], the script was actually very slow to develop in German-speaking areas. It developed first in those areas closest to France and then spread to the east and south in the 13th century. However, the German-speaking areas are where black letter remained in use the longest.

[[Schwabacher]] typefaces dominated in Germany from about 1480 to 1530, and the style continued in use occasionally until the 20th century. Most importantly, all of the works of [[Martin Luther]], leading to the [[Protestant Reformation]], as well as the [[Apocalypse (Dürer)|Apocalypse]] of [[Albrecht Dürer]] (1498) used this typeface. [[Johann Bämler]], a printer from [[Augsburg]], probably first used it as early as 1472. The origins of the name remain unclear; some assume that a typeface-carver from the village of Schwabach—one who worked externally and who thus became known as the ''Schwabacher''—designed the typeface.

====''Textualis''====
German ''Textualis'' is usually very heavy and angular, and there are few features that are common to all occurrences of the script. One common feature is the use of the letter "w" for Latin "vu" or "uu". ''Textualis'' was used in the 13th and 14th centuries, afterward becoming more elaborate and decorated and used for liturgical works only.

[[Johann Gutenberg]] used a ''textualis'' [[typeface]] for his famous [[Gutenberg Bible]], possibly the first book ever to be printed with movable type, in 1455. [[Schwabacher]], a black letter with more rounded letters, soon became the usual printed [[typeface]], but it was replaced by [[Fraktur (typeface)|Fraktur]] in the early 17th century.

[[File:Fraktur walbaum.png|left|frame|[[Fraktur (typeface)|Fraktur]] lettering]]
Fraktur came into use when Emperor [[Maximilian I, Holy Roman Emperor|Maximilian I]] (1493–1519) established a series of books and had a new typeface created specifically for this purpose. In the 19th century, the use of antiqua alongside Fraktur increased, leading to the [[Antiqua-Fraktur dispute]], which lasted until the [[Nazism|Nazis]] abandoned Fraktur in 1941. Since it was so common, all kinds of black letter tend to be called ''Fraktur'' in German.

====''Cursiva''====
[[File:Gebrochene Schriften klein.png|thumb]] German ''cursiva'' is similar to the cursive scripts in other areas, but forms of "a", "s" and other letters are more varied; here too, the letter "w" is often used. A ''hybrida'' form, which was basically ''cursiva'' with fewer looped letters and with similar square proportions as ''textualis'', was used in the 15th and 16th centuries.

In the 18th century, the pointed quill was adopted for black-letter handwriting. In the early 20th century, the [[Sütterlin]] script was introduced in the schools.

===Italy===

====''Rotunda''====
:''Full article at [[Rotunda (script)]]''
[[Italian language|Italian]] black letter also is known as [[rotunda (script)|rotunda]], as it was less angular than in northern centres. The most usual form of Italian ''rotunda'' was ''littera bononiensis'', used at the [[University of Bologna]] in the 13th century. Biting is a common feature in ''rotunda'', but breaking is not.

Italian ''Rotunda'' also is characterized by unique abbreviations, such as q with a line beneath the bow signifying "qui", and unusual spellings, such as x for s ("milex" rather than "miles").

====''Cursiva''====
[[Italian cursive]] developed in the 13th century from scripts used by notaries. The more calligraphic form is known as ''minuscola cancelleresca italiana'' (or simply ''cancelleresca'', [[chancery hand]]), which developed into a [[book hand]], a script used for writing books rather than charters, in the 14th century. ''Cancelleresca'' influenced the development of ''[[bastarda]]'' in France and [[Secretary hand]] in England.

===The Netherlands===
[[File:DutchFraktur.tif|thumb|right|Textualis mixed with select use of Antiqua in an 1853 Dutch edition of the [[New Testament]].]]

====''Textualis''====
A textualis form, commonly known as ''Gotisch'' or 'Gothic script' was used for general publications from the fifteenth century on, but became restricted to official documents and religious publications during the seventeenth century. Its use persisted into the nineteenth century for editions of the [[Statenvertaling|State Translation]] of the [[Bible]], but had otherwise become obsolete.

==Unicode==
Mathematical black-letter characters are separately encoded in [[Unicode]] in the [[Mathematical alphanumeric symbols]] range at U+1D504-1D537 and U+1D56C-1D59F (bold), except for individual letters already encoded in the [[Letterlike Symbols]] range (plus [[long s]] at U+017F).

This block of characters should be used only for setting mathematical text, as mathematical texts use black-letter symbols contrastively to other letter styles. For stylized black-letter prose, the normal Latin letters should be used, with font choice or other markup used to indicate black-letter styling. The character names use "Fraktur" for the mathematical alphanumeric symbols, while "black-letter" is used for those symbol characters in the letterlike symbols range.
:{{SMP|de-Latf|𝔄 𝔅 ℭ 𝔇 𝔈 𝔉 𝔊 ℌ ℑ 𝔍 𝔎 𝔏 𝔐 𝔑 𝔒 𝔓 𝔔 ℜ 𝔖 𝔗 𝔘 𝔙 𝔚 𝔛 𝔜 ℨ 𝔞 𝔟 𝔠 𝔡 𝔢 𝔣 𝔤 𝔥 𝔦 𝔧 𝔨 𝔩 𝔪 𝔫 𝔬 𝔭 𝔮 𝔯 𝔰 𝔱 𝔲 𝔳 𝔴 𝔵 𝔶 𝔷}}
:{{SMP|de-Latf|𝕬 𝕭 𝕮 𝕯 𝕰 𝕱 𝕲 𝕳 𝕴 𝕵 𝕶 𝕷 𝕸 𝕹 𝕺 𝕻 𝕼 𝕽 𝕾 𝕿 𝖀 𝖁 𝖂 𝖃 𝖄 𝖅 𝖆 𝖇 𝖈 𝖉 𝖊 𝖋 𝖌 𝖍 𝖎 𝖏 𝖐 𝖑 𝖒 𝖓 𝖔 𝖕 𝖖 𝖗 𝖘 𝖙 𝖚 𝖛 𝖜 𝖝 𝖞 𝖟}}
{{small|Note: (The above may not render fully in all web browsers.)}}

Fonts supporting the range include [[Code2001]].

For normal text writing, the ordinary Latin code points are used. The black-letter style is then determined by a font with black-letter glyphs. The glyphs in the SMP should only be used for mathematical typesetting, not for ordinary text. They are of limited use for writing German, as they lack [[umlaut (diacritic)|umlaut diacritics]] and the ligature [[ß]].

==See also==
*[[Black letter law]]
*[[Breitkopf Fraktur]]
*[[Calligraphy]]
*[[Fraktur (script)|Fraktur]]
*[[Gaelic type|Gaelic script]]
*[[Typefaces]]
*[[Typography]]

==References==
{{More footnotes|date=June 2009}}
{{reflist|2}}

==Further reading==
*Bernhard Bischoff, ''Latin Palaeography: Antiquity and the Middle Ages'', Cambridge University Press, 1989.
* {{Cite book|title=Blackletter: type and national identity |editor1-first=Peter |editor1-last=Bain |editor2-first=Paul |editor2-last=Shaw |others=[[Cooper Union for the Advancement of Science and Art]] |publisher=Princeton Architectural Press|year=1998 |isbn=978-1-56898-125-3}}

==External links==
{{Commons|Blackletter}}
{{Wiktionary|black letter}}
*[http://www.theiling.de/schrift/ Learn Blackletter Online]
*[http://www.bfds.org Association for the German Script and Language]
* [http://robert-pfeffer.net/schriftarten/englisch/pfeffer_simpelgotisch.html Pfeffer Simpelgotisch] A simple OpenType blackletter font setting ſ and s by itself

{{European calligraphy}}
{{Typography terms}}

[[Category:Blackletter| ]]
[[Category:Palaeography]]
[[Category:Typography]]
[[Category:Medieval scripts]]

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[[File:Bingham_mayo.jpg|thumb|right|302px|[[Mayonnaise]] is a Bingham plastic. The surface has ridges and peaks because Bingham plastics mimic solids under low shear stresses.]]
A '''Bingham plastic''' is a [[viscoplastic]] material that behaves as a rigid body at low stresses but flows as a [[viscosity|viscous]] [[fluid]] at high stress. It is named after [[Eugene C. Bingham]] who proposed its mathematical form.<ref>E.C. Bingham,(1916) ''U.S. Bureau of Standards Bulletin'', 13, 309-353 "An Investigation of the Laws of Plastic Flow"</ref>

It is used as a common [[mathematical model]] of [[mud]] flow in [[drilling engineering]], and in the handling of [[slurry|slurries]]. A common example is [[toothpaste]],<ref name=Steffe>J. F. Steffe (1996) ''Rheological Methods in Food Process Engineering'' 2nd ed ISBN 0-9632036-1-4</ref> which will not be [[extruded]] until a certain [[pressure]] is applied to the tube. It then is pushed out as a solid plug.

==Explanation==
[[File:Bingham1a.svg|thumb|left|302px|Figure 1. Bingham Plastic flow as described by Bingham]]
'''Figure 1''' shows a graph of the behaviour of an ordinary viscous (or Newtonian) fluid in red, for example in a pipe. If the pressure at one end of a pipe is increased this produces a stress on the fluid tending to make it move (called the [[shear stress]]) and the volumetric flow rate increases proportionally. However for a Bingham Plastic fluid (in blue), stress can be applied but it will not flow until a certain value, the [[yield stress]], is reached. Beyond this point the flow rate increases steadily with increasing shear stress. This is roughly the way in which Bingham presented his observation, in an experimental study of paints.<ref>E. C. Bingham (1922) ''Fluidity and Plasticity'' McGraw-Hill (New York) page 219</ref> These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a Newtonian fluid.
[[File:Bingham2a.svg|thumb|right|302px|Figure 2. Bingham Plastic flow as described currently]]
'''Figure 2''' shows the way in which it is normally presented currently.<ref name=Steffe/> The graph shows [[shear stress]] on the vertical axis and [[shear rate]] on the horizontal one. (Volumetric flow rate depends on the size of the pipe, shear rate is a measure of how the velocity changes with distance. It is proportional to flow rate, but does not depend on pipe size.) As before, the Newtonian fluid flows and gives a shear rate for any finite value of shear stress. However, the Bingham Plastic again does not exhibit any shear rate (no flow and thus no velocity) until a certain stress is achieved. For the Newtonian fluid the slope of this line is the [[viscosity]], which is the only parameter needed to describe its flow. By contrast the Bingham Plastic requires two parameters, the '''yield stress''' and the slope of the line, known as the '''plastic viscosity'''.

The physical reason for this behaviour is that the liquid contains particles (e.g. clay) or large molecules (e.g. polymers) which have some kind of interaction, creating a weak solid structure, formerly known as a '''false body''', and a certain amount of stress is required to break this structure. Once the structure has been broken, the particles move with the liquid under viscous forces. If the stress is removed, the particles associate again.

==Definition==
The material is an elastic solid for [[shear stress]] ''τ'', less than a critical value <math>\tau_0</math>. Once the critical shear [[shear stress|stress]] (or "[[yield (engineering)|yield stress]]") is exceeded, the material flows in such a way that the [[shear rate]], ∂''u''/∂''y'' (as defined in the article on [[viscosity]]), is directly proportional to the amount by which the applied shear stress exceeds the yield stress:

:<math>\frac {\partial u} {\partial y} = \left\{\begin{matrix} 0 &, \tau < \tau_0 \\ (\tau - \tau_0)/ {\mu_\infty} &, \tau \ge \tau_0 \end{matrix}\right.</math>

==Friction Factor Formulae==
In fluid flow, it is a common problem to calculate the pressure drop in an established piping network.<ref>{{Cite book| title=Chemical Engineering Fluid Mechanics. | first1=Ron | last1=Darby | publisher=Marcel Dekker | year=1996 | isbn=0-8247-0444-4| postscript= }}. See Chapter 6.</ref> Once the friction factor, ''f'', is known, it becomes easier to handle different pipe-flow problems, viz. calculating the pressure drop for evaluating pumping costs or to find the flow-rate in a piping network for a given pressure drop. It is usually extremely difficult to arrive at exact analytical solution to calculate the friction factor associated with flow of non-Newtonian fluids and therefore explicit approximations are used to calculate it. Once the friction factor has been calculated the pressure drop can be easily determined for a given flow by the [[Darcy–Weisbach equation]]:
:<math> \ f = \ {2 h_f g D \over L V^2}</math>

where:
* <math>{\bold \ h_f}</math> is the frictional head loss ([[SI units]]: m)
* <math>{\bold \ f}</math> is the [[Darcy friction factor]] ([[SI units]]: Dimensionless)
* <math>{\bold \ L}</math> is the pipe length ([[SI units]]: m)
* <math>{\bold \ g}</math> is the gravitational acceleration ([[SI units]]: m/s²)
* <math>{\bold \ D}</math> is the pipe diameter ([[SI units]]: m)
* <math>{\bold \ V}</math> is the mean fluid velocity ([[SI units]]: m/s)

===Laminar flow===
An exact description of friction loss for Bingham plastics in fully developed laminar pipe flow was first published by Buckingham.<ref>Buckingham, E. (1921). "on Plastic Flow through Capillary Tubes". ''ASTM Proceedings'' '''21''': 1154–1156.</ref> His expression, the ''Buckingham-Reiner'' equation, can be written in a dimensionless form as follows:
:<math> \ f_L = \ {64 \over Re}\left[1 + {He\over 6 Re} - {64\over3}\left({He^4\over {f}^3 Re^7}\right)\right]</math>

where:
* <math>{\bold \ f}</math> is the laminar flow friction factor ([[SI units]]: Dimensionless)
* <math>{\bold \ Re}</math> is the [[Reynolds number]] ([[SI units]]: Dimensionless)
* <math>{\bold \ He}</math> is the Hedstrom number ([[SI units]]: Dimensionless)

The [[Reynolds number]] and the Hedstrom number are respectively defined as:
:<math> \mathrm{Re} = { \rho {\ V} D \over {\mu}} </math>, and

:<math> \mathrm{He} = { \rho {\ D^2 } {\ \tau_o} \over {{\mu}^2}} </math>

where:
* <math>{\bold \rho}</math> is the mass density of fluid ([[SI units]]: kg/m3)
* <math>{\bold \ \mu}</math> is the dynamic [[viscosity]] of fluid ([[SI units]]: kg/m s)

===Turbulent flow===
Darby and Melson developed an empirical expression <ref name=Darby>Darby, R. and Melson J.(1981). "How to predict the friction factor for flow of Bingham plastics". ''Chemical Engineering'' '''28''': 59–61.</ref>
that was then refined, and is given by:<ref name=Darbyetal>Darby, R. et al. (1992). "Prediction friction loss in slurry pipes." ''Chemical Engineering'' '''September''': .</ref>
:<math> \ f_T = \ {10^a} \ {Re^{-0.193}} </math>
where:
* <math>{\bold \ f_T}</math> is the turbulent flow friction factor ([[SI units]]: Dimensionless)
* <math> \ a = -1.47\left[1 + 0.146{\ e^{-2.9\times {10^{-5}}He}}\right] </math>

==Approximations of the ''Buckingham-Reiner'' equation==
Although an exact analytical solution of the ''Buckingham-Reiner'' equation can be obtained because it is a fourth order polynomial equation in ''f'', due to complexity of the solution it is rarely employed. Therefore, researchers have tried to develop explicit approximations for the ''Buckingham-Reiner'' equation.

===Swamee-Aggarwal Equation===
The ''Swamee Aggarwal'' equation is used to solve directly for the Darcy–Weisbach friction factor ''f'' for laminar flow of Bingham plastic fluids.<ref>Swamee, P.K. and Aggarwal, N.(2011). "Explicit equations for laminar flow of Bingham plastic fluids". ''Journal of Petroleum Science and Engineering''. {{doi|10.1016/j.petrol.2011.01.015}}.</ref> It is an approximation of the implicit ''Buckingham-Reiner'' equation, but the discrepancy from experimental data is well within the accuracy of the data.
The ''Swamee-Aggarwal'' equation is given by:
:<math> \ f_L = \ {64 \over Re} + {10.67 + 0.1414{({He\over Re})^{1.143}}\over {\left[1 + 0.0149{({He\over Re})^{1.16}}\right]Re }}\left({He\over Re}\right)</math>

===Danish-Kumar Solution===
Danish ''et al.'' have provided an explicit procedure to calculate the friction factor ''f'' by using the Adomian decomposition method.<ref>Danish, M. ''et al.'' (1981). "Approximate explicit analytical expressions of friction factor for flow
of Bingham fluids in smooth pipes using Adomian decomposition method". ''Communications in Nonlinear Science and Numerical Simulation'' '''16''': 239–251.</ref> The friction factor containing two terms through this method is given as:
:<math> f_L = \frac{K_1 + \dfrac{4 K_2}{\left( K_1 + \frac{K_1 K_2}{K_1^4 + 3 K_2}\right)^3}}{1+ \dfrac{3 K_2}{\left(K_1 + \frac{K_1 K_2}{K_1^4 + 3 K_2}\right)^4}}</math>
where:
:<math> \ K_1 = \ {16 \over Re} + {16 He \over 6{Re^2}}</math>, and
:<math> \ K_2 = \ - {16 {He^4} \over 3{Re^8}}</math>

==Combined Equation for friction factor for all flow regimes==
===Darby-Melson Equation===
In 1981, Darby and Melson, using the approach of Churchill<ref>Churchill, S.W. (1977). "Friction factor equation spans all fluid-flow regimes". ''Chemical Engineering'' '''Nov. 7''': 91–92.</ref> and of Churchill and Usagi,<ref>Churchill, S.W. and Usagi, R.A. (1972). "A general expression for the correlation of rates of transfer and other phenomena". ''AIChE Journal'' '''18(6)''': 1121-1128.</ref> developed an expression to get a single friction factor equation valid for all flow regimes:<ref name=Darby/>
:<math> \ f = \ {\left[{f_L}^m + {f_T}^m\right]}^{1\over m}</math>
where:
:<math> \ m = \ 1.7 + {40000\over Re} </math>

Both ''Swamee-Aggarwal'' equation and the ''Darby-Melson'' equation can be combined to give an explicit equation for determining the friction factor of Bingham plastic fluids in any regime. Relative roughness is not a parameter in any of the equations because the friction factor of Bingham plastic fluids is not sensitive to pipe roughness.

==See also==

* [[Bagnold number]]
* [[Bernoulli's principle]]
* [[Rheology]]

==References==
{{reflist}}

{{DEFAULTSORT:Bingham Plastic}}
[[Category:Materials]]
[[Category:Non-Newtonian fluids]]
[[Category:Viscosity]]
[[Category:Offshore engineering]]

Multiple integral

2013-09-22T04:31:19Z

182.64.74.1: /* Multiple improper integral */

== Summary ==
Created in [[Mathematica]]. Plots:

<math>y = \sqrt[3]{x}</math>

I'm not sure about the copyright status since this seems awfully mundane...
== Licensing ==
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