|
|
| Line 1: |
Line 1: |
| In [[fluid mechanics]] and [[mathematics]], a '''capillary surface''' is a [[surface]] that represents the interface between two different [[fluid]]s. As a consequence of being a [[surface]], a capillary surface has no thickness in slight contrast with most real fluid interfaces.
| | Moncler down jackets include the most common trendy jackets that a specific could locate today led to particular individual that owns Moncler coats understands what statements of fashion are regarding. It is actually straight to discover the kind which possess to been looking for, because of the fact all the patterns have been made maintaining in current tendencies and vogue necessities.<br><br>Zipper: Some down-filled garments can unzip from the bottom. This most people to take bigger strides, and offer more options in staying warmer or cooler. Interior zip pocket stores cards, keys, additional small essentials, so also it not afraid the things will disappear after you zip the pocket. Some special zippers will always make the coat look more eye-catching.<br><br>How do they really be known as cheap? The reason is that some shopping to do several times a year, handling registration, especially in the off winter. Thus, due to seasonal changes with these [http://www.iprs.it/includes/italia/ moncler outlet] shopping stores, yow will discover incredible gives.<br><br>They are longing for every new Louis handbag, but extravagant price scares these items. For the purpose of meeting ordinary women`s desires, replica Louis vuitton handbags are available in to being. Manufacturers take quality as priority and take note . on each detail. Might be similar for the authentic you. What`s morn, the buying is competitive and appealing. Women can buy chaap moncler handbags without the worries at the back of.<br><br>Moncler different types, different color coat and attractive male and feminine. For men they halloween night costumes and with warm to clothes, it keep man glamour and warm body bloom in cold the winter months. Pink, brown and black for women, to let look fashionable, rich and colorful in the street. Also with exquisite workmanship, it became popular Ugg boots after selling price.<br><br>You won't have regarding troubled about losing balance due. Instead, you'd walk in comfort and style. Moncler boots have special designs and materials which ensures you keep a comfortable temperature around your feet in 25 centigrade. If you can get warmy and stylish at one time with such cheap but chic moncler online, why not do that? You needn't to pay such money to obtain a high-end Jimmy Choo clothes, but you might have a widely selection with colourful Moncler Women without bankrupt.<br><br>Monclershop.hk has several style of moncler kids clothing. There have a jacket with a natural color which attracted my eyes while i first saw it, its appearance rrs extremely cull as well as have many size fit for 28 years old children, if you buy this green moncler winter down, only need your $238 while the rip is $800,you may possibly enjoy will shipping on all orders, 5-7 days to worldwide,100% satisfaction and funds back make sure.<br><br>There were some impressive west in navy ($ 1,700) along with the other overall blue and white stripes ($ 1530) in the design of Browne was halted. Royal blue corduroy trousers were together with legs ($ 850), are usually employed in ski boots, they aren't werent attractive enough to divert attention from the truth that the collection of clothing and boots. |
| | |
| Capillary surfaces are of interest in mathematics because the problems involved are very nonlinear and have interesting properties, such as discontinuous dependence on boundary data at isolated points.<ref name="Finn">{{cite paper
| |
| | author = Robert Finn
| |
| | title = Capillary Surface Interfaces
| |
| | version =
| |
| | publisher = [[American Mathematical Society]]
| |
| | year = 1999
| |
| | url = http://www.ams.org/notices/199907/fea-finn.pdf
| |
| | format =
| |
| | accessdate = }}</ref> In particular, static capillary surfaces with gravity absent have constant [[mean curvature]], so that a [[minimal surface]] is a special case of static capillary surface.
| |
| | |
| They are also of practical interest for fluid management in space (or other environments free of [[body force]]s), where both flow and static configuration are often dominated by capillary effects.
| |
| | |
| ==The stress balance equation==
| |
| The defining equation for a capillary surface is called the stress balance equation,<ref>[http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/ Surface Tension Module], by John W. M. Bush, at [[MIT OCW]]</ref> which can be derived by considering the forces and stresses acting on a small volume that is partly bounded by a capillary surface. For a fluid meeting another fluid (the "other" fluid notated with bars) at a surface <math>S</math>, the equation reads
| |
| | |
| :<math>(\sigma_{ij} - \bar{\sigma}_{ij}) \mathbf{\hat{n}} = - \gamma \mathbf{\hat{n}} (\nabla_{\!S} \cdot \mathbf{\hat{n}}) + \nabla_{\!S} \gamma \qquad ; \quad \nabla_{\!S} \gamma = \nabla \gamma - \mathbf{\hat{n}} (\mathbf{\hat{n}} \cdot \nabla \gamma)</math> | |
| | |
| where <math>\scriptstyle \mathbf{\hat{n}}</math> is the unit [[Surface normal|normal]] pointing toward the "other" fluid (the one whose quantities are notated with bars), <math>\scriptstyle \sigma_{ij}</math> is the [[Derivation of the Navier–Stokes equations#The stress tensor|stress tensor]] (note that on the left is a tensor-vector [[tensor product|product]]), <math>\scriptstyle \gamma</math> is the [[surface tension]] associated with the interface, and <math>\scriptstyle \nabla_S</math> is the [[surface gradient]]. Note that the quantity <math>\scriptstyle -\nabla_{\!S} \cdot \mathbf{\hat{n}}</math> is twice the [[mean curvature]] of the surface.
| |
| | |
| In [[fluid mechanics]], this equation serves as a [[boundary condition]] for interfacial flows, typically complementing the [[Navier–Stokes equations]]. It describes the discontinuity in [[stress (physics)|stress]] that is balanced by forces at the surface. As a boundary condition, it is somewhat unusual in that it introduces a new variable: the surface <math>S</math> that defines the interface. It's not too surprising then that the stress balance equation normally mandates its own boundary conditions.
| |
| | |
| For best use, this vector equation is normally turned into 3 scalar equations via dot product with the unit normal and two selected unit tangents:
| |
| | |
| :<math>((\sigma_{ij} - \bar{\sigma}_{ij}) \mathbf{\hat{n}}) \cdot \mathbf{\hat{n}} = -\gamma \nabla_{\!S} \cdot \mathbf{\hat{n}}</math>
| |
| | |
| :<math>((\sigma_{ij} - \bar{\sigma}_{ij}) \mathbf{\hat{n}}) \cdot \mathbf{\hat{t}_1} = \nabla_{\!S} \gamma \cdot \mathbf{\hat{t}_1}</math>
| |
| | |
| :<math>((\sigma_{ij} - \bar{\sigma}_{ij}) \mathbf{\hat{n}}) \cdot \mathbf{\hat{t}_2} = \nabla_{\!S} \gamma \cdot \mathbf{\hat{t}_2}</math>
| |
| | |
| Note that the products lacking dots are [[tensor product]]s of tensors with vectors (resulting in vectors similar to a matrix-vector product), those with dots are [[dot product]]s. The first equation is called the '''normal stress equation''', or the normal stress boundary condition. The second two equations are called '''tangential stress equations'''.
| |
| | |
| ===The stress tensor===
| |
| | |
| The stress tensor is related to [[velocity]] and pressure. Its actual form will depend on the specific fluid being dealt with, for the common case of incompressible Newtonian flow the stress tensor is given by
| |
| | |
| :<math> | |
| \begin{align}
| |
| | |
| \sigma_{ij} &=
| |
| -\begin{pmatrix}
| |
| p&0&0\\
| |
| 0&p&0\\
| |
| 0&0&p
| |
| \end{pmatrix} +
| |
| | |
| \mu \begin{pmatrix}
| |
| 2 \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} & \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \\
| |
| \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} & 2 \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \\
| |
| \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} & \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} & 2\frac{\partial w}{\partial z}
| |
| \end{pmatrix} \\
| |
| | |
| &= -p I + \mu (\nabla \mathbf{v} + (\nabla \mathbf{v})^T)
| |
| | |
| \end{align}
| |
| </math>
| |
| | |
| where <math>p</math> is the [[pressure]] in the fluid, <math>\scriptstyle \mathbf{v}</math> is the velocity, and <math>\mu</math> is the [[viscosity]].
| |
| | |
| ==Static interfaces==
| |
| | |
| In the absence of motion, the stress tensors yield only [[hydrostatic pressure]] so that <math>\scriptstyle \sigma_{ij} = -pI</math>, regardless of fluid type or compressibility. Considering the normal and tangential equations,
| |
| | |
| :<math>\bar p - p = \gamma \nabla \cdot \mathbf{\hat{n}}</math>
| |
| | |
| :<math>0 = \nabla \gamma \cdot \mathbf{\hat{t}}</math>
| |
| | |
| The first equation establishes that curvature forces are balanced by pressure forces. The second equation implies that a static interface cannot exist in the presence of nonzero surface tension gradient.
| |
| | |
| If gravity is the only [[body force]] present, the [[Navier–Stokes equations]] simplify significantly:
| |
| | |
| :<math>0 = -\nabla p + \rho \mathbf{g}</math>
| |
| | |
| If coordinates are chosen so that gravity is nonzero only in the <math>z</math> direction, this equation degrades to a particularly simple form:
| |
| | |
| :<math>\frac{d p}{d z} = \rho g \quad \Rightarrow \quad p = \rho g z + p_0</math>
| |
| | |
| where <math>p_0</math> is an integration constant that represents some reference pressure at <math>z = 0</math>. Substituting this into the normal stress equation yields what is known as the [[Young-Laplace equation]]:
| |
| | |
| :<math>\bar \rho g z + \bar p_0 - (\rho g z + p_0) = \gamma \nabla \cdot \mathbf{\hat{n}} \quad \Rightarrow \quad \Delta \rho g z + \Delta p = \gamma \nabla \cdot \mathbf{\hat{n}}</math>
| |
| | |
| where <math>\Delta p</math> is the (constant) pressure difference across the interface, and <math>\Delta \rho</math> is the difference in [[density]]. Note that, since this equation defines a surface, <math>z</math> is the <math>z</math> coordinate of the capillary surface. This [[nonlinear]] [[partial differential equation]] when supplied with the right boundary conditions will define the static interface.
| |
| | |
| The pressure difference above is a constant, but its value will change if the <math>z</math> coordinate is shifted. The linear solution to pressure implies that, '''unless the gravity term is absent''', it is always possible to define the <math>z</math> coordinate so that <math>\Delta p = 0</math>. [[Scale analysis|Nondimensionalized]], the Young-Laplace equation is usually studied in the form <ref name="Finn"/>
| |
| | |
| :<math>\kappa z + \lambda = \nabla \cdot \mathbf{\hat{n}}</math>
| |
| | |
| where (if gravity is in the negative <math>z</math> direction) <math>\kappa</math> is positive if the denser fluid is "inside" the interface, negative if it is "outside", and zero if there is no gravity or if there is no difference in density between the fluids.
| |
| | |
| This [[nonlinear]] equation has some rich properties, especially in terms of existence of unique solutions. For example, the nonexistence of solution to some [[boundary value problem]] implies that, physically, the problem can't be static. If a solution does exist, normally it'll exist for very specific values of <math>\lambda</math>, which is representative of the pressure jump across the interface. This is interesting because there isn't another physical equation to determine the pressure difference. In a capillary tube, for example, implementing the contact angle boundary condition will yield a unique solution for exactly one value of <math>\lambda</math>. Solutions often aren't unique, this implies that there are multiple static interfaces possible; while they may all solve the same boundary value problem, the minimization of energy will normally favor one. Different solutions are called ''configurations'' of the interface.
| |
| | |
| ===Energy consideration===
| |
| | |
| A deep property of capillary surfaces is the [[surface energy]] that is imparted by surface tension:
| |
| | |
| :<math>E_S = \gamma_S A_S\,</math>
| |
| | |
| where <math>A</math> is the area of the surface being considered, and the total [[energy]] is the summation of all energies. Note that ''every'' interface imparts energy. For example, if there are two different fluids (say liquid and gas) inside a solid container with gravity and other energy potentials absent, the energy of the system is
| |
| | |
| :<math>E = \sum \gamma_S A_S = \gamma_{LG} A_{LG} + \gamma_{SG} A_{SG} + \gamma_{SL} A_{SL}\,</math>
| |
| | |
| where the subscripts <math>LG</math>, <math>SG</math>, and <math>SL</math> respectively indicate the liquid–gas, solid–gas, and solid–liquid interfaces. Note that inclusion of gravity would require consideration of the volume enclosed by the capillary surface and the solid walls.
| |
| | |
| [[Image:ForcesAtContactLine.png|frame|right|400px|Illustration of distributed forces at the contact line, with the contact line perpendicular to the image. The vertical part of <math>\gamma_{LG}</math> is balanced by a slight deformation of the solid (not shown and inconsequential to this context).]]
| |
| | |
| Typically the surface tension values between the solid–gas and solid–liquid interfaces are not known. This does not pose a problem; since only changes in energy are of primary interest. If the net solid area <math>A_{SG} + A_{SL}</math> is a constant, and the [[contact angle]] is known, it may be shown that (again, for two different fluids in a solid container)
| |
| | |
| :<math>E = \gamma_{SL}(A_{SL} + A_{SG}) + \gamma_{LG} A_{LG} + \gamma_{LG} A_{SG} \cos(\theta)\,</math>
| |
| | |
| so that
| |
| | |
| :<math>\frac{\Delta E}{\gamma_{LG}} = \Delta A_{LG} + \Delta A_{SG} \cos(\theta) = \Delta A_{LG} - \Delta A_{SL} \cos(\theta)\,</math>
| |
| | |
| where <math>\theta</math> is the [[contact angle]] and the capital delta indicates the change from one configuration to another. To obtain this result, it's necessary to sum (distributed) forces at the contact line (where solid, gas, and liquid meet) in a direction tangent to the solid interface and perpendicular to the contact line:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| 0 &= \sum F_{\mathrm{Contact \ line}} \\
| |
| &= \gamma_{LG} \cos(\theta) + \gamma_{SL} - \gamma_{SG}
| |
| \end{align}
| |
| </math> | |
| | |
| where the sum is zero because of the [[Statics|static]] state. When solutions to the Young-Laplace equation aren't unique, the most physically favorable solution is the one of minimum energy, though experiments (especially low gravity) show that [[metastable]] surfaces can be surprisingly persistent, and that the most stable configuration can become metastable through mechanical jarring without too much difficulty. On the other hand, a metastable surface can sometimes spontaneously achieve lower energy without any input (seemingly at least) given enough time.
| |
| | |
| ==Boundary conditions==
| |
| Boundary conditions for stress balance describe the capillary surface at the '''contact line''': the line where a solid meets the capillary interface; also, volume constraints can serve as boundary conditions (a suspended drop, for example, has no contact line but clearly must admit a unique solution).
| |
| | |
| For static surfaces, the most common contact line boundary condition is the implementation of the [[contact angle]], which specifies the angle that one of the fluids meets the solid wall. The contact angle condition on the surface <math>S</math> is normally written as:
| |
| | |
| :<math>\mathbf{\hat{n}} \cdot \mathbf{\hat{v}} = \cos(\theta)\,</math>
| |
| | |
| where <math>\theta</math> is the contact angle. This condition is imposed on the boundary (or boundaries) <math>\scriptstyle \partial S</math> of the surface. <math>\scriptstyle \hat v</math> is the unit outward normal to the solid surface, and <math>\scriptstyle \hat n</math> is a unit normal to <math>S</math>. Choice of <math>\scriptstyle \hat n</math> depends on which fluid the contact angle is specified for.
| |
| | |
| For dynamic interfaces, the boundary condition showed above works well if the contact line velocity is low. If the velocity is high, the contact angle will change ("dynamic contact angle"), and as of 2007 the mechanics of the moving contact line (or even the validity of the contact angle as a parameter) is not known and an area of active research.<ref>{{cite paper
| |
| | author = E. B. Dussan V, Enrique Ramé, and Stephen Garoff
| |
| | title = On identifying the appropriate boundary conditions at a moving contact line: an experimental investigation
| |
| | version =
| |
| | publisher = CJO
| |
| | year = 2006
| |
| | url =
| |
| | format =
| |
| | accessdate = }}</ref>
| |
| | |
| ==See also==
| |
| * [[Capillary pressure]]
| |
| * [[Surface energy]]
| |
| * [[Surface tension]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| [[Category:Fluid mechanics]]
| |
| [[Category:Fluid dynamics]]
| |
| [[Category:Fluid statics]]
| |
Moncler down jackets include the most common trendy jackets that a specific could locate today led to particular individual that owns Moncler coats understands what statements of fashion are regarding. It is actually straight to discover the kind which possess to been looking for, because of the fact all the patterns have been made maintaining in current tendencies and vogue necessities.
Zipper: Some down-filled garments can unzip from the bottom. This most people to take bigger strides, and offer more options in staying warmer or cooler. Interior zip pocket stores cards, keys, additional small essentials, so also it not afraid the things will disappear after you zip the pocket. Some special zippers will always make the coat look more eye-catching.
How do they really be known as cheap? The reason is that some shopping to do several times a year, handling registration, especially in the off winter. Thus, due to seasonal changes with these moncler outlet shopping stores, yow will discover incredible gives.
They are longing for every new Louis handbag, but extravagant price scares these items. For the purpose of meeting ordinary women`s desires, replica Louis vuitton handbags are available in to being. Manufacturers take quality as priority and take note . on each detail. Might be similar for the authentic you. What`s morn, the buying is competitive and appealing. Women can buy chaap moncler handbags without the worries at the back of.
Moncler different types, different color coat and attractive male and feminine. For men they halloween night costumes and with warm to clothes, it keep man glamour and warm body bloom in cold the winter months. Pink, brown and black for women, to let look fashionable, rich and colorful in the street. Also with exquisite workmanship, it became popular Ugg boots after selling price.
You won't have regarding troubled about losing balance due. Instead, you'd walk in comfort and style. Moncler boots have special designs and materials which ensures you keep a comfortable temperature around your feet in 25 centigrade. If you can get warmy and stylish at one time with such cheap but chic moncler online, why not do that? You needn't to pay such money to obtain a high-end Jimmy Choo clothes, but you might have a widely selection with colourful Moncler Women without bankrupt.
Monclershop.hk has several style of moncler kids clothing. There have a jacket with a natural color which attracted my eyes while i first saw it, its appearance rrs extremely cull as well as have many size fit for 28 years old children, if you buy this green moncler winter down, only need your $238 while the rip is $800,you may possibly enjoy will shipping on all orders, 5-7 days to worldwide,100% satisfaction and funds back make sure.
There were some impressive west in navy ($ 1,700) along with the other overall blue and white stripes ($ 1530) in the design of Browne was halted. Royal blue corduroy trousers were together with legs ($ 850), are usually employed in ski boots, they aren't werent attractive enough to divert attention from the truth that the collection of clothing and boots.