Viscosity

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Viscosity - Wikipedia, the free encyclopedia
Fluid dynamics · Viscosity · Newtonian fluids. Non-Newtonian fluids. Surface tension ... Viscosity is a measure of the resistance of a fluid which is being ...
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Viscosity is a measure of the Drag (physics) of a fluid to deform under either shear stress or extensional stress. It is commonly perceived as "thickness", or resistance to flow. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. All real fluids (except superfluids) have some resistance to stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid.{{cite book].

Etymology The word "viscosity" derives from the Latin word "" for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds. The Online Etymology Dictionary

Viscosity coefficients When looking at a value for viscosity, the number that one most often sees is the coefficient of viscosity. There are several different viscosity coeffients depending on the nature of applied stress and nature of the fluid. They are introduced in the main books on hydrodynamics Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965), Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press,(1959) and rheology Barnes, H.A. "A Handbook of Elementary Rheology", Institute of Non-Newtonian Fluid mechanics, UK (2000)

Shear and dynamic viscosity are much more known than two others. That is why they are often reffered to as simply viscosity. Simply put, this quantity is the ratio between the pressure exerted on the surface of a fluid, in the lateral or horizontal direction, to the change in velocity of the fluid as you move down in the fluid (this is what is referred to as a velocity gradient). For example, at "room temperature", water has a nominal viscosity of 1.0 x 10-3 Pa∙s and motor oil has a nominal apparent viscosity of 250 x 10-3 Pa∙s.

Extensional viscosity is widely used for characterizing polymers. Volume viscosity is essentual for Acoustics in fluids, see Stokes' law (sound attenuation) Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)

Newton's theory .

In general, in any flow, layers move at different velocity and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.

Isaac Newton postulated that, for straight, Parallel (geometry) and uniform flow, the shear stress, τ, between layers is proportional to the velocity gradient, ∂u/∂y, in the direction perpendicular to the layers.

\tau=\eta \frac{\partial u}{\partial y}.

Here, the constant η is known as the coefficient of viscosity, the viscosity, the dynamic viscosity, or the Newtonian viscosity. Many fluids, such as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity.

The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance y, and separated by a heterogeneous substance. Assuming that the plates are very large, with a large area A, such that edge effects may be ignored, and that the lower plate is fixed, let a force F be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow (as opposed to just deformation elasticity (solid mechanics) until the shear stress in the substance balances the applied force), the substance is called a fluid. The applied force is proportional to the area and velocity of the plate and inversely proportional to the distance between the plates. Combining these three relations results in the equation F = η(Au/y), where η is the proportionality factor called the absolute viscosity (with units Pa·s = kg/(m·s) or slugs/(ft·s)). The absolute viscosity is also known as the dynamic viscosity, and is often shortened to simply viscosity. The equation can be expressed in terms of shear stress; τ = F/A = η(u/y). The rate of shear deformation is u/y and can be also written as a shear velocity, du/dy. Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.

James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.

Viscosity Measurement Dynamic viscosity is measured with various types of viscometer. Close temperature control of the fluid is essential to accurate measurements, particularly in materials like lubricants, whose viscosity (-40 < sample temperature |-| [air| 111| 300.55| 17.81|-| [oxygen| 240| 293.15| 14.8|-| [carbon monoxide| 72| 293.85| 8.76|-| [ammonia| 416| 293.65| 12.54|}

Viscosity of a dilute gas The Chapman-Enskog equation may be used to estimate viscosity for a dilute gas. This equation is based on semi-theorethical assumption by Chapman and Enskoq. The equation requires three empirically determined parameters: the collision diameter (σ), the maximum energy of attraction divided by the Boltzman constant (є/к) and the collision integral (ω(T*)).

{\eta}_0 {x 10^7}= {266.93}\frac {(MT)^{1/2--> {\sigma^{2}\omega(T^*)} ; T*=κT/ε



Liquids In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial. Thus, in liquids:



The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.

Viscosity blending of liquids The viscosity blending of two or more liquids having different viscosities is a three-step procedure. The first step is to calculate the Viscosity Blending Index (VBI) of each component of the blend using the following equation (known as a Refutas equation):C.T. Baird (1989), Guide to Petroleum Product Blending, HPI Consultants, Inc. HPI website

(1)   VBN = 14.534 × ln + 0.8) + 10.975

where v is the viscosity in centistokes (cSt) and ln is the natural logarithm (Loge). It is important that the viscosity of each component of the blend be obtained at the same temperature.

The next step is to calculate the VBN of the blend, using this equation:

(2)   VBNBlend = × VBNA + × VBNB + ... + × VBNX

where w is the weight fraction (i.e., % ÷ 100) of each component of the blend.

Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determine the viscosity of the blend by using the invert of equation (1):

(3)   v = ee(VBN - 10.975) ÷ 14.534 − 0.8


where VBN is the viscosity blending number of the blend and e is the transcendental number 2.71828, also known as Euler's number.

Viscosity of materials The viscosity of air and water are by far the two most important materials for aviation aerodynamics and shipping fluid dynamics. Temperature plays the main role in determining viscosity.

Viscosity of air The viscosity of air depends mostly on the temperature.At 15.0 °C, the viscosity of air is 1.78 × 10−5 kg/(m·s). You can get the viscosity of air as a function of altitude from the eXtreme High Altitude Calculator

Viscosity of water The viscosity of water is 8.90 × 10−4 Pa·s or 8.90 × 10−3 dyn·s/cm² at about 25 °C.
As a function of temperature T (K):μ(Pa·s) = A × 10B/(T−C)
where A=2.414 × 10−5 Pa·s ; B = 247.8 K ; and C = 140 K.

Viscosity of various materials

being drizzled. is a semi-solid and so can hold peaks.

Some dynamic viscosities of Newtonian fluids are listed below:

Gases (at 0 °celsius):{| class="wikitable"|- bgcolor="#efefef"!!viscosity|-|hydrogen|17.4 × 10−6|-|[xenons (at 25 °[celsius):

{| class="wikitable"|- bgcolor="#efefef"!!viscosity!viscosity|-|liquid nitrogen @ 77K]*|0.306 × 10−3|0.306|-|methanol*]*|0.604 × 10−3|0.604|-|ethanol*]|0.894 × 10−3|0.894|-|mercury (element)*|1.526 × 10−3|1.526|-|nitrobenzene*]*|1.945 × 10−3|1.945|-|Ethylene glycol*|24.2 × 10−3|24.2|-|[olive oil*|.934|934|-|[castor bean|985 × 10−3|985|-|HFO-380 (oil)|2.022|2022|-|pitch (resin)|2.3 × 108|2.3 × 1011|}

* Data from CRC Handbook of Chemistry and Physics, 73rd edition, 1992-1993. [Fluids with variable compositions, such as [honey, can have a wide range of viscosities. A more complete table can be found http://xtronics.com/reference/viscosity.htm here, including the following: {| class="wikitable" |- bgcolor="#efefef" ! !viscosity cP |- |[honey |2,000–10,000 |- |[molasses |5,000–10,000 |- |molten [glass |10,000–1,000,000 |- |[chocolate syrup |10,000–25,000 |- |[chocolate* | 45,000–130,000 http://www.brookfieldengineering.com/education/applications/laboratory-chocolate-processing.asp |- |[ketchup* |50,000–100,000 |- |[peanut butter |~250,000 |- |[shortening* |~250,000 |} * These materials are highly non-Newtonian fluid.

Viscosity of solids On the basis that all solids flow to a small extent in response to shear stress some researchers The Physics Hypertextbook, by Glen Elert, retrieved on August 1, 2007. The Properties of Glass , page 6, retrieved on August 1, 2007 have contended that substances known as amorphous solids, such as glass and many polymers, may be considered to have viscosity. This has led some to the view that solids are simply liquids with a very high viscosity, typically greater than 1012 Pa•s. This position is often adopted by supporters of the widely held misconception that Glass#Glass as a liquid can be observed in old buildings. This distortion is more likely the result of glass making process rather than the viscosity of glass."Antique windowpanes and the flow of supercooled liquids", by Robert C. Plumb, (Worcester Polytech. Inst., Worcester, MA, 01609, USA), J. Chem. Educ. (1989), 66 (12), 994-6

However, others argue that solids are, in general, elastic for small stresses while fluids are not.{{cite web]s flow at higher stresses, they are characterized by their low-stress behavior. Viscosity may be an appropriate characteristic for solids in a plasticity (physics) regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example Maxwell materials, to describe the relationship between stress and the rate of change of strain, rather than rate of shear.

These distinctions may be largely resolved by considering the constitutive equations of the material in question, which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and their elasticity are important in a particular range of deformation and deformation rate are called viscoelasticity. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids.

Viscosity of amorphous materials Viscous flow in Amorphous solid (e.g. in Glass and melts) is a thermally activated process:

\eta = A \cdot e^{Q/RT}

where Q is activation energy, T is temperature, R is the molar gas constant and A is approximately a constant.

The viscous flow in amorphous materials is characterised by a deviation from the Arrhenius equation behaviour: Q changes from a high value Q_H at low temperatures (in the glassy state) to a low value Q_L at high temperatures (in the liquid state). Depending on this change, amorphous materials are classified as either



The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:

R_D = Q_H/Q_L

and strong material have R_D < 2\; whereas fragile materials have R_D \ge 2

The viscosity of amorphous materials is quite exactly described by a two-exponential equation:

\eta = A_1 \cdot T \cdot + A_2 \cdot e^{B/RT} \cdot + C \cdot e^{D/RT}

with constants A_1 , A_2 , B, C and D related to thermodynamic parameters of joining bonds of an amorphous material.

Not very far from the glass transition temperature, T_g, this equation can be approximated by a Vogel-Tammann-Fulcher (VTF) equation or a Kohlrausch-type stretched-exponential law.

If the temperature is significantly lower than the glass transition temperature, T\ll T_g\;, then the two-exponential equation simplifies to an Arrhenius type equation:

\eta = A \cdot e^{Q_H/RT}

with:

Q_H = H_d + H_m

where H_d is the enthalpy of formation of broken bonds (termed configurons) and H_m is the enthalpy of their motion.

When the temperature is less than the glass transition temperature, T < T_g, the activation energy of viscosity is high because the amorphous materials are in the glassy state and most of their joining bonds are intact.

If the temperature is highly above the glass transition temperature, T \gg Tg, the two-exponential equation also simplifies to an Arrhenius type equation:

\eta = A\cdot e^{Q_L/RT}

with:

Q_L = H_m

When the temperature is higher than the glass transition temperature, T > T_g, the activation energy of viscosity is low because amorphous materials are melt and have most of their joining bonds broken which facilitates flow.

Volume (Bulk) viscosity The negative-one-third of the Trace (linear algebra) of the Stress (physics) tensor is often identified with the thermodynamic pressure,

-{1\over3}T_a^a = p,

which only depends upon the equilibrium state potentials like temperature and density (equation of state). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution plus another contribution which is proportional to the divergence of the velocity field. This constant of proportionality is called the volume viscosity.

Eddy viscosity In the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation). Values of eddy viscosity used in modeling ocean circulation may be from 5x104 to 106 Pa·s depending upon the resolution of the numerical grid.

Fluidity The reciprocal of viscosity is fluidity, usually symbolized by \phi = 1/\eta or F=1/\eta, depending on the convention used, measured in reciprocal poise (centimetre·second·gram-1), sometimes called the rhe. Fluidity is seldom used in engineering practice.

The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components a and b, the fluidity when a and b are mixed is

F \approx \chi_a F_a + \chi_b F_b

which is only slightly simpler than the equivalent equation in terms of viscosity:

\eta \approx \frac{1}{\chi_a /\eta_a + \chi_b/\eta_b}

where \chi_a and \chi_b is the mole fraction of component a and b respectively, and \eta_a and \eta_b are the components pure viscosities.

The linear viscous stress tensor (See Hooke's law and strain tensor for an analogous development for linearly elastic materials.)

Viscous forces in a fluid are a function of the rate at which the fluid velocity is changing over distance. The velocity at any point \mathbf{r} is specified by the velocity field \mathbf{v}(\mathbf{r}). The velocity at a small distance d\mathbf{r} from point \mathbf{r} may be written as a Taylor series:

\mathbf{v}(\mathbf{r}+d\mathbf{r}) = \mathbf{v}(\mathbf{r})+\frac{d\mathbf{v-->{d\mathbf{r-->d\mathbf{r}+\ldots

where \frac{d\mathbf{v-->{d\mathbf{r--> is shorthand for the dyadic product of the del operator and the velocity:

\frac{d\mathbf{v-->{d\mathbf{r--> = \begin{bmatrix}\frac{\partial v_x}{\partial x} & \frac{\partial v_x}{\partial y} & \frac{\partial v_x}{\partial z}\\\frac{\partial v_y}{\partial x} & \frac{\partial v_y}{\partial y} & \frac{\partial v_y}{\partial z}\\\frac{\partial v_z}{\partial x} & \frac{\partial v_z}{\partial y}&\frac{\partial v_z}{\partial z}\end{bmatrix}

This is just the Jacobian matrix of the velocity field. Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function of the velocity field. In other words, the forces at \mathbf{r} are a function of \mathbf{v}(\mathbf{r}) and all derivatives of \mathbf{v}(\mathbf{r}) at that point. In the case of linear viscosity, the viscous force will be a function of the Jacobian tensor alone. For almost all practical situations, the linear approximation is sufficient.

If we represent x, y, and z by indices 1, 2, and 3 respectively, the i,j component of the Jacobian may be written as \partial_i v_j where \partial_i is shorthand for \partial /\partial x_i. Note that when the first and higher derivative terms are zero, the velocity of all fluid elements is parallel, and there are no viscous forces.

Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix, and this decomposition is independent of coordinate system, and so has physical significance. The velocity field may be approximated as:

v_i(\mathbf{r}+d\mathbf{r}) = v_i(\mathbf{r})+\frac{1}{2}\left(\partial_i v_j-\partial_j v_i\right)dr_i + \frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)dr_i

where Einstein notation is now being used in which repeated indices in a product are implicitly summed. The second term on the left is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluid about \mathbf{r} with angular velocity \omega where:

\omega=\frac12 \mathbf{\nabla}\times \mathbf{v}=\frac{1}{2}\begin{bmatrix} \partial_2 v_3-\partial_3 v_2\\\partial_3 v_1-\partial_1 v_3\\\partial_1 v_2-\partial_2 v_1\end{bmatrix}

For such a rigid rotation, there is no change in the relative positions of the fluid elements, and so there is no viscous force associated with this term. The remaining symmetric term is responsible for the viscous forces in the fluid. Assuming the fluid is isotropic (i.e. its properties are the same in all directions), then the most general way that the symmetric term (the rate-of-strain tensor) can be broken down in a coordinate-independent (and therefore physically real) way is as the sum of a constant tensor (the rate-of-expansion tensor) and a traceless symmetric tensor (the rate-of-shear tensor):

\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right) = \frac{1}{3}\partial_k v_k \delta_{ij}+\left( \frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)-\frac{1}{3}\partial_k v_k \delta_{ij}\right)

where \delta_{ij} is the Kronecker delta. The most general linear relationship between the stress tensor \mathbf{\sigma} and the rate-of-strain tensor is then a linear combination of these two tensors:

\sigma_{visc;ij} = \zeta\partial_k v_k \delta_{ij}+ \eta\left(\partial_i v_j+\partial_j v_i-\frac{2}{3}\partial_k v_k \delta_{ij}\right)

where \zeta is the coefficient of bulk viscosity (or "second viscosity") and \eta is the coefficient of (shear) viscosity.

The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may be thought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above gives the force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of the molecules is just the hydrostatic pressure. This pressure term (p\delta_{ij}) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.

\sigma_{ij} = p\delta_{ij}+\sigma_{visc;ij}\,

The infinitesimal force dF_i on an infinitesimal area dA_i is then given by the usual relationship:

dF_i=\sigma_{ij}dA_j\,

See also

References

Additional reading | author = Massey, B. S.| title = Mechanics of Fluids| edition = Fifth Edition| publisher= Van Nostrand Reinhold (UK)| year = 1983| id = ISBN 0-442-30552-4 -->

External links



Viscosity - Wikipedia, the free encyclopedia
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In general terms it is the resistance of a liquid to flow ...

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Teacher Page: Viscosity
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Kinematic Viscosity - DiracDelta Science & Engineering Encyclopedia
Science Engineering Encyclopedia ... Kinematic Viscosity The dynamic viscosity of a fluid divided by the fluid density.





 
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