Surface Tension ,Absolute, Dynamic and Kinematic Viscosity

 Surface Tension ,Absolute, Dynamic and Kinematic Viscosity

Surface Tension

The force of attraction between molecules in a liquid gives rise to what is termed surface tension.

The surface tension 𝝈 of a liquid is the force per unit length acting in the surface perpendicular to one side of a line in the surface.

The free surface energy 𝜸 is the energy required to create an additional unit area against the attractive forces of the molecules. The surface tension 𝝈 and the free surface energy 𝜸 are numerically the same

Consider a wire frame as shown in Figure 1 on which there is a soap film. XY is a sliding wire. The length of soap film in contact with the sliding wire is l. The force F due to surface tension on the wire XY is 2𝝈l, the factor 2 occurring because there are two surfaces to the soap film. 𝝈 is the surface

tension of the soap film. If the wire is moved a distance x to the right, the work done against the force of surface tension is 2𝝈lx.

FIG 1

Thus the increase in the surface tension is 2lx. Hence the energy required to create an additional unit area of film is:

2𝝈lx / 2𝝈x = 𝝈

But this is the definition of free surface energy  𝜸. Thus 𝝈= 𝜸 numerically. Because of differences between the cohesive force between molecules of liquid and the adhesive force between molecules of liquids and molecules of solids, a liquid surface is usually curved where it makes contact with a solid.

For example, the surface of water in a glass tube is concave and the surface of mercury in a glass tube is convex, as shown in Figure 2

FIG 2

The angle of contact 𝞠 is defined as the angle between the solid surface and the tangent to the liquid surface.𝞠 is measured through the liquid as shown in Figure 3.

FIG 3

If 𝞠 < 90° the liquid is said to ‘wet’ the solid surface. Liquids for which 𝞠 < 90° rise in a tube with a small internal diameter (such as a capillary tube).

Figure 4 shows a liquid that has risen a height h up a capillary tube of radius r.

The force due to the surface tension acting on the meniscus depends upon the circumference of the meniscus and the surface tension 𝝈

FIG 4

The upward vertical component of the force due to surface tension is:

𝝈 × circumference×  cos 𝞠 = 𝝈 (2𝝅r ) cos𝞠

The downward vertical force on the column of liquid is due to:

weight of liquid = volume × density × earth’s gravitational field

= (𝝅r²h ) 𝝆g

These two forces are equal, i.e.

 𝝈 (2𝝅r cos𝞠) = (𝝅r²h ) 𝝆g

from which

height h =(2 𝝈 cos 𝞠) / r𝝆g

Excess Pressure

It may be shown that there is a pressure inside a spherical drop of liquid that exceeds the surrounding air pressure by an amount equal to 2𝝈/ R as shown in figure  5

where

 R is the radius of the drop. This is called the excess pressure.

(i) For a spherical drop of liquid in air the excess pressure is 2𝝈/ R

(ii) For a bubble of gas in a liquid the excess pressure is 2𝝈/ R

(iii) For a soap bubble in air the excess pressure is 4𝝈/ R since a soap bubble has two surfaces.

FIG 5

Viscosity

Viscosity is an important fluid property when analyzing liquid behavior and fluid motion near solid boundaries. The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. The shear resistance in a fluid is caused by inter-molecular friction exerted when layers of fluid attempt to slide by one another.

       ◾viscosity is the measure of a fluid's resistance to flow◾molasses is highly viscous

       ◾water is medium viscous

       ◾gas is low viscous

Liquids (and gases) in contact with a solid surface stick to that surface. If a liquid flows on a solid surface we can consider the liquid to consist of layers. The bottom layer remains in contact with the solid and at rest. The other layers slide on one another and travel with velocities that increase the further the layer is from the solid, as shown in Figure 6. This is a description of streamline flow. If the velocity increases to beyond a critical value the flow becomes turbulent and the description in terms of layers no longer applies. In Figure 6, the arrows indicate the velocities of different layers.

This condition will exist when the liquid is subjected to a shear force. The opposition to this is called the viscosity of the liquid.

FIG 6

There are two related measures of fluid viscosity

           ◾ dynamic (or absolute)

           ◾ kinematic

Dynamic (absolute) Viscosity

Absolute viscosity - coefficient of absolute viscosity - is a measure of internal resistance. Dynamic (absolute) viscosity is the tangential force per unit area required to move one horizontal plane with respect to an other plane - at an unit velocity - when maintaining an unit distance apart in the fluid.

Consider two parallel layers of liquid separated by a distance dy travelling at velocities v and v +dv. The lower layer tends to impede the flow of the upper layer and exerts a retarding force F on it, whereas the lower layer itself experiences an accelerating force F exerted on it by the upper layer.

The tangential stress between the two layers is F / A

where

 A is the area of contact between the layers. The ratio dv/ dy is called the velocity gradient.

The shearing stress between the layers of a non turbulent fluid moving in straight parallel lines can be defined for a Newtonian fluid as

𝝉=F/A=  𝛍 (dv/dy)

where

𝝉= shearing stress in fluid (N/m²)

𝛍= dynamic viscosity of fluid (N s/m²)

dv = unit velocity (m/s)

dy = unit distance between layers (m)

where 𝛍 is a constant called the coefficient of viscosity. Thus,

𝛍= 𝝉 dy/dv

FIG 7

𝛍 usually decreases with increasing temperature although ‘viscostatic’ oils are almost temperature independent. The units of the coefficient of viscosity are N s m⁻² or, 

alternatively, kg m⁻¹s⁻¹ (since 1 N = 1 kg m s⁻²)

Dynamic viscosity may also be expressed in the metric CGS (centimeter-gram-second) system as g/(cm s), dyne s/cm2 or poise (p) where
         ⋆1 poise = 1 dyne s/cm² = 1 g/(cm s) = 1/10 Pa s = 1/10 N s/m²

For practical use the Poise is normally too large and the unit is therefore often divided by 100 - into the smaller unit centipoise (cP) - 

where
        ∎ 1 P = 100 cP
        ∎ 1 cP = 0.01 poise = 0.01 gram per cm second = 0.001 Pascal second = 1 milli Pascal second = 0.001 N s/m²
Water at 20.2⁰C (68.4⁰F) has the absolute viscosity of one - 1 - centiPoise.

Poiseulle’s Formula

Poiseulle’s formula for streamline flow through a circular pipe gives an expression for the volume V of liquid passing per second:

V= (𝝅 P r⁴)/ (8𝛍 l)

where r is the radius of the pipe, p is the pressure difference between the ends of the pipe, l is the length of the pipe and 𝛍 is the coefficient of viscosity of the liquid.

Stoke’s Law

Stoke’s law gives an expression for the force F due to viscosity acting on a sphere moving with streamline flow through a liquid

force F = 6 𝝅 𝛍 r v

where r is the radius of the sphere, and v its velocity.

Kinematic Viscosity
Kinematic viscosity is the ratio of - absolute (or dynamic) viscosity to density - a quantity in which no force is involved. Kinematic viscosity can be obtained by dividing the absolute viscosity of a fluid with the fluid mass density like

𝝊 = 𝛍/𝝆

where
𝝊 = kinematic viscosity (m²/s)
μ = absolute or dynamic viscosity (N s/m²)
ρ = density (kg/m³)

In the SI-system the theoretical unit of kinematic viscosity is m2/s - or the commonly used Stoke (St) where
                 ∎1 St (Stokes) = 10-4 m²/s = 1 cm²/s
Stoke comes from the CGS (Centimetre Gram Second) unit system.
Since the Stoke is a large unit it is often divided by 100 into the smaller unit centiStoke (cSt) - where
                 ∎ 1 St = 100 cSt

                 ∎ 1 cSt (centiStoke) = 10-6 m²/s = 1 mm²/s 

                 ∎ 1 m2/s = 106 centiStokes

The specific gravity for water at 20.2⁰C (68.4⁰F) is almost one, and the kinematic viscosity for water at 20.2⁰C(68.4⁰F) is for practical purpose 1.0 mm²/s (cStokes). A more exact kinematic viscosity for water at 20.2⁰C 

A conversion from absolute to kinematic viscosity in Imperial units can be expressed as

𝝊= 6.7197 10⁻ ⁴ μ / γ 

𝝊 = kinematic viscosity (ft²/s)

μ = absolute or dynamic viscosity (cP)

γ = specific weight (lb/ft³)

Viscosity and Reference Temperature

The viscosity of a fluid is highly temperature dependent - and for dynamic or kinematic viscosity to be meaningful the reference temperature must be quoted. In ISO 8217 the reference temperature for a residual fluid is 100⁰C. For a distillate fluid the reference temperature is 40⁰C.

           ∎ for a liquid - the kinematic viscosity decreases with higher temperature

           ∎ for a gas - the kinematic viscosity increases with higher temperature

FIG 8

Viscosity of some Common Liquids