Fluid Mechanics Concept Page - 17

Definition

Excess pressure inside a soap bubble

Excess pressure due to surface tension in liquid bubble is given by P=4TR$P={\displaystyle \frac{4T}{R}}$
Proof:
Work done by excess pressure, W=Fs=PAs=PπR2ΔR$W=Fs=PAs=P\pi R^2\mathrm{\Delta }R$
Increase in surface area, ΔA=2×4π(R+ΔR)2−4πR2=16πRΔR$\mathrm{\Delta }A=2\times 4\pi (R+\mathrm{\Delta }R{)}^2-4\pi R^2=16\pi R\mathrm{\Delta }R$
Increase in free surface energy, S=ΔAT=16πRΔRT$S=\mathrm{\Delta }AT=16\pi R\mathrm{\Delta }RT$
Work done by excess pressure=Increase in surface energy
⟹P=4TR$⟹P={\displaystyle \frac{4T}{R}}$

Example

Problem on surface tension

Example:
A straw 6cm$6cm$ long floats on water. The water film on one side has surface tension of 50$50$ dyne/cm. On the other slide, camphor reduces the surface tension to 40$40$dyne/cm. Find the resultant force acting on the straw. Solution:
The resultant force , F=l(T1−T2)=6(50−40)=300−240=60$F=l(T_1-T_2)=6(50-40)=300-240=60$dyne.

Definition

Understand interaction of molecules in bulk of liquid and on surface

At liquid-air interfaces, surface tension results from the greater attraction of liquid molecules to each other (due to cohesion) than to the molecules in the air (due to adhesion). The net effect is an inward force at its surface that causes the liquid to behave as if its surface were covered with a stretched elastic membrane. Thus, the surface becomes under tension from the imbalanced forces, which is probably where the term "surface tension" came from. Because of the relatively high attraction of water molecules for each other, water has a higher surface tension (72.8 mN per meter at 20o${}^o$C) compared to that of most other liquids. Surface tension is an important factor in the phenomenon of capillarity.

Definition

Surface Energy and its explanation at molecular level

Surface energy, or interface energy, quantifies the disruption of intermolecular bonds that occur when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favorable than the bulk of a material (the molecules on the surface have more energy compared with the molecules in the bulk of the material), otherwise there would be a driving force for surfaces to be created, removing the bulk of the material. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces.

Definition

Shape of liquid drop

Contact angles are extremely sensitive to contamination; values reproducible to better than a few degrees are generally only obtained under laboratory conditions with purified liquids and very clean solid surfaces. If the liquid molecules are strongly attracted to the solid molecules then the liquid drop will completely spread out on the solid surface, corresponding to a contact angle of 0. This is often the case for water on bare metallic or ceramic surfaces.
On the other hand, the cohesive force between the molecules of mercury is more than the adhesive force between the molecules of mercury and those of glass.This results in the formation of a convex meniscus for the surface of mercury. So, the mercury drops don't spread out on the glass surface.

Definition

Wetting and nonwetting liquids

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting (wettability) is determined by a force balance between adhesive and cohesive forces. Wetting deals with the three phases of materials: gas, liquid, and solid. It is now a center of attention in nanotechnology and nanoscience studies due to the advent of many nanomaterials in the past two decades.
Precisely, for wetting liquid angle of contact is acute and for nonwetting liquid angle of contact is obtuse.

Example

Use of relation of surface energy with heat of vaporization

Example: Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is T$T$, density of liquid is ρ$\rho$ and L$L$ is its latent heat of vaporization.

Solution:
ρ4πR2ΔRL=T4π[R2−(R−ΔR)2]$\rho 4\pi R^2\mathrm{\Delta }RL=T4\pi [R^2-(R-\mathrm{\Delta }R{)}^2]$
ρR2ΔRL=T[R2−R2+2RΔR−ΔR2]$\rho R^2\mathrm{\Delta }RL=T[R^2-R^2+2R\mathrm{\Delta }R-\mathrm{\Delta }{R}^2]$
ρR2ΔRL=T2RΔR$\rho R^2\mathrm{\Delta }RL=T2R\mathrm{\Delta }R$ ( ΔR$\mathrm{\Delta }R$ is very small)
R=2TρL${\displaystyle R=\frac{2T}{\rho L}}$

Definition

Weight correction of liquid in meniscus and rise in capillary tube

The weight of the liquid inside the capillary tube is given as : W=πr2hρg$W=\pi r^{2}h\rho g$
and height of the liquid is h=2Scosθrρg$h={\displaystyle \frac{2Scos\theta }{r\rho g}}$
The correction due to the weight of the liquid contained in the meniscus can be easily made if the contact angle is zero. The meniscus is then hemispherical.
Its volume is (πr2)r−12(43)=13πr3$(\pi r^2)r-{\displaystyle \frac{1}{2}}({\displaystyle \frac{4}{3}})={\displaystyle \frac{1}{3}}\pi r^3$
The weight of the liquid contained in the meniscus is 13πr3ρg${\displaystyle \frac{1}{3}}\pi r^3\rho g$ and weight balance gives:
πr2hρg+13πr3ρg=2πrS$\pi r^{2}h\rho g+{\displaystyle \frac{1}{3}}\pi r^3\rho g=2\pi rS$ 
h=2Srρg−r3$h={\displaystyle \frac{2S}{r\rho g}}-{\displaystyle \frac{r}{3}}$

Definition

Surface energy and its units

The surface energy is defined as the sum of all intermolecular forces that are on the surface of a material, the degree of attraction or repulsion force of a material surface exerts on another material. In the case of liquids this same definition is applied to define the surface tension as a result of this surface tension liquid with low surface tends to contract and form droplets. Surface energy density has unit  J/m2.$J/m^2.$

Definition

Angle of contact

The surface of liquid near the plane of contact, with another medium is, in general, curved. The angle between tangent to the liquid surface inside the liquid is termed as angle of contact. It is denoted by θ$\theta$. It is different at interfaces of different pairs of liquids and solids. The value of θ$\theta$ determines whether a liquid will spread on the surface of a solid or it will form droplets on it.

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