Properties of Fluids

SURFACE TENSION

The force of attraction between the molecules of the same substance is called cohesion.

In case of solids, the force of cohesion is very large and due to this solids have definite shape and size. On the other hand, the force of cohesion in case of liquids is weaker than that of solids. Hence liquids do not have definite shape but have definite volume. The force of cohesion is negligible in case of gases. Because of this fact, gases have neither fixed shape nor volume.

Properties of surface Tension

MATHEMATICAL DESCRIPTION OF SURFACE TENSION

surface tension is defined as the force per unit length in the plane of a liquid surface, acting at right angles on either side of the imaginary line drawn in the surface

T = F / $\ell$

SURFACE TENSION AND WORK DONE

Surface tension and work done in increasing surface area

F $\propto 2\ell$

$F=T\times 2\ell$

\begin{align}  W=F\times dx \\   \,\,\,\,\,\,\,=F\times 2\ell \times dx \\ \,\,\,\,\,\,\,=T\times dA\\\end{align}

$T=W/dA$

ANGLE OF CONTACT

Angle of Contact

The angle which the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid is called angle of contact. Those liquids which wet the walls of the container (say in case of water and glass) have meniscus concave upwards and their value of angle of contact is less than 90° (also called acute angle). However, those liquids which don't wet the walls of the container (say in case of mercury and glass) have meniscus convex

upwards and their value of angle of contact is greater than 90° (also called obtuse angle). The angle of contact of mercury with glass about 140°, whereas the angle of contact of water with glass is about 8°. But, for pure water, the angle of contact q with glass is taken as 0°.

Case I: When q < 90°:

The liquid surface curves up towards the solid. This happens when the force of cohesion between two liquid molecules is less than force of adhesion between the liquid and the solid. If such a liquid is poured into a solid tube, it will have a concave meniscus. For example, a glass rod dipped in water, or water inside a glass tube.

Case II: When q > 90°:

The liquids surfaces get curved downward in contact with a solid. In this case the force of cohesion is greater than the force of adhesion. In such cases, solids do not get "wet". When such liquids are put into a solid tube, a convex meniscus is obtained.

For example, a glass rod dipped in mercury or mercury within a solid glass tube.

Capillarity: A molecule at the surface of a liquid is attracted by other molecules in the surface in all directions. If the surface is convex, then a resultant component of all the forces of attraction acting on every molecules act normal to the surface and is directed downward. Similarly, if the surface is concave, then every molecule experiences a resultant force due to surface tension acting normally outward. For balance the resultant force due to surface tension. Hence the pressure on the concave side must be greater than the pressure on the convex side. This difference in pressure is equal to 2T/R, where T is the surface tension and R is the radius of curvature of the surface.

Example : What is the surface energy of a soap bubble of radius r.

Solution: E = TA = T $\times$ 4$\pi$r² $\times$ 2 (as it has two surfaces)

= 8$\pi$r² T.

EXCESS PRESSURE

(i) Inside a bubble: Consider a soap bubble of radius r. Let p be the pressure inside the bubble and p_a outside. The excess pressure = p – p_a. Imagine the bubble broken into two halves, and consider one half of it as shown in Fig. Since there are two surfaces, inner and outer, so the force due to surface tension is

F = surface tension x length = Tx2 (circumference of the bubble)

= T x 2 (2 T $\pi$ r)

The excess pressure (p – p_a) acts on a cross-sectional area $\pi$ r², so the force due to excess pressure is $\Rightarrow$ $F=\left(p-p_a\right)\pi r^2$

or $\left(p-p_a\right)=\frac{4T}{r}=p_{excess}$

(ii) Inside the drop: In a drop, there is only one surface and hence excess pressure can be written as

$\left(p-p_a\right)=\frac{2T}{r}=p_{excess}$

(iv) A charged bubble: If bubble is charged, it's radius increases. Bubble has pressure excess due to charge too.

Initially pressure inside the bubble = $p_a+\frac{4T}{r_1}$

For charge bubble, pressure inside = $p_a+\frac{4T}{r_2}-\frac{{\sigma }^2}{2{\in }_0}$ where s surface is surface charge density. Taking temperature remains constant then from Boyle's law

$\left(p_a+\frac{4T}{r_1}\right)\frac{4}{3}\pi r_1^3=\left[p_a+\frac{4T}{r_2}-\frac{{\sigma }^2}{2{\in }_0}\right]\frac{4}{3}\pi r_2^3$

From above expression the radius of charged drop may be calculated. It can conclude that radius of charged bubble increases, i.e. r₂ > r₁.

(v) Excess of Pressure inside a Curved Surface

a. Plane Surface: If the surface of the liquid is plane [as shown in Fig.(a)], the molecule on the liquid surface is attracted equally in all directions. The resultant force due to surface tension is zero. The pressure, therefore, on the liquid surface is normal.

b. Concave Surface: If the surface is concave upwards [as shown in Fig.(b)], there will be upward resultant force due to surface tension acting on the molecule. Since the molecule on the surface is in equilibrium, there must be an excess of pressure on the concave side in the downward direction to balance the resultant force of surface tension $p_A-p_B=\frac{2T}{r}$.

c. Convex Surface: If the surface is convex [as shown in Fig.(c)], the resultant force due to surface tension acts in the downward direction. Since the molecule on the surface are in equilibrium, there must be an excess of pressure on the concave side of the surface acting in the upward direction to balance the downward resultant force of surface tension, Hence there is always as excess of pressure on concave side of a curbed surface over that on the convex side. $p_A-p_B=\frac{2T}{r}$ )

Effect of Temperature and Impurities of Surface Tension

The surface tension of a liquid decreases with the rise in temperature and vice versa. According to Ferguson, $T=T_0{\left(1-\frac{\theta }{{\theta }_c}\right)}^n$ where T₀ is surface tension at 0°C, q is absolute temperature of the liquid,${\theta }_c$ is the critical temperature and n is a constant varies slightly from liquid and has mean value 1.21. This formula shows that the surface tension becomes zero at the critical temperature.

The surface tension of a liquid changed appreciably with addition of impurities. For example, surface tension of water increases with addition of highly soluble substances like NaCl, ZnSO₄ etc. On the other hand surface tension of water gets reduced with addition of sparingly soluble substances like phenol, soap etc.

VISCOSITY

The property of a liquid by virtue of which an opposing force ( internal friction) comes into play whenever there is a relative motion between the diffirent layers of the liquid is called viscosity.

Suppose that a glass plate in contact with a water column of height h is moved with constant velocity v. Forces of viscosity appear between the solid surface and the layer in contact.

F = -$\eta$ A $\frac{dv}{dz}$.

$\eta$ is coefficient of viscosity. Negative shows that the direction of viscous force(F) is just opposite to the direction of the motion of the liquid.

its CGS unit is poise. Dimension is ML^-1T^-1. The SI units of viscosity is kg/m$\times$sec.

POISEUILLE'S FORMULA

Flow of viscous liquid through a capillary tube (Poiseuille's formula)

The velocity v at a distance y from the capillary axis for a flow of liquid of viscosity $\eta$ in a capillary tube of length L and radius r under a pressure difference p across it is given by

v = $\frac{P}{4\eta L}$ (r² – y²)

and the volume of liquid flowing per second is given by

$\left(\frac{dQ}{dt}\right)=\frac{\pi {\mathrm{Pr}}^4}{8\eta L}$.

STOKE'S FORMULA

Stoke's Formula: Stokes proved that the viscous drag (F) on a spherical body of radius r moving with terminal velocity v in a fluid of viscosity $\eta$ is given by

$F=6\pi \eta rv$

Terminal velocity: When a body is dropped in a viscous fluid, it is first accelerated and then its acceleration becomes zero and it attains a constant velocity called terminal velocity.

$v=\frac{2}{9}{r}^2({\rho }_2-{\rho }_1)\frac{g}{\eta }$

Where r is the radius of the penetrating body $\rho$₁ is density of body, $\rho$₂ is density of liquid, $\eta$ is coefficient of viscosity.