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Sunday, 10 December 2017

SURFACE TENSION IN FLUID MECHANICS

We were discussing the basic definition and significance of Kinematic viscosity, Dynamic viscosity, various properties of fluid, type of fluids, Newton’s law of viscosity, Compressibility and bulk modulus in our previous posts. 

Now we will understand here the basic concept of surface tension in fluid mechanics and simultaneously we will find out here the expression for surface tension on liquid droplet, surface tension on a hollow bubble and surface tension on liquid jet. 

So, let us see first the basics of surface tension

Surface tension

Surface tension is basically defined as the tensile force acting on the surface of a liquid in contact with gas such as air or on the surface between two immiscible liquids. 

Surface tension is basically the tensile force per unit length of the surface of liquid and therefore unit of surface tension will be N/m. 

Surface tension will be displayed by the symbol σ.

Let us explain the surface tension phenomenon with the help of following figure

Let us consider three molecules of water A, B and C as displayed in following figure. 
Molecule A is well within water and therefore molecule A will be attracted equally in all direction by its surrounding molecules of liquid i.e. water. Therefore, resultant force acting on molecule A will be zero. 

Let us see now the molecule B which is located at the surface film. A portion of molecule B is in the air i.e. above the water surface and rest portion is inside the water. Resultant force acting on the molecule B will not be zero but also resultant force acting on molecule B will be towards downward direction. 

Similarly let us see now the molecule C which is located on the surface. Half portion of molecule C is in the air i.e. above the water surface and rest half portion is inside the water. Resultant force acting on the molecule C will not be zero but also resultant force acting on molecule C will also be towards downward direction. 

Therefore, it is concluded that all molecules of water located at the free surface will feel a resultant downward force. Hence free surface of water, in above figure, will act like a thin film under tension.

Let us see now the surface tension on liquid droplet

Let us consider one liquid droplet of diameter d as displayed in following figure. Liquid droplet entire surface will be under the influence of tensile force due to surface tension.
Tensile force due to surface tension = Surface tension x Circumference
Tensile force due to surface tension = σ x П d

Force due to pressure inside the droplet = Pressure intensity inside the droplet x Area
Force due to pressure inside the droplet = P x (П/4) d2

We must note it here that pressure intensity inside the droplet is basically the difference in pressure between inside and outside of the liquid droplet.

For equilibrium situation, above two forces must be equal and opposite and hence we will have following equation.
σ x П d = P x (П/4) d2
P = 4 σ /d

Let us see now the surface tension on hollow bubble

Now we will see the expression for surface tension and pressure intensity inside a Hollow bubble.  Soap bubble is one good example of hollow bubble.

In case of hollow bubble, inner surface and outer surface of hollow bubble will be subjected with surface tension.

Therefore for hollow bubble
2 x σ x П d = P x (П/4) d2
P = 8 σ /d

Let us see now the surface tension on liquid jet

Let us consider a liquid jet of diameter d and length L as displayed in following figure. Let us consider the pressure intensity inside the liquid jet which is basically the difference in pressure between inside and outside of the liquid jet is P.
Let us consider the equilibrium situation of semi jet and we will have following equation as mentioned here

Force due to pressure = Force due to surface tension
P X L X d = σ x 2L
P = σ x 2L/ (L X d)
Do you have any suggestions? Please write in comment box.

We will now discuss the basic concept of capillarity, in the category of fluid mechanics, in our next post.

Reference:

Fluid mechanics by R. K. Bansal
Image Courtesy: Google

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