Viscosity And Equation Of Continuity


In a stationary fluid, the atoms or molecules have two motions:

  1. Diffusion due to Brownian motion and
  2. Vibratory motion around their mean positions.

The forces on the fluid molecules are such that the molecule of fluid can easily slide by the side of another molecule. Hence diffusion is predominant in liquids and gases.

When a pressure difference is applied to a liquid, the molecules of liquid start moving from higher pressure to a lower pressure following the process of diffusion. Here this motion of a large number of molecules moving together, alongside the other molecules, constitutes the process of flow in a fluid.

When a layer of a fluid slips or tends to slip on another layer in contact, the two layers exert tangential forces on each other. The directions are such that the relative motion between the layers is opposed. This property of the fluid to oppose relative motion between its layers is called viscosity.

The forces between the two layers which oppose relative motion are termed forces of viscosity. Mechanical energy is lost against such viscous forces.

Viscosity and viscous drag in fluids:

Consider a steady flow of a liquid through a tube when a pressure difference is maintained across the ends of a tube. If we imagine a liquid to be divided into a large number of thin layers, it will be observed that the layer of liquid immediately in contact with the side of a tube is for all purposes stationary, while the layers near the axis of the tube move with larger velocities. In such steady flow, there is therefore a velocity gradient in the tube at right angles to the direction of flow.

The velocity gradient can be written as dv/dy. The force will act in the opposite direction to the direction of the flow of the liquid. The viscosity is defined quantitatively in terms of this force acting on the layers of flowing liquid and the velocity gradient.

The coefficient of viscosity (η) is given by
F/A = -η dv/dy.

F/A = force per unit area acting on the liquid layer due to viscosity. The negative sign is due to the force due to viscosity is in a opposite direction to that of the flow of liquid. One has to apply an equal and opposite force from outside to make the liquid flow. External pressure on liquid may provide this.

The unit of viscous in the CGS system is poise and in the MKS, it is Ns/m2. The coefficient of viscosity η has dimensions [M-1 L-1 T-1].

1 poise = 0.1 Ns/m2 .

The analysis of the flow of fluid becomes much simplified if we consider the fluid to be incompressible and nonviscous and that the flow is irrotational. Incompressibility means that the density of a fluid is constant at all the points and at all the time. The assumption of nonviscous fluid will mean that the internal friction when a layer of fluid slips over another layer is negligible. The assumption of irrotational flow means there is no net angular velocity of fluid particles. Now onwards, we shall consider only the fluids to be incompressible, nonviscous, and irrotational.

Equation Of Continuity:

If a fluid is flowing, the total mass of fluid going into the tube through any cross-section is equal to the total mass coming out of the same tube from any other cross-section in the same time. This leads to equation of continuity.

Liquid flowing in and out of tube

Consider the flow of liquid in a pipe as shown in Fig. Consider two cross sections at A and B. Let the area of the cross-section at A be A1 and that at B is A2. Let the speed of liquid is v1 at A and v2 at B. Let p be the density of liquid. Then the mass of the liquid Amy, flowing through point A in time At is given by
Δm1= A1 v1 p Δt.

Similarly, the mass of the liquid Amy flowing through the point B in time At is given by
Δm2= A2 v2 p Δt.

Since the liquid in between the two cross-sections is not absorbed nor the liquid is
being created. Hence we have
A1 v1 p = A2 v2 p
or for any cross section A.p.v= constant.
For incompressible fluid p= constant
hence, A1 v1 = A2 v2.
or AV= constant.
i.e. The product of the area of cross-section and the speed remains the same at all points of a tube of flow. This is called an equation of continuity and expresses the law of conservation of mass in fluid dynamics. This also indicates that in a pipe of varying cross-sections, the liquid will move slowly when the cross-sectional area of the tube is large and will move faster where the cross-sectional area is small. Hence v ∝ 1/A for a steady flow of a liquid through a tube.

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