Viscocity : Definition, Units, Dimension and Solved Examples

The study of Physics Topics involves the exploration of matter, energy, and the forces that govern the universe.

What are the Properties of Streamline Flow and Turbulent Flow?

Laminar And Turbulent Flow

The flow of fluids (liquids and gases) is of two types-

1. laminar or streamline flow and
2. turbulent flow.

Laminar or Streamline Flow

Definition: A smooth, uninterrupted flow in ordered layers, without any energy transfer between the layers, is called a laminar or streamline or steady flow.

The velocity of a fluid along its flow, in general, depends on its position and on time. This means that the fluid velocity may be different at different points, and at any particular point, it may also change with time. Mathematically speaking, for an one-dimensional fluid flow along the x- direction, the fluid velocity \(\vec{v}\) is a function of position x and time t, i.e., \(\vec{v}\) = f(x, t).

For a laminar or streamline motion, the fluid satisfies the condition that v is a function of x only, and not of t. It means that, at any particular point along the fluid flow, the magnitude and the direction of the fluid velocity do not change with time, although the velocity may be different at different points. In short, \(\vec{v}\) = f(x), but \(\vec{v}\) ≠ f(t).

In Fig., a laminar flow for a liquid is shown. At points A, B, C and D, let the flow velocities at any instant be v_A, v_B, v_C and v_D respectively. Also at any subsequent time, a

liquid particle that reaches the point A will have the velocity v_A; similarly, at point B, the velocity will be v_B; at point C, the velocity will be v_C and, at point D, the velocity will be, v_D. It means that each particle of the liquid follows the velocity of its preceding particle and moves along the same path.

Streamline: In the case of streamline motion, the paths along which the particles of the fluid move are called streamlines. A tangent drawn at any point on this path indicates the direction of motion of the fluid at that point.

Properties of streamline:

i) Two streamlines never intersect each other. Otherwise, at the point of intersection of two streamlines, two tangents can be drawn and hence two directions of motion of the particle are possible. But, in streamline motion, any particle can move in one direction only and hence two streamlines can never intersect.

ii) In the flow tube, where the streamlines are crowded together, the velocity of flow is higher. Where they are spaced, the velocity of flow is lower.

A special case of streamline flow is a steady flow, for which the fluid velocity is a constant at all points along this flow at all times. So this velocity in neither a function of time, nor of position. Example: a sufficiently slow liquid flow along a narrow uniform horizontal tube.

Turbulent Flow

In general, the motion of a fluid is streamlined, if its velocity does not exceed a definite limiting value. The limiting value of velocity is called the critical velocity. If the velocity of a fluid exceeds the critical velocity, then the flow becomes turbulent and in some regions, eddies and vertices are formed. This kind of flow is called a turbulent flow.

Definition: If the velocity of a fluid along its flow continu-ously and randomly changes in magnitude and direction, then it is called a turbulent or disorderly flow.

The path of fluid particles in turbulent flow is shown in Fig. At every point along the flow, both the magnitude and the direction of fluid velocity change with time.

Experiment: Reynolds demonstrated the difference between streamline and turbulent flow by a simple experiment. In Fig., a discharge pipe Q is attached horizontally to a vertical water-filled cylinder P. The flow rate may be varied with a valve at the end of the pipe. For making the flow behaviour visible, we use KMnO₄ solution, which is injected centrally into the horizontal pipe through a very narrow tube (diameter < 1 mm), positioned to avoid additional turbulence.

If the velocity of the transparent liquid is low, the coloured liquid is observed to travel continuously in the form of a thread, indicating a streamline flow. As the flow of the transparent liquid is increased gradually, the coloured thread gets disrupted and later on the coloured liquid begins to move randomly, or forms eddies and vortices and mixes with the transparent liquid. The velocity of the transparent liquid at which this disturbance starts is called the critical velocity. It depends on the nature of the liquid, the cross section of the tube, etc.

Viscosity

When a liquid flows slowly over a fixed horizontal surface, i.e., when the flow is laminar, the layer of the liquid in contact with the fixed surface remains at rest due to adhesion. The layer just above it moves slowly over the lower one, the third layer moves faster over the second one and so on. The velocities of the layers of liquid increase with the increase in distance from the horizontal rigid surface [Fig.(a)],

For two consecutive horizontal layers inside the liquid, the upper layer moves with a velocity greater than that of the lower one. The upper layer tends to accelerate the lower layer, while the lower layer tends to retard the upper one. In this way, the two adjacent layers tend to decrease their relative velocity—as if a tangential force acts on the upper layer and tries to oppose its motion. This tangential force is called viscous force. Therefore, to maintain a constant relative motion between the layers, an external force must act. If no external force acts, then the relative motion between the layers will cease and the flow of the liquid will stop.

Definition: The property by virtue of which a liquid opposes the relative motion between its adjacent layers is called viscosity of the liquid.

Comparison of viscosity with friction: Viscosity is a general property of a fluid. The frictional force acting between two solid surfaces resembles in many ways the viscosity of a liquid. Hence, viscosity is called internal friction of a liquid. Like friction, the viscous force is absent if a liquid is at rest. The difference between the frictional force in solids and viscosity in liquids is that the viscous force depends on the area of liquid surface while the frictional force does not.

Viscosity and mobility of different liquids:
Viscosities of different liquids are different. If alcohol and oil are poured separately into two identical vessels and stirred, then oil will come to rest earlier. This shows that the viscosity of oil is greater. The greater the viscosity of a liquid, the lesser is its mobility. For example, the viscosity of honey is more than that of water and hence honey flows much slower than water. Coal tar has the least mobility.

velocity profile: The surface formed by joining the end points of the velocity vectors of different layers of any
section of a flowing liquid is called its velocity profile [Fig.]. Velocity profile for flow above a horizontal surface is shown in Fig.(b).

Velocity profile of a non-viscous liquid: An ideal liquid is non-viscous. For such a liquid, there is no resistance due to viscosity. The velocities of the different layers are the same. Every particle in a given cross section of the liquid moves forward with the same velocity. On joining the ends of these velocity vectors, we get a plane surface. Therefore, we can say that the velocity profile of a non-viscous liquid is linear (on 2D graph) [Fig.(c)].

Velocity profile of a viscous liquid: When a viscous liquid flows through a horizontal tube, the layer of liquid in contact with the wall of the tube remains stationary due to adhesion. So the velocity of that layer is zero. The layer of the liquid which flows along the axis of the tube has the maximum velocity. As we progress from the centre towards the walls, the velocity decreases. Therefore, on joining the ends of the velocity vectors, we get a parabolic surface. The velocity profile of a viscous liquid is a parabola (on 2D graph) [Fig.(d)].

Coefficient of Viscosity

Let PQ be a solid horizontal surface [Fig.]. A liquid is in streamline motion over the surface PQ. Two liquid surfaces CD and MN are at distances x and (x + dx) respectively from the fixed solid surface. The velocity of layer CD is y and that of layer MN is v + dv.

Due to the viscosity of the liquid, an opposing force acts between these two layers and tries to slow down the relative motion of the layers. If this opposing viscous force is F, then for streamline motion of the liquid, Newton proved that

1. F ∝ A; A = area of cross section of the liquid surface, and
2. F ∝ \(\frac{d \nu}{d x}\); \(\frac{d \nu}{d x}\) = velocity gradient = rate of changeof velocity
   with distance perpendicular to the direction of flow.

∴ F ∝ A\(\frac{d v}{d x}\) or, F = -ηA\(\frac{d v}{d x}\) ……. (1)
Here, η is a constant known as the coefficient of viscosity.
Its value depends on the nature of the liquid.
Equation (1) is known as Newton’s formula for the streamline flow of a viscous liquid. Liquids which obey this law are called Newtonian liquids and liquids that do not obey this law are called non-Newtonian liquids.

From equation (1), we get, η = \(\frac{F}{A \frac{d v}{d x}}\)
If A = 1 and \(\frac{d v}{d x}\) = 1, then η = F; from this, we can define the coefficient of viscosity.

Definition: The coefficient of viscosity of a liquid is defined as the required tangential force acting per unit area to maintain unit relative velocity between two liquid layers unit distance apart.

Untis of coefficient of viscosity:

Poise and decapoise: The coefficient of viscosity of a liquid is 1 poise, when a tangential force of 1 dyn is required to maintain a relative velocity of 1 cm ᐧ s^-1 between two parallel layers of the liquid 1 cm apart where each layer has an area of 1 cm².
So, 1 poise is the CGS unit of the coefficient of viscosity η.
1 poise = 1 dyn ᐧ s ᐧ cm^-2 = 1 g ᐧ cm^-1 ᐧ s^-1.
As, 1 kg ᐧ m^-1 ᐧ s^-1 = 10g ᐧ cm^-1ᐧs^-1 = 10 poise,
the SI unit of η is called 1 decapoise = 10 poise.

The coefficient of viscosity of a liquid is 1 decapoise, when a tangential force of 1 newton is required to maintain a relative velocity of 1 m ᐧ s^-1 between two parallel layers separated by a distance of 1 m, where each layer has an area of 1 m².

Dimension of coefficient of viscosity:

Effect of pressure and temperature on the coefficient of viscosity:

Effect of pressure: Usually, viscosity increases with pressure. In less viscous liquids, the viscosity increases at a low rate with pressure. But for highly viscous liquids, an increase in pressure results in a rapid rise in its viscosity. However, water behaves differently and, with an increase in pressure, its viscosity decreases.

From the kinetic theory of gases, it is known that a change in pressure does not affect the viscosity of a gas. But for a large increase (or decrease) in pressure, viscosity is affected.

Effect of temperature: Usually, the coefficient of viscosity of liquids decreases with a rise in temperature. The relation between temperature and coefficient of viscosity is rather complicated. One commonly used equation relating these two is
η_t = \(\frac{A}{(1+B t)^C}\)
where, η_t= coefficient of viscosity of a liquid at t°C and A, B and C are constants for a particular fluid.
For gases, the coefficient of viscosity increases with an increase in temperature.

Critical Velocity and Reynolds Number

Critical velocity: The maximum velocity of a fluid, up to which the flow of the fluid is streamlined and beyond which the flow becomes turbulent, is regarded as the critical velocity for that fluid.

On gradually increasing the velocity of a fluid, the streamline flow does not become turbulent abruptly. Rather this change occurs gradually.

With the help of experimental demonstration and also by dimensional analysis, it can be proved that the critical velocity (v_c) of a fluid is

1. inversely proportional to the density (ρ) of the fluid,
2. directly proportional to the coefficient of viscosity (η) of the fluid, and
3. inversely proportional to the characteristic length (l) of the channel. So,
   V_c ∝ \(\frac{\eta}{\rho l}\) 0r v_c = N_c ᐧ \(\frac{\eta}{\rho l}\) ……… (1)

In the case of a tube, the characteristic length is the diameter of the tube while, for a canal, the characteristic length is its breadth.

If, for a liquid, ρ and q are known and its critical velocity v can be determined experimentally during its flow through a tube of diameter l, then from equation (1), the value of the constant N_c for that liquid can be determined. This value is nearly 2300.
For any velocity v of the fluid flow, equation (1) can also be written in an equivalent form as
v = Nᐧ\(\frac{\eta}{\rho l}\) or, N = \(\frac{\rho l v}{\eta}\) ……… (2)
N is called the Reynolds number.

Special cases:

i) If v < v_c, i.e., the velocity of fluid flow is less than the critical velocity, then comparing equations (1) and (2), we can say that N < N_c. It means that the value of Reynolds number is less than 2300. So, if the value of Reynolds number is less than 2300, then the flow will be streamlined.

ii) On the other hand, if v > v_c, i.e., the velocity of the fluid is greater than the critical velocity, then N > N_c and hence the value of Reynolds number will be greater than 2300. If Reynolds number is greater than 2300, then the flow will be turbulent.

Dimension of Reynolds number: From equation (2) we get,

So, N is a dimensionless quantity; it is a pure number.

Reynolds number: A dimensionless number N = \(\frac{\rho l v}{\eta}\) can be formed by combining the characteristic length (l) of a fluid channel and the velocity (v), density (ρ) and coefficient of viscosity (v) of the fluid; the magnitude of N determines whether the fluid flow is streamline or turbulent. This number N is called the Reynolds number.

In the above discussion, 2300 is an approximate value of N_c. Usually, for N < 2000, the fluid flow is streamline, and for N > 3000 the fluid flow is turbulent. If N lies between 2000 and 3000, the streamline flow of a fluid gradually changes into turbulent flow.

As N is a pure number, its value does not depend on the system of units chosen. For a particular flow, the value of N remains the same.

If the radius of a tube of flow is considered, instead of its diameter, then the effective value is, N_c ≈ 1150.

Numerical Examples

Example 1.
A plate of area 100 cm² is floating on an oil of depth 2 mm. The coefficient of viscosity of oil is 15.5 poise. What horizontal force is required to move the plate horizontally with a velocity of 3 cm ᐧ s^-1?
Solution:
The viscous force, F = ηA\(\frac{d v}{d x}\)
Here, A = 100 cm², η = 15.5 poise,
dv = 3 cm ᐧ s^-1 and dx = 2 mm = 0.2 cm.
∴ F = 15.5 × 100 × \(\frac{3}{0.2}\) = 23250 dyn
So the required horizontal force is 23250 dyn.