Wednesday, December 31, 2014

Coefficient of Viscosity and Terminal Velocity


The viscous force acting between two adjacent layers of a liquid is directly proportional  to the surface area of the layers in contact and the velocity gradient. The viscous force acts tangential to the liquid and in opposite direction to the direction of flow of liquid.  Hence negative sign is used. It is like a frictional force acting against the relative motion.

It can  be defined as the tangential force per unit area required to maintain unit velocity gradient. (or) It is the ratio between tangential stress and velocity gradient.

Effect of temperature on Viscosity

In the case of liquids, coefficient of viscosity decreases with increase  of temperature as the  cohesive forces decrease with increase of temperature.In the case of gases,coefficient of viscosity increases with increase of temperature because the change in momentum of molecules increases with increase of temperature.

In the case of liquids with increase in temperature, distance between molecules increases . This leads to  the decrease of force of attraction .

In the case of gases, with the increase in temperature, random motion of molecules increases and collisions also increases. This increases viscous nature.

Effect of pressure

In a liquids the value of increases with increase of pressure. For gases, its value of  increases with increase of pressure at low pressure.  But at high pressure, it is independent of pressure.

Raynold’s Number

We can decide the flow of the fluid as is a streamlined flow or not basing on a value called Raynold’s number.

The velocity at which the streamlined flow turns into a non-streamline flow is called as critical velocity. Critical velocity of a fluid is directly proportional to coefficient of viscosity, inversely proportional to diameter of the fluid flow and also inversely proportional to the density of the fluid. We can write the equation by keeping all of these things together. The proportionality can be eliminated with a constant as shown below.

Terminal velocity

A constant velocity is acquired by a body when the resultant force acting on it is zero and it is called as terminal velocity. When a spherical body is moving in a fluid its weight always acts in downward direction and it’s upthrust always acts in upward direction. The medium applies a force against its motion and it is called viscous force. The direction of the viscous forces always against the relative motion of the body in the fluid.

When the body is initially in the state of rest there is no viscous force acting on it. When the body starts coming down it’s velocity increases due to the gravitational force and automatically viscous force also starts increasing against  the gravitational force.

At a certain stage the downward force is balanced by the up ward force therefore the body will acquire a constant velocity and that velocity is called as terminal velocity. By drawing the condition for equilibrium we can derive the equation further terminal velocity as shown below.

Special Case 

When multiple drops are falling with a different terminal velocities and if the drops are combined together to form a big drop we can derive a equation further terminal velocity of the big drop as shown below.

While solving this problem we are going to depend on the simple concept that volume of the big drop is equal to the sum of the volume of all the small drops together. We can also express the terminal velocity in terms of the mass is shown in the below diagram.

Poisellie’s Equation

This equation helps in identifying the volume of the fluid flowing through a given hole per second.It depends on radius of tube with power four,pressure and inversely proportional to  its length and coefficient of viscosity.Keeping only pressure in the numerator, every thing can be shifted to denominator and hence it oppose the flow of fluid.Then it is being called as fluid resistance.

When the pipes are connected in series, the same volume of the fluid flows through them and pressure at different points is going to be different.

When the pipes are connected in parallel, volume of fluid will be shared across them but pressure across two pipes is going to be the same.

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