Dimesional Formula of Kinematics Viscosity

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Dimensions

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. Dimensions of any given quantity tell us about how and which way different physical quantities are related. Finding dimensions of different physical quantities has many real-life applications and is helpful in finding units and measurements. Imagine a physical quantity X which depends mainly on base mass(m), length(L), and time(T) with their respective powers, then we can represent dimensional formula as [MaLbTc]


Dimensional Formula 

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

E.g: Dimension formula of mass is: (M)


Dimensional Equation 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 


Application of Dimensional Analysis 

1. To convert a physical quantity from one system of the unit to the other:

It is based on a fact that magnitude of a physical quantity remain same whatever system is used for measurement i.e magnitude = numeric value(n) multiplied by unit (u) = constant

n1u1= n1u2

2. To check dimensional correctness of a given physical relation: 

If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

3. To derive a relationship between different physical quantities: 

Using the principle of homogeneity of dimension, the new relation among physical quantities can be derived if the dependent quantities are known. 


Limitation of this Method

  •  This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term, which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

  •  The relation derived from this method gives no information about the dimensionless constants. 


Kinematic Viscosity

Viscosity: Viscosity is the property of the fluid ( liquid or gas ) by virtue of which it opposes the relative motion between its adjacent layers. It is the fluid friction or internal friction. The internal tangential force which tries to retard the relative motion between the layers is called viscous force. 


Properties of Viscosity:

  • It opposes motion.

  •  It acts tangentially in a direction opposite to that of motion.

  •  It comes into play when the two layers of a liquid are in relative motion. 


Dependence of Viscosity Fluids

On the temperature of fluid:

  • Since cohesive force decreases with an increase in temperature. Therefore with the rise in temperature the viscosity of liquid decreases.

  • The viscosity of gases results from the diffusion of gas molecules from one moving layer to another moving layer. Now with an increase in temperature the rate of diffusion increases. So, the viscosity also increases. Thus the viscosity of gases increases with the rise of temperature. 

  On the pressure of fluid:

  •  The viscosity of liquids increases with an increase in pressure.

  •  The viscosity of gas increases is practically independent of pressure. 


Dimensional Formula of Kinematic Viscosity

The dimensional formula of Kinematic Viscosity is written as M0 L2 T-1

Where M represents mass, L represents length and T represents time. 


Derivation of the Dimensional Formula of Chemical Kinematic

The chemical formula can be formulaically written as:

Kinematic viscosity (ν) = Dynamic viscosity × [Density]-1.  . . . (1)

As, Density = Mass × [Volume]-1

⇒ ρ(density) = [M1 L0 T0] × [M0 L3 T0]-1

∴ The dimensional formula of density = [M1 L-3 T0] . . . . (2)

As, Dynamic viscosity (η) = Tangential Force × distance between layers × [Area × velocity]-1. . . .(3)

Now, Tangential Force = M × a = M × [L T-2]

∴ The dimension of force = M1 L1 T-2 . . . . (4)

And, the dimensional formula of the area and velocity = L2 and L1 T-1 . . . . (5)

On substituting equation (4) and (5) in equation (3) we get,

Dynamic viscosity (η) = [M L T-2] × [L] × [L2]-1 × [L1 T-1]-1 = [M1 L-1 T-1].

Therefore, the dimensions of dynamic viscosity = [M1 L-1 T-1] . . . .(6)

On putting equation (2) and (6) in equation (1) we get,

Kinematic viscosity (ν) = Dynamic viscosity × [Density]-1

Or, ν = [M1 L-1 T-1] × [M1 L-3 T0]-1 = [M0 L2 T-1].

Therefore, the Kinematic viscosity is dimensionally represented as [M0 L2 T-1].

FAQ (Frequently Asked Questions)

1. Explain the Properties on Which Viscosity Depends?

On the temperature of fluid:

  • Since cohesive force decreases with an increase in temperature. Therefore with the rise in temperature the viscosity of liquid decreases.

  • The viscosity of gases is the result of the diffusion of gas molecules from one moving layer to another moving layer. Now with an increase in temperature the rate of diffusion increases. So, the viscosity also increases. Thus the viscosity of gases increases with the rise of temperature. 

On the pressure of fluid:

  • The viscosity of liquids increases with an increase in pressure.

  • The viscosity of gas increases is practically independent of pressure.

2. Write a Few Limitations of Dimensional Formula?

  • This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

  • The relation derived from this method gives no information about the dimensionless constants.

3. Write a Few Properties of Viscosity?

Few properties of viscosity are discussed below:

1. It opposes motion.

2. It acts tangentially in a direction opposite to that of motion.

3. It comes into play when the two layers of a liquid are in relative motion.