Question

from a consideration of he flow of liquid through narrow tube, define viscosity n in terms of te internal frictional force F, surface area A of the liquid and velocity gradient dv and derive its units using dx analysis dimensional

Answer 1

Explanation:

Have you ever noticed that some liquids like water flow very rapidly while some others like castor oil do not flow fast? Why is it so? Didn’t that question occur to you yet? Well, if it did, we have the answer to it! This is the concept of Viscosity. In this chapter, we will study all about the topic and look at the laws and examples of the same.

Answer 2

The definition of viscosity (η) in terms of the internal frictional force (F), surface area of the plane (A), and velocity gradient ($\frac{dv}{dx}$) along with the units of viscosity is given below.

Given:

The coefficient of viscosity = η.

The internal frictional force = F.

The surface area of the plane = A.

The velocity gradient = $\frac{dv}{dx}$.

To Find:

We have to find the following:

1. Definition of viscosity (η) in terms of the internal frictional force (F), surface area of the plane (A), and velocity gradient ($\frac{dv}{dx}$).

2. The units of coefficient of viscosity.

Solution:

The viscosity, fluid friction, or internal friction of a fluid is the property that prevents its many layers from moving in unison or the flow of the fluid.

The expression for internal frictional force or viscous force (F) acting between different layers of a liquid is given by,

$F =$ -η × A × $\frac{dv}{dx}$

The negative sign indicates that the direction of the force is opposite to the direction of flow of the liquid.

On rearranging the above equation, we get the expression for viscosity.

i.e., η = $- \frac{F(\frac{dv}{dx} )}{A}$.

The coefficient of viscosity is directly proportional to the viscous force and velocity gradient, whereas it is inversely proportional to the area of the plane.

The dimensional formula is the expression of a physical quantity in terms of its fundamental unit.

From the expression $Force = mass$ × $acceleration$, the dimensional formula of force is obtained as: $[MLT^{-2} ]$.

The dimensional formula of velocity (v) = $[LT^{-1} ]$

The dimensional formula of distance (x) = $[L ]$

The dimensional formula of velocity gradient ($\frac{dv}{dx}$) = $[T ]$

The dimensional formula of area (A) = $[L^{2} ]$

∴, The dimensional formula of viscosity (η) = $\frac{[MLT^{-2} ].[T]}{[L^{2}] }$

On simplifying, the dimensional formula of viscosity becomes: $[ML^{-1} T^{-1} ]$.

∴, The unit of viscosity (η) using dimensional analysis = Kg/ms.

Hence, the definition and units of coefficient of viscosity (η) is stated above.

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