Question

State newton's law of viscosity and hence derive expression for coefficient viscosity give its C.G.S. Unit.?

Answer 1

Newton’s viscosity law’s states that, the shear stress between adjacent fluid layers is proportional to the velocity gradients between the two layers.

The ratio of shear stress to shear rate is a constant, for a given temperature and pressure, and is defined as the viscosity or coefficient of viscosity.

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Answer 2

Newton's law of viscosity states that shear stress is directly proportional to the velocity gradient. The expression for coefficient of viscosity is $\[\eta = FL/vA\]$ and its C.G.S. unit is $poise$.

Explanation:

Newton's law of viscosity:

• According to Newton’s law of viscosity, the shear stress acting between the two adjacent layers of a fluid is directly proportional to the negative value of the velocity gradient between the same two adjacent layers of the fluid.
• In mathematical form, it can be expressed as:

        $\[{\tau _{yx}} = \mu \frac{{d{v_x}}}{{dy}}\]$

       or

       $\[\tau = - \mu \frac{{dv}}{{dy}}\]$

       where, $\[\tau = \]$ shear stress, $\[\mu = \]$ viscosity, $\[\frac{{dv}}{{dy}} = \]$ rate of shear deformation

Expression for the coefficient of viscosity:

• Consider a liquid column having height $L$ between the two plates. The contact area of the plate with the liquid is $A$.
• A force $F$ pulls the upper plate that produces a velocity $v$. It is given in the figure below.
• The force $F$ is directly proportional to $A$, i.e., the area of the plate.
• The force $F$ is also directly proportional to speed $v$ as more speed needs more force.
• Force $F$ is inversely proportional to the distance $L$ between the plates as a smaller force will be required if the distance is large.
• Force $F$ is directly proportional to the coefficient of viscosity $\[\eta \]$ as greater force is required for greater viscosity.
• All these relations give the following equation:

       $\[F = \eta vA/L\]$            ... (i)

• Solving equation (i) for $\[\eta \]$, we have,

       $\[\eta = FL/vA\]$

• The S.I. unit of coefficient of viscosity is $\[(N/{m^2})s\]$ or $\[Pa \cdot s\]$.
• The C.G.S unit of coefficient of viscosity is $\[Poise\]$.

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