What is the Dimensional Formula of Coefficient of Viscosity?

According to Newton, the Viscous Force (F) between two layers of liquid is directly proportional to the Area (A) and the velocity gradient (dv/dx). The formula is: F = η × A × (dv/dx)

Step 1: Rearrange to isolate Eta (η): η = F / [A × (dv/dx)]

Step 2: Plug in the known dimensions:

• Force (F) = [M¹ L¹ T⁻²]
• Area (A) = [L²]
• Velocity Gradient (dv/dx) = (Distance/Time) / Distance = 1 / Time = [T⁻¹]

Step 3: Calculate: η = [M L T⁻²] / [L² × T⁻¹] η = [M] × [L¹⁻²] × [T⁻²⁻⁽⁻¹⁾] η = [M¹ L⁻¹ T⁻¹]

This is an easier alternative method often taught in schools. Stokes' Law calculates the massive drag force on a tiny sphere falling through a fluid: F = 6πηrv. (Where r is radius, v is velocity, and 6π is a dimensionless constant).

Step 1: Rearrange for η: η = F / (rv)

Step 2: Plug in dimensions:

• Force (F) = [M¹ L¹ T⁻²]
• Radius (r) = Length = [L¹]
• Velocity (v) = [L¹ T⁻¹]

Step 3: Calculate: η = [M L T⁻²] / [L × L T⁻¹] η = [M L T⁻²] / [L² T⁻¹] η = [M¹ L⁻¹ T⁻¹]

Looking at the derived dimension [M¹ L⁻¹ T⁻¹], the SI unit is kg/(m·s). However, in physics, the official SI unit is famously called the Poiseuille (Pl) or Pascal-second (Pa·s).