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Introduction
The coefficient of viscosity, denoted by η, is a measure of the internal friction within a fluid. It quantifies the resistance of a fluid to flow. In the given relation F = 6πηrv, where F is the force of viscosity, r is the radius of the tube, and v is the velocity of the fluid, we can determine the dimensions of the coefficient of viscosity using dimensional analysis.

Method
Dimensional analysis involves examining the dimensions of all the quantities involved in an equation to determine the dimensions of an unknown quantity. In this case, we need to determine the dimensions of the coefficient of viscosity (η).

Dimensions of Force of Viscosity (F)
The force of viscosity (F) can be expressed in terms of mass (M), length (L), and time (T) as follows:
F = MLT^(-2)

Dimensions of Radius (r)
The radius (r) can be expressed in terms of length (L) as follows:
r = L

Dimensions of Velocity (v)
The velocity (v) can be expressed in terms of length (L) and time (T) as follows:
v = LT^(-1)

Dimensions of Coefficient of Viscosity (η)
Substituting the dimensions of force of viscosity (F), radius (r), and velocity (v) into the given equation F = 6πηrv, we can determine the dimensions of the coefficient of viscosity (η).

MLT^(-2) = 6πη(L)(LT^(-1))(L)

Simplifying the equation, we get:
MLT^(-2) = 6πηL^2T^(-1)

Comparing the dimensions on both sides of the equation, we can equate the corresponding powers of mass, length, and time:

M: 1 = 0
L: 1 = 2
T^(-2): 1 = -1

From the above equations, we can determine the dimensions of the coefficient of viscosity (η):

M^0L^1T^(-2) = 1

Therefore, the dimensions of the coefficient of viscosity (η) are:
[M^0L^1T^(-2)]

Conclusion
The dimensions of the coefficient of viscosity (η) are [M^0L^1T^(-2)]. This implies that the coefficient of viscosity is dimensionless, as it does not have any physical dimensions of mass, length, or time. The coefficient of viscosity represents the ratio of the force of viscosity to the product of radius and velocity, and its numerical value depends on the specific fluid being considered.