# Viscosity

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Entry by Robin Kirkpatrick

## Introduction

Viscosity is a measure of the fluid's internal resistance to flow. Formally, for Newtonian fluids, the shear stress is related to the shear strain via: $\tau _{ij} = \mu (\frac{\delta u_i}{\delta x_j} + \frac{\delta u_j}{\delta x_i})$

Stress is a 2nd rank tensor, where we keep the standard convention that $\tau_{ij}$ refers to the stress on the 'i' face in the 'j' direction (when doing force balances) and is related to the shear strain via the dynamic viscosity. Note that inviscid fluids cannot support shear stress, as evident by the above relation.

For non-Newtonian fluids, this relationship can be nonlinear. Some fluids can exhibit shear thinning (viscosity deacreases with shear rate) and shear thickening. Viscosity can be measured using a rheometer, such as the cone an plate viscometer which creates a relatively uniform value of shear stress.

Kinematic viscosity is useful for characterizing the ratio of the intertial to viscous forces (via the Reynold's number), where the kinematic viscosity is $\nu = \mu / \rho$ and the Reynold's number is

$Re = Lv/ \nu$ where L is the characteristic length of the systems, and v is the flow velocity.

The viscosity of a fluid can be estimated from the elastic modulus, G (Energy per volume) and the characteristic time of the system $\tau$ via $\mu = G \tau /2$

See also: