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Viscosity of a Fluid

Force applied on a matter creates stresses on it. Stress is simply force per unit area:{\displaystyle \displaystyle \tau ={\frac {F}{A}}\left[{\frac {N}{m^{2}}}=Pa\right]}

{\displaystyle \displaystyle \tau={\frac {F}{A}}\left[{\frac {N}{m^{2}}}=Pa\right]}
{\displaystyle Pa}

Hence the unit of stress is {\displaystyle Pa}. There can be normal and shear stresses in and on the matter.

Shear stress is proportional to the deformation rate of the matter, i.e. strain rate:{\displaystyle \displaystyle \tau \propto {\frac {\delta \theta }{\delta t}}}

{\displaystyle \displaystyle \tau \propto {\frac {\delta \theta }{\delta t}}}

{\displaystyle \displaystyle \tan {\delta \theta }={\frac {\delta u\delta t}{\delta y}}}

{\displaystyle \displaystyle \tan {\delta \theta }={\frac {\delta u\delta t}{\delta y}}}
{\displaystyle \displaystyle u}

{\displaystyle \displaystyle u} is the deformation speed. For very small deformation angles{\displaystyle \displaystyle \delta \theta ={\frac {\delta u\delta t}{\delta y}}\ \rightarrow \ {\frac {\delta \theta }{\delta t}}={\frac {\delta u}{\delta y}}\ \rightarrow \ \tau \propto {\frac {\delta u}{\delta y}}\ \rightarrow \ \tau =\mu {\frac {\delta \theta }{\delta t}}=\mu {\frac {\delta u}{\delta y}}}

{\displaystyle \displaystyle \delta \theta={\frac {\delta u\delta t}{\delta y}}\ \rightarrow \ {\frac {\delta \theta }{\delta t}}={\frac {\delta u}{\delta y}}\ \rightarrow \ \tau \propto {\frac {\delta u}{\delta y}}\ \rightarrow \ \tau=\mu {\frac {\delta \theta }{\delta t}}=\mu {\frac {\delta u}{\delta y}}}
{\displaystyle \displaystyle \mu }

{\displaystyle \displaystyle \mu } is the dynamic viscosity of the fluid.

{\displaystyle \displaystyle \tau }
{\displaystyle \displaystyle \mu }


For the same {\displaystyle \displaystyle \tau } and fluid having higher viscosity {\displaystyle \displaystyle \mu }, the deformation rate, i.e. velocity gradient is smaller.

Dynamic viscosity is a thermodynamic property of the material and it depends on temperature and pressure. In general, viscosity of liquids drop by increasing temperature, whereas that of gases increases. The viscosities of liquids and gases increase with increasing pressure.{\displaystyle \displaystyle \mu =f\left(T,P\right)\left[Pa\cdot s\right]}

{\displaystyle \displaystyle \mu=f\left(T,P\right)\left[Pa\cdot s\right]}

Often dynamic viscosity is normalized by the density of the fluid and this quantity is called “kinematic viscosity”:{\displaystyle \displaystyle \upsilon ={\frac {\mu }{\rho }}\left[{\frac {m^{2}}{s}}\right]}

{\displaystyle \displaystyle \upsilon={\frac {\mu }{\rho }}\left[{\frac {m^{2}}{s}}\right]}

One can judge the dominance of inertial effects to viscous effects by using a dimensionless number, namely Reynolds number:{\displaystyle \displaystyle Re={\frac {\rho U_{c}l_{c}}{\mu }}={\frac {U_{c}l_{c}}{\upsilon }}}

{\displaystyle \displaystyle Re={\frac {\rho U_{c}l_{c}}{\mu }}={\frac {U_{c}l_{c}}{\upsilon }}}
{\displaystyle \displaystyle U_{c}}
{\displaystyle \displaystyle l_{c}}

{\displaystyle \displaystyle U_{c}} and {\displaystyle \displaystyle l_{c}} are characteristic velocity and length scales of the flow.