Parth Ajmera

Shear stress

\tau = \frac{F}{A}

$\tau$ — shear stress (tangential force per unit area, resisting relative sliding between adjacent fluid layers), in pascals ( $\text{Pa}$ ) or newtons per square metre ( $\text{N/m}^2$ )
$F$ — tangential force, in newtons ( $\text{N}$ )
$A$ — area, in square metres ( $\text{m}^2$ )

\dot{\gamma} = \frac{v}{h}

$\dot{\gamma}$ — shear rate (velocity gradient), in reciprocal seconds ( $\text{s}^{-1}$ )
$v$ — velocity of the topmost fluid layer, in metres per second ( $\text{m/s}$ )
$h$ — height of the fluid layer, in metres ( $\text{m}$ )

\nu = \frac{\mu}{\rho}

$\nu$ — kinematic viscosity (momentum diffusivity — viscous effect on the fluid, independent of its mass or force), in square metres per second ( $\text{m}^2/\text{s}$ )
$\mu$ — dynamic viscosity, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )

\tau = K \dot{\gamma}^n

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$K$ — consistency index (generalized viscosity), in $\text{Pa} \cdot \text{s}^n$ (or $\text{N} \cdot \text{s}^n / \text{m}^2$ ): $K \uparrow \rightarrow$ viscosity $\uparrow$ ; $K \downarrow \rightarrow$ viscosity $\downarrow$
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )
$n$ — flow behaviour index (exponent that classifies shear-thinning vs. shear-thickening behaviour), dimensionless

\mu_a = K \dot{\gamma}^{\,n-1}

$\mu_a$ — apparent viscosity (the effective, shear-rate-dependent viscosity of a non-Newtonian fluid at an instant — stands in for constant $\mu$ in equations built for Newtonian fluids), in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$K$ — consistency index, in $\text{Pa} \cdot \text{s}^n$
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )
$n$ — flow behaviour index (exponent that classifies shear-thinning vs. shear-thickening behaviour), dimensionless

\tau = \mu \dot{\gamma} \quad \text{or} \quad \frac{\tau}{\dot{\gamma}} = \mu

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\mu$ — dynamic viscosity (constant for a Newtonian fluid — independent of the shear stress or shear rate applied), in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )

\tau = \tau_y + \mu_p \dot{\gamma}

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\tau_y$ — yield stress (threshold stress that must be exceeded before the material begins to flow at all), in pascals ( $\text{Pa}$ )
$\mu_p$ — plastic viscosity (also called apparent viscosity), in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )

\sqrt{\tau} = \sqrt{\tau_0} + \sqrt{K_c \dot{\gamma}}

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\tau_0$ — Casson yield stress (threshold stress below which the fluid behaves like a Bingham plastic), in pascals ( $\text{Pa}$ )
$K_c$ — Casson viscosity coefficient, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )

\tau = \tau_y + K \dot{\gamma}^n

$\tau$ — shear stress, in pascals ( $\text{Pa}$ )
$\tau_y$ — yield stress (threshold stress that must be exceeded before the material begins to flow at all), in pascals ( $\text{Pa}$ )
$K$ — consistency index, in $\text{Pa} \cdot \text{s}^n$
$\dot{\gamma}$ — shear rate, in reciprocal seconds ( $\text{s}^{-1}$ )
$n$ — flow behaviour index (exponent that classifies shear-thinning vs. shear-thickening behaviour), dimensionless

Very important

Re = \frac{d_i^2 N_i \rho}{\mu}

Re = \frac{d v \rho}{\mu}

Fr = \frac{N_i^2 d_i}{g}

$Fr$ — Froude number (ratio of inertial to gravitational/buoyancy forces), dimensionless
$N_i$ — frequency of the impeller, in reciprocal seconds ( $\text{s}^{-1}$ )
$d_i$ — diameter of the impeller, in metres ( $\text{m}$ )
$g$ — acceleration due to gravity, $9.8 \ \text{m/s}^2$