PX285 - H7 - quantifying the effects of viscosity

the stress:

τ = μ \frac{\partial u_{x}}{\partial y}

it is expressed in the units of pressure

[τ] = \frac{[μ] [L]}{[T^{2}] [L^{2}]} = \frac{[M]}{[T] [L]}

[μ] = [M] [L]^{- 1} [T]^{- 1}

typical scales in a fluid:
- density: $ρ_{0} = [M] / [L]^{3}$
- characteristic speed: $u_{0} = [L] / [T]$
- characteristic spatial scale: $L_{0} = [L]$
constructing a combination of those typical scales of the same dimensions as viscosity:

ρ_{0} u_{0} L_{0} \sim μ

to assess whether $m u$ is big or small, compare it with $ρ_{0} u_{0} L_{0}$ in a specific problem
$ρ_{0} u_{0} L_{0} / μ$ is a dimensionless quantity
if the fraction is big, $μ$ can be neglected
reynold's number:

R_{e} = \frac{ρ_{0} u_{0} L_{0}}{μ}

a small $R_{e}$ means the flow is viscous dominant (laminar)
a large $R_{e}$ means the flow is turbulent
specific values of $R_{e}$ comes from experiments
note: this estimation is rough
eg:
- in the flow in a cylinder, $L_{0}$ could be either $a$ or $2 a$
- furthermore, $u_{0}$ could be the $u_{m a x}$ , or $u_{m a x} / 2$ , or the average speed $⟨ u ⟩$

⟨ u ⟩ = \frac{1}{π a^{2}} \int_{0}^{2 π} \int_{0}^{a} u_{z} (r) r d r d ϕ = \frac{Q a^{2}}{a μ}

R_{e} = \frac{Q d_{0}^{3} ρ_{0}}{32 μ^{2}}

flows in this case are laminar, $R_{e} < 3200$
a pipe of $5$ cm in diameter, $ρ_{w a t e r} = 10^{3}$ kg m $^{- 3}$ , $μ_{w a t e r} = 10^{- 3}$ N s m $^{- 2}$
- $u_{0} \approx 5$ cm s $^{- 1}$ is the threshold
- if $u > u_{0}$ , the flow is turbulent
- if $u < u_{0}$ , the flow is laminar