A microfluidic chip carries an aqueous buffer of dynamic viscosity and density along…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity

Fluids and Continuum Physics, Ch The Navier-Stokes equations and exact viscous solutions

Problem

A microfluidic chip carries an aqueous buffer of dynamic viscosity

\mu = 1.2\times10^{-3}\ \mathrm{Pa\,s}

and density

\rho = 1000\ \mathrm{kg\,m^{-3}}

along a straight horizontal channel of circular cross-section, radius

a = 60\ \mu\mathrm{m}

and length

L = 25\ \mathrm{mm}

. The flow is steady, smooth, layered, incompressible, unidirectional and fully developed, with no slip at the wall, so

u(r) = (G/4\mu)(a^{2}-r^{2})

with

G

the uniform axial pressure gradient. The required volume flow rate is

Q = 2.0\times10^{-9}\ \mathrm{m^{3}\,s^{-1}}

. In the plane containing the axis take

x

along the flow and

y

across it measured from the axis, so that there

u_x = (G/4\mu)(a^{2}-y^{2})

and

u_y = 0

. ((a)) Find the pressure difference needed between the ends of the channel. ((b)) At

y = a/2

, find the rate-of-strain component

e_{xy} = \tfrac12(\partial u_x/\partial y + \partial u_y/\partial x)

and the viscous shear stress

\tau_{xy}

, giving magnitudes. ((c)) The vorticity

\boldsymbol{\omega} = \nabla\times\mathbf{u}

of this planar flow varies linearly with

y

. Using Chapter 8's result that the circulation round a flat loop equals the flux of vorticity through it,

\Gamma = \int_A\omega_z\,\mathrm{d}A

, find the magnitude of the circulation round a square loop of side

d = 4.0\ \mu\mathrm{m}

lying in this plane, with its centre at

y = a/2

and its sides parallel to the axes. In your working, state which terms of the Navier-Stokes equation vanish for this flow and why.

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