Theoritical background
A Poiseuille flow of an incompressible Newtonian fluid through a
straight channel will exhibit a parabolic velocity flow profile with a
maximum velocity at the center and zero velocity at the walls of the
channel, i.e. a no-slip boundary condition. The flow of a fluid in the
channel is characterized using a dimensionless number Reynolds number
(Re), defined as \(Re=\frac{D_{h}\text{Uρ}}{\mu}\) ,where\(D_{h}=\frac{2\text{wh}}{w+h}\) is the hydraulic diameter with
‘w ’ and ‘h ’ being the width and height of the channel
respectively, ‘U ’ is the bulk velocity of the fluid, ‘\(\rho\)’
is the fluid density and ‘µ ’ is the dynamic viscosity. The
equilibrium position of the particle suspended in the fluid is mainly
due to the combined effect of shear induced lift force
(FLS) and wall induced lift force (FW).
The parabolic flow profile of the fluid induces FLS,
pushing the particle away from the center of the channel while
FW pushes the particle away from the wall. The
combination of these two forces, FL, defined as\(F_{L}=f_{l}\ \left(Re,x\right)\ \rho U^{2}\frac{a^{4}}{{D_{h}}^{2}}\),
will result in an equilibrium focusing position of the particle. Here,\(f_{l}\ \left(R_{e},x\right)\) is the lift coefficient, which is a
function of Re, ‘x ’ is the position of the particle and
‘a ’ is the size of the particle 22. The number
of equilibrium positions of the particles depend upon the geometry of
the channel. Segre & Silberberg in 1961, showed that, particles attain
their equilibrium positions along the annulus of a circular channel at
0.6 radius 46. In 2007, Di Carlo et al. observed that
in a square channel, particles occupy equilibrium positions at the faces
of each wall of the channel 22 while Bhagat et al.
demonstrated equilibrium positions along the longer sides of a
rectangular channel 47. In case of a fluid flowing
through a curved channel, the curvature of the channel creates a radial
pressure gradient creating secondary fluid flows leading to the
formation of two counter-rotating Dean vortices, normal to the bulk
flow. The drag force, FD , caused due to Dean vortices
is defined as \(F_{D}\sim\frac{\rho U^{2}a{D_{h}}^{2}}{R}\)22, where ‘R ’ is the radius of curvature of the
channel. The strength of the Dean vortices is determined by a
dimensionless Dean number defined as\(\text{De}=R_{e}\sqrt{\frac{D_{h}}{2R}}\) 48. In
contrast to a Newtonian fluid, when a viscoelastic (non-Newtonian) fluid
flows through a channel, an additional elastic force ,
FE , arises due to the first normal stress difference of
the fluid which pushes the particle toward the center of the channel.
This force is defined as \(F_{E}=a^{3}\nabla N_{1}\) where
‘N1 ’ is the first normal stress difference44, 49, 50.The elastic property of the fluid is
defined by the dimensionless Weissenberg number,\(W_{i}=\frac{2\text{λQ}}{hw^{2}}\), where ‘Q ’ is the
volumetric flow rate of the fluid and ‘\(\lambda^{\prime}\) is the relaxation
time of the fluid 51, 52.
In this paper, the complex interaction and balance between the above
forces are utilized to demonstrate focusing and differential migration
of bacteria from fully focused blood cells to achieve size-based
separation at high flow rates.