Now, the liquid during this time from (1.81) will lose a fraction of its linear
momentum density, which can be given as
piρU 0 ð 1 : 82 ÞThe lost fraction of momentum (pd)in(1.82) will be transferred as a force,
named inertial/centrifugal force, which is directed outwards in the same direction
the liquid was initially flowing prior entering the curve. This inertial force density
can be calculated as
fipi=ti ð 1 : 83 ÞBy replacing (1.81) and (1.82)in(1.83) for pdand Tiwe will getfiρU 0 =tiρU^20 =w ð 1 : 84 ÞThese three equations form the basis of particle separation in non-circulating
fluidic chips.
2.5.2 Pẻclet Number: Diffusivities Across Channel Width
and No-Membrane Dynamic Filtering
In day-to-day life turbulent fluid mixing is crucial. To elaborate the time scale vs
length scale in the absence of this mixing, let us consider that we are holding a cup
piz= ρUρU0wpix= 0~ wfipfz= 0
pfx= 0Fig. 1.10 Inertial separation of the particles moving with a velocityU 0 and following a curved
path approximately equal to the width of the channel. Due to the conservation of momentum,
particle will experience an outward push known as inertial centrifugal force with a densityfias
described in (1.84).Piis inertial flow pressure where x represents in x direction and z represents in
z direction. Initially, prior to the curve, along x axis, the momentum is 0 while all the momentum is
focused along z axis. At the curve particle will lose z momentum that will translate to the
x-momentum. During this transition, the particle will experiencefinormally outwards towards
the putter wall of the microchannel
28 C.K. Dixit