Motion of a Fiber in a Bifurcating Tube Flow

On internal fluid dynamics

“…This reproduces (12), (13) in effect. The validity of the boundary layer system (12, 13) relies mostly on a balance |u| 2 /|x| ∼ Re −1 |u|/|y| 2 being struck between inertial and viscous contributions in the sense of orders of magnitude, provided that the acceleration and pressure contributions respond passively to the others.…”

“…The velocity u in (26) along with zero v and constant dp/dx gives an exact solution not only of the Navier-Stokes equations (4, 5) but also of the boundary layer equations (12,13) by virtue of the balances remaining in (23)(24)(25) as well as the subset in (21). The boundary layer system reinforces the importance of the O(Re) length scale, while the lubrication system tends to correspond to longer length scales.…”

“…So provided σ 4 is not too small and σ 5 not too large the system (22) is inferred; this can be confirmed by a strict transformation and limiting process as above which yield (22) in terms of the primed variables. The lubrication approximation (4,13,21) can still apply here for thin vessels.…”

“…The latter leads to the orders displayed in (15) but now the scaling σ 2 = σ 6 Re −1/2 is implied, followed by a strict transformation and limit process which yield (21) effectively. The relevant boundary conditions for (12,13) or (21) depend on the specific physical setting as we shall see later.…”

“…These aspects are of special interest in practical [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] as well as theoretical terms [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”