Flow in a slit with uniform cross flow (Fig. 3B.16). A fluid flows in the positive x-direction through a long flat duct of length L,

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Flow in a slit with uniform cross flow (Fig. 3B.16). A fluid flows in the positive x-direction through a long flat duct of length L, width W, and thickness B, where L > > W >> B. The duct has porous walls at y = 0 and y = B, so that a constant cross flow can be maintained, with v_y = v₀, a constant, everywhere. Flows of this type are important in connection with separation processes using the sweep-diffusion effect. By carefully controlling the cross flow, one can concentrate the larger constituents (molecules, dust particles, etc.) near the upper wall.   

(a) Show that the velocity profile for the system is given by in which A = Bv₀p/μ 

(b) Show that the mass flow rate in the x direction is

(c) Verify that the above results simplify to those of Problem 2B.3 in the limit that there is no cross flow at all (that is, A → 0). 

(d) A colleague has also solved this problem, but taking a coordinate system with y = 0 at the mid plane of the slit, with the porous walls located at y = _b. His answer to part (a) above is in which a = bv₀P/μ, and η = y/b. Is this result equivalent to Eq. 3B.16-1?

y = B v,ly) y = 0 Part (a) (Po P)B 1 (y ehy/B (.16-1) v, = eA 1 4 (B L Part (b) (Po P)BWp 1 (1 A (2 (.16-2) L Part (d) ean - n sinh a cosh a cosh a (.16-3) (v,) (1/a) sinh a