(Solved): Flow in a tube with suction Consider laminar flow in a circular tube. The tube wall is made of porou ...
Flow in a tube with suction Consider laminar flow in a circular tube. The tube wall is made of porous material through which the fluid enters with uniform radial velocity
v_(w)to join the mainstream flow inside the duct. Because of axisymmetry, the velocity profile may be considered two-dimensional - i.e.,
v=
v_(r)(r,z),v_(z)(r,z),0. Use the following variable,
\eta =((r)/(R))^(2)and
\psi (r,z)=v_(b)ar(z)R^(2)f(\eta ), where
Ris the radius of the pipe,
v_(bar)(z)=v_bar
(0)+2v_(w)((z)/(R))is the average axial velocity at any axial position,
z, (which is consistent with the global continuity equation) and
\psi is the stream function in cylindrical coordinates defined as
v_(r)=-(1)/(r)(del\psi )/(delz),v_(z)=+(1)/(r)(del\psi )/(delr)(a) Show that the continuity is satisfied by the introduction of above stream function. (b) Write down the simplified form of the
rand
zmomentum equations. (c) Show that they can be reduced to the following single equation for
f(\eta ).
2\eta f^('''')+4f^(''')+Re[f^(')f^('')-ff^(''')]=0where
Reis based on
R,v_(w)and
v. If you are pressed for time, show at least the steps involved in getting the above equation for partial credit.
