A wire of radius Ri is coated by a process described above. It is fed to a reservoir filled with a Newtonian fluid with viscosity %u03BC(mu) and density %u03C1(rho). It is then pulled through a circular hole (or die) of radius Ro and length L with a velocity V. Assuming a steady laminar flow in the annulus, derive an expression for the coating thickness t as a function of material properties and V. (Assume that the viscous force is dominant and the effect of gravity is negligible. Because both ends of the annular flow region is open to the air, the pressure in the annular space is constant throughout L.)
(a) Write down the continuity equation, Navier-Stokes equations and the boundary conditions for the flow in the annular region using a cylindrical coordinate system. (The equations should contain only the non-zero terms.)
(b) Solve the equations in (a) to determine the velocity field in the annulus.
(c) Calculate the flow rate Q of the fluid carried by the moving wire.
The following integral may be useful:
\(\int x ln x dx =(1/2)x^2 ln x - (1/4) x^2\)
(d) Using the mass balance, calculate the coating thickness t.
(e) What is the force required to pull the wire at the velocity V?