Question 2 - Flow Through Narrow Slit
Viscous stresses exerted by laminar flow in a narrow slit. See below figure:
Fluid in
Fluid out
Figure 1: Flow through a slit, with B < W << L.
A Newtonian fluid (of viscosity μ and density ρ) is in laminar flow in a narrow slit formed between
two parallel walls with a distance 2B apart. The z-component of velocity in this narrow slit is:
2L.
Where P = ρ + gh = ρ gz, and all other velocity components are equal to 0
a) Compute divergence of the velocity field, ∇.u, and the xx, xz, zx, and zz components of the gradient of the velocity field (∇). Note: divergence of a vector is a scalar, gradient of a vector is a tensor. We can look at this problem in two dimensions (x, z)
b) Using results from (a), calculate viscous stresses Tx, z, zx, and zz everywhere in the domain.
c) Evaluate normal viscous stresses Tyx and Tzz at the slit walls: x = +/- B (as shown in figure, 2B is the slit aperture)
d) Let's assume the following values: L = 6.5 cm, W = 3 cm, 2B = 0.02 cm, water viscosity = 1 cP, water density = 1 g/cm³, and applied pressure difference of (ΔP) = 180 (g/cm/s²). Please compute total shear stress and total force exerted by the fluid on both plates. On the plates means x = +/- B. Note: if the shear stress is constant over the relevant area, you can get the force simply by multiplying stress and the area. If the shear stress varies, then you need to calculate the surface integral of the shear stress over that area. Be careful about whether you need to take positive or negative stress in the force calculation.
Hint: See appendix B. I for formulas. The inside cover of the textbook has some useful formulas as well (for Cartesian coordinate system).
∇ = (∂v/∂x + (∂v/∂y + (∂v/∂z) Cartesian coordinates (x, y, z):
(B.1-1)*
(B.1-2)
(∂v/∂t) -
(B.1-3)*
(B.1-4)
(B.1-5)
B.1-6)
in which
(B.1-7)