CHBE 501 – Exam
CHBE 501 – Exam
Question 2 – 30 points
(a) Consider a wedge formed by two flat plates that meet along the z-axis with an opening angle of θ0 as shown. Where the plates meet, a small slit allows a volumetric
flow Q0 to enter or leave the wedge per unit width in z. The flow is characterized by steady-state and fully-developed conditions in the absence of gravity and
in which v = vrˆr, where vr(r, θ) is a function only of r and θ. Assuming that
vr(r, θ) = f(r)g(θ) can be written as a product of functions f(r) and g(θ), write an
integral expression in terms of vr for the full volumetric flow (per width in z) Q(r)
through the surface defined at r (i.e., the dashed boundary shown). Assuming the
fluid is incompressible, find f(r) up to a multiplicative constant.
(b) In class, we calculated the steady-state, fully-developed laminar flow in a pipe of
radius R, with the result that
whereP is the z-gradient of the dynamic pressure and µ is the viscosity of
the fluid. What is the vorticity w = ∇ × v?
(c) Consider planar flow of a Newtonian liquid of density ρ, viscosity µ and with vx =
x
2 − y
2 + x and vy = −(2x + 1)y. Verify that the flow is incompressible. Is the flow
irrotational? Find P(x, y), assuming that P(0, 0) = 0.
Question 3 – 20 points
(a) Show that the condition for the vectors a, b, and c to be coplanar is: εijkaibj ck = 0.
(b) The Stokes theorem can be stated as
for a vector field v. This is discussed in section A.5. Use this equality between
surface (S) and bounding contour (C) integrals to prove that ∇ × (∇f) = 0 for any
single-valued twice-differentiable scalar f. Hint: consider various surface regions S
and orientations nˆ.
DETAILED ASSIGNMENT
