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Hi i need help with this practice exam, can you help me with it? 3 questions and mostly it is vector and linear equations.

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CHBE 501 – Exam #1

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

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r

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Q0

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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 = vr r̂, 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

vz (r) = −

where P 0 =

∂

P

∂z

P0

R2 − r 2 ,

4µ

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 =

x2 − 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.

1

CHBE 501 – Exam #1

Question 3 – 20 points

(a) Show that the condition for the vectors a, b, and c to be coplanar is: εijk ai bj ck = 0.

(b) The Stokes theorem can be stated as

Z

Z

t · vdC =

C

n̂ · (∇ × v) dS

S

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 n̂.

1

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Tags:

fluid mechanics

vector field

Stokes Theorem

tensors and vectors

fluid viscosity

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