CHBE 441 UCLA Stokes Theorem Tensors & Vectors Fluid Mechanics Practice Exam Practice

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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
✓ = ✓0
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r
✓
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Q0
✓=0
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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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