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Continuum Mechanics M`aster en Enginyeria de Camins, Canals i Ports M`aster en Enginyeria Geol`ogica i de Mines
Rehearsal Exam AP4 (Example A) - 16/01/2017
Name and surnames: ............................................................. National Identification Number (DNI/NIE):
STATEMENTS sheet
............................................................. Observations: 1. Each question may have one or more correct answers. The student must mark them filling with dark ink the whole space of the corresponding box in the answers sheet. 2. Each correct answer in a same question adds 1/n times the value of the question (where n={1,2,3,4} is the number of correct answers of the question). 3. Each incorrect answer in a same question subtracts 1/(4-n) times the value of the question (where n={1,2,3,4} is the number of correct answers of the question). 4. Any unanswered question will be given 0 marks. Answers must be given exclusively on the answers sheet: answers given on the statements sheets will be ignored. Question 1
The mean pressure and thermodynamic pressure in an isotropic Newtonian fluid coincide if:
A The fluid has a uniform velocity.
C The fluid is incompressible.
B The fluid’s bulk viscosity is zero, κ=0.
D The fluid’s viscosity parameter is zero, µ = 0.
Question 2
Indicate which of the following statement/s is/are true for a Newtonian fluid:
A For an incompressible fluid, the recoverable power is null. B The external mechanical power entering the medium is used only in modifying the kinetic energy.
C For an incompressible fluid, if the external mechanical power entering the medium is zero, the kinetic energy will never increase. D The recoverable power is always null.
p 1 ∂ χ¯ ¯ 2+ ∇ χ) + φ + (∇ = 0 (∀x, ∀t) with v = ∇ χ¯ to be valid for the ρ0 2 ∂t motion of a fluid, it is enough that one of these situations be fulfilled: Question 3
For the equation
A Incompressible fluid in transitory regime. B Perfect and incompressible fluid with potential body forces in steady-state regime.
C Incompressible fluid and irrotational flow with potential body forces in transitory regime. D None of the other answers.
The figure shows a disc with radius R rotating with a constant angular velocity ω at a distance a from a
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horizontal surface. Between the disc and the surface is a Newtonian fluid with viscosity µ and subjected to body forces b = [0, 0, −g]T . The pressure just below the disc (at z = a) is Patm ≈ 0. HYPOTHESES • Steady state regime. • Inertial forces are negligible. • Incompressible flow. • The friction effects between the fluid and the lateral walls are negligible. • The plate has radial symmetry. • The velocity field varies linearly with the distance to the inferior surface, which means: vθ = vθ (r, z) = (C1 z +C2 ) f (r) ∂ 2 vθ =0 therefore, ∂ z2 vr = vz = 0 Question 4
The θ - component of the velocity field of the fluid satisfies: ω ω A vθ = C vθ = zr. z + 1 r. a a ω 1 ω B vθ = 2π r. D vθ = zr + . a a r
Question 5
The pressure field of the fluid satisfies:
A p = ρg (ωa − z).
C p = ρga.
B p = ρg (a − z).
D p = ρgz.
Question 6
Considering the following stress field for the fluid, −p 0 0 ω 0 −p µ r . σ = a ω 0 µ r −p a
the moment M shown in the figure above that must be applied on the plate for the described motion to be maintained satisfies: 1 ω A M = π µ R4 . 2 a ω B M = 2π µ R4 . a
C M = −2π pR2 . ω 1 D M = πµ − p R2 . 2 a
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Question 7
Indicate which of the following statement/s is/are true for the Navier-Stokes equations:
A They are valid only for potential flows, both in steady-state and transient regimes.
C They are obtained by introducing the mechanical constitutive equation of Newtonian fluids into the momentum balance equations.
B They take into account the inertial forces. D They are the Cauchy equations expressed solely in terms of velocity and pressure.
Question 8 Indicate which of the following statement/s is/are true in relation to the boundary conditions in a fluid mechanics problem: A The particles belonging to a free surface have no acceleration because they belong to a material surface.
C The adherence condition implies that the relative velocity of the fluid with respect to the wall is zero.
B The impenetrability condition implies that the relative velocity of the fluid with respect to the wall is zero.
D Mixed boundary conditions prescribe velocity fields and body forces acting on the fluid.
Question 9 A container filled with water up to a height H is accelerated with a constant value a = (ax , 0) in addition to being subjected to the gravitational field b = (0, −g). The angle of the free surface with respect to the x-axis is:
A tan(θ ) = −
ax . g
ax . g g D tan(θ ) = − . ax C tan(θ ) =
B tan(θ ) = 0.
Question 10 A continuous medium with density ρ, volume V and boundary ∂V = Γu ∪ Γσ is subjected to certain body forces, b on V , prescribed contact forces, t∗ on Γσ , and prescribed displacements, u∗ on Γu . The stress tensor is σ and the acceleration, a. Considering the Cauchy equations and equilibrium equations on the boundary, the associated variational principle is Z Z δ W(u; δ u) = E · δ u dV + T · δ u dΓ, ∀δ u(x) δ u x∈Γu = 0. V
∂V
The values E and/or T are: A T = t∗ .
C E = ∇ ·σ σ (u) + ρb.
B T = σ (u) · n.
D E = ∇ ·σ σ (u) + ρ (b − a(u)).
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Continuum Mechanics M`aster en Enginyeria de Camins, Canals i Ports M`aster en Enginyeria Geol`ogica i de Mines
Rehearsal Exam AP4 (Example A) - 16/01/2017
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Please encode your National Identification Number (DNI/NIE) using the columns to the left. Name, surnames and DNI/NIE must also be written in the box above.
Observations: 1. Each question may have one or more correct answers. The student must mark them filling with dark ink the whole space of the corresponding box in the answers sheet. 2. Each correct answer in a same question adds 1/n times the value of the question (where n={1,2,3,4} is the number of correct answers of the question). 3. Each incorrect answer in a same question subtracts 1/(4-n) times the value of the question (where n={1,2,3,4} is the number of correct answers of the question). 4. Any unanswered question will be given 0 marks. Answers must be given exclusively on this answers sheet: answers given on the statements sheets will be ignored. Q UESTION 1:
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