A newtonian fluid having a specific gravity of 0.92

NUCE502 Homework 1

Name: Score: of 100 1. (15 pts) A Newtonian fluid having a specific gravity (SG) of 0.92 and a kinematic viscosity of

4x10-4 m2/s flows past a fixed surface as shown in the figure below. The “non-slip” condition suggests that the velocity of fluid at the fixed surface is zero. The fluid velocity profile away from the fixed surface is given by

u U

= 3 2 y δ − 1 2

y δ

⎛ ⎝⎜

⎞ ⎠⎟ 3

, where U is the constant maximum velocity.

(a) Express the shear stress as a function of U and δ ; (b) Determine the magnitude of the shear stress at y=0 and at y=δ . (c) Present discussion on how shear stress vary with varying U and δ . 2. (10 pts) Refer to the figure on the right. Assume that steady-state flow

is established and the fluid velocity is a linear function of x. Determine the acceleration at points A, B and C. Recall that the total

acceleration is given by

Dv Dt

= ∂ v ∂t

+ v ⋅∇v

Answer: 720 m/s2; 1,440 m/s2; 2,160 m/s2 3. (15 pts) The velocity vector for a two-dimensional flow is given by

 V = 2xt î − 2yt ĵ

where t is time in second. Determine the (a) local acceleration and (b) convective acceleration for this flow. (c) What is the magnitude and direction of the velocity and the acceleration at points x = y = 2 m at t = 0. Answer:

alocal = 2x î − 2y ĵ ; aconv = 4xt

2 î + 4yt 2 ĵ ;  V = 0;

atotal = 4 î − 4 ĵ ; atotal = 5.657 m/s

2

4. (5 pts) The velocity component in the y-direction for an incompressible, two-dimensional flow is given by

v = 3xy − x2y Determine the velocity component in the x-direction, so that the continuity equation is satisfied.

Answer: u = 1 3 x3 − 3

2 x2 + f y( )

5. (10 pts) The velocity components of a two-dimensional fluid flow are given by u = y2 − x 1+ x( ) and v = y 2x +1( ) . Show if (a) the flow is incompressible and (b) the flow is irrotational. Show all the steps in proof for full credit.

6. (10 pts) The momentum balance equation is given by

∂ρv ∂t

+∇ ⋅ ρvv( ) = −∇p − ∇ ⋅τ + ρg ................................................... (2) (1) (5 pts) Explain in words what each term represents (or describes) physically. (2) (5 pts) From Eq. (2), derive the “Equation of Motion” given by

ρ D v

Dt = −∇p − ∇ ⋅τ + ρg ................................................................... (3)

Show all the steps in derivation for full credit.

7. (10 pts) Starting from Dsm = De 3

Ds 2 ; Show that the Sauter-mean diameter of a fluid particle can

be related to the void fraction and the interfacial area concentration by Dsm = 6α ai

. Here De

and Ds denote volume-equivalent and surface-equivalent diameters, respectively. Show all the steps in derivation for full credit.

8. (5 pts) Starting from the definition of the drift flux velocity, Vgj = vg − j ; Show that it can be also written as Vgj = 1−α( )vr . Show all the steps in derivation for full credit.

9. (10 pts) Starting from the definition of the flow quality, χ = Wg Wm

; Show that it can be also

written as a function of void fraction as

χ = 1

1+ ρ f ρg

⎛

⎝⎜ ⎞

⎠⎟ 1−α α

⎛ ⎝⎜

⎞ ⎠⎟ 1 s

⎛ ⎝⎜

⎞ ⎠⎟

.

Show all the steps in derivation for full credit.

10. (10 pts): In problem 9, show that the flow quality can be also written as a function of volumetric flux and as a function of mass flux as

χ = ρg jg

ρg jg + ρ f j f and χ =

Gg Gg +Gf

.

Show all the steps in derivation for full credit.