Mid Term 10 Answer

Part I Choose TEN terms from the following list and explain each briefly(1.5 mark each, total 15 marks).

1) Ideal fluid: =0 or inviscid fluid.

2) Incompressible fluid: d/dP=0 or density does not vary with flow field.

3) Gage pressure: Pg=Pabs-Patm, where Patm is the ambient pressure.

4) Specific weight: =g, it is the weight per volume.

5) Time line: a set of adjacent fluid particles that were marked at the same instant in time, or the particles that traveled the same time with regard to a position marked at the same time upstream.

6) Stagnation point: A point on a surface where the approaching fluid velocity vanishes.

7) Path line: The actual path traveled by an individual fluid particle over the time period.

8) Substantial deviation: .

9) Contact angle: The angle that the tangent to the liquid surface makes with the solid surface at the point of contact.

10) Surface tension: It is the energy required to bring the liquid molecules from the bulk to the surface.

11) Bingham fluid: some materials can resist a finite shear stress and thus behave as a solid, but deform continuously when the shear stress exceeds the yield stress and behave like a liquid.

12) Velocity head: It is an expression of kinetic energy, or it is kinetic energy per weight of fluid, its value is V2/2g.

13) Non-slip condition: It is a condition frequently used as a boundary condition. It means that the fluid velocity next to a solid surface (or interface) is equal to the velocity of the solid surface (or interface).

14) Fluid: A substance is in gas or liquid state, which will continously deform under shear stress, no matter how small the stress is.

15) Velocity gradient: Velocity variation in space in the direction normal to the velocity, du/dy, etc.

16) Newtonian fluid: It is the fluid that obeys Newton’s law of viscosity with the viscosity being independent of the shear rate. =(du/dy), f(du/dy).

II-1. Bernoulli equation can be applied to describe the flow through a Venturi tube as shown in Figure II-1. For a gas flow, the pressure drop between the two tabs is measured by a manometer as indicatedFind the flow rate of air whose density can be taken as 1.2 kg/m3. (15 marks).

One form of Bernoulli equation is

Figure II-1

Solution:

(1) (3 marks)

(2) (5 marks)

(3) (2 marks)

Substitute (3) into (1): (2 marks)

(3 marks)

II-2. A vertically placed glass plate is used to create a thin flow film of water with some pollutants. The other side of the glass has light source for the advanced oxidation process to degrade the pollutants in water. If the glass has the dimension of 1m in width and 2m in height, neglect the end and edge effects, for a steady state flow,

a) Find the velocity distribution of the water film (10 marks).b) Find the thickness of the water film if the flow rate of water is 12 liters per

minute, density of water 1000 kg/m3, viscosity of water 1 mPaS. (3 marks)

c) Find the shear force exerting on the glass surface. (2 marks)

a)

Force balance on the differential control volume between x and x+Δx

(2 marks)Where ‘a’ is the width and ‘L’ is the height of the glass plate and ‘W’ is the weight of the control volume

Divide the whole equation by the volume and take limit →0

(1 marks)

Integration gives: (2 marks)

Using the boundary condition at gas-liquid interface, ,

. (1 marks)

Assume the fluid is Newtonian fluid,

(1 marks)

Integration gives:

x

Glass plate surface

Water film

x+Δx

z

x

(2 marks)

Using the boundary condition at solid-liquid interface, , Where is the thickness of the water film.

(1 marks)

Then we obtain the velocity profile of vz at different x,

b) The flowrate Q can be calculate by:

(2 marks)

Q =12L/min = 2x10-4 m3s-1

a = 1m, = 1000kgm-3, g = 9.81ms-2, = 1x10-3 Pas

= 3.9x10-4 m (1 marks)

d) Shear fore acting on the glass surface (1 marks)

(1 marks)

II-3. Consider the following steady, two-dimensional velocity field:

a) Is there a stagnation point? If so, where is it? (5 marks)b) Is the flow rotational? If so, which direction does it rotate, clockwise or

counterclockwise? (5 marks)Solution:

a) at the stagnation point u=-(1-x)2=0 x=1; v=2y(x-1)=0,y=0. So the stagnation point lies at (1,0) (5 marks)

b) at s-dimensional flow:

(3 marks) When y>0, counter clockwise (1 marks) When y<0, clockwise (1 marks)

c)

(3 marks)

III-1. If B is an extensive property of fluid with , show that:

where sys stands for system, cv for control volume, cs for the control surface, t is time. (20 marks)

III-1Define (see also the figure):1) The system is the quantity of matter of fixed identity (2 marks)2) The control volume (fixed in this case) is the region in space chosen for study, it allows mass to flow in and out across its boundaries (2 marks)

At time t, (2 marks)

At time t+Δt,

(2 marks)

Subtracting the first equation from the second one and dividing by Δt gives:

Taking the limit Δt→0, we get:

(2 marks)

Given that dB = b dm,

(4 marks)

Define the unit outer normal on control surface as shown in the figure (2 marks)

,

where the is the angle between the velocity and the outer normal. Inflow is negative, outflow is positive. (2 marks)

(2 marks)

Combined above:

III-2. For a flat plate of area A placed outside a submarine under water at an inclined angle of with respect to the water free surface. If the centroid of area is hc

meters below the sea surface, a) Show that the hydrostatic force acting on this plate is ghcA with being

the water density. (10 marks)b) Find the position of center of the pressure, (xp , yp ) in relationship with

(xc, yc), centroid of the area. (10 marks)

Solution:

a) (5 marks)

(3 marks)

(2 marks)

b)

(5 marks)

(2 marks)

(2 marks)

(1 marks)

III-3. When a horizontal pipe flow suddenly expands from A1 to A2, as shown in Figure III-3, low-speed, low-friction eddies appear in the corners and the flow gradually expands to A2 downstream. (Hint: use linear momentum equation, pressure is uniform at any given longitudinal position across the diameter, one can choose the CV as the volume of the inside of the pipe between positions 1 and 2). a) Find the pressure difference between positions 1 and 2 in terms of V1, A1

and A2. (10 marks) b) Find the force necessary to hold the pipe in position. (5 marks)

c) Obviously Bernoulli equation does not apply here because of the energy loss at this pipe contraction. Find the total head loss from position 1 to position 2 as a function of upstream velocity head. (5 marks)

Figure III-3Solution:

Mass balance gives: A1V1=A2V2 (2 marks)

Momentum balance gives: (3 marks)

(3 marks)

(2 marks)

b) Force necessary to hold the pipe(3 marks)

(2 marks)

c) Energy equation gives:

(3 marks)

(2 marks)