2141-365 - 2010 - HW1 (Solution) - Fundamental Concepts, Pathlines and Streamlines

2141-365 Fluid Mechanics for International Engineers HW #1: Fundamental concepts: dimensions, field concepts, velocity field. Pathlines and streamlines. Newton’s Viscosity Law Due: Mon, Aug 30, 2010. Drop the homework in the ISE’s course box.

1

2141-365 Fluid Mechanics for International Engineers HW #1: Fundamental concepts: dimensions, field concepts, velocity field.

Pathlines and streamlines. Newton’s Viscosity Law.

Problem 1. The Principle of Dimensional Homogeneity

A valid physical relation for the change in thermal/internal energy of a substance ( 12 UUU −=Δ ) and the specific heat (at constant volume, c ) can be expressed as

TmcU Δ=Δ where m is the mass of the substance and 12 TTT −=Δ is the change in temperature. 1.1. What are the dimensions of the specific heat c in terms of the primary dimensions

• EAtT (E = Energy, M = Mass, t = time, T = Temperature), and • MLtT (M = Mass, L = Length, t = time, T = Temperature)?

1.2. What should be the physical interpretation of the specific heat c ? Problem 2. Field Concept. Classification of Velocity Field. 2.1. Steadiness and Uniformity of the Property Field φ

Let φ be a property whose field representation is ),( txEvφφ = .

We are interested in • the uniformity of the field φ over a region of interest (it can be line, area, or volume) at any one instant, • the steadiness of the field φ over a period of time of interest (and over the region of interest).

Let the shading represent the value of the property φ , say light = high value of φ , dark = low value of φ . Answer the following questions.

(b) Sketch a similar diagram for a uniform but unsteady field.

(a)

t t+dt

Over the region of interest, is the property field φ steady over the period of time of interest? At the instant t, is the property field φ uniform over the region of interest? At the instant t+dt, is the property field φ uniform over the region of interest?


2

2.2. Classification of Velocity Field

(a) Fill in the table below. ba, , and c are non-zero, finite constants.

tV ∂∂ /v

Unsteady or Steady ..)(...,...,.VV

vv=

1-, 2-, or 3-dim.

Vvv

×∇=:ω Rotational

or Irrotational

Vv

⋅∇ [a] [b] [c] [d]

A kctjbyiaxV ˆ)(ˆ)(ˆ)( ++=v

kctV ˆ/ =∂∂v

Unsteady ),,( tyxVVvv

= 2 0vv

=×∇ V Irrotational baV +=⋅∇v

1/t 1/t L/t2 -

B ieayV bt ˆ)( 2 −=v

C kcjbyiaxyV ˆ)(ˆ)(ˆ)( +−=v

D ( ) jdicybtaV ˆ)(ˆsin +−=v

(b) Can you initially guess physically what the quantity b in Problem B should be related to? If so, what?


3

2.3. Dimensionality and Functionality of A Velocity Field Assuming that the following velocity fields are steady. State the dimensionality of the fields and write down

its functional relation (e.g., ),,,( tzyxVv

, ),,( tyxVv

, ),( zxVv

, or etc.)

(d) Similar to above, sketch an example of the velocity field in which iyVV x

ˆ)(=v

Problem 3. Velocity Field and Its Boundary Conditions. Shear deformation and shear stress.

1. Qualitatively sketch the velocity profile along the transverse AB. Be mindful about physical constraints at the solid and free boundaries. Assume for simplicity first that there is only one dominant velocity component.

2. For the marked surface (marked by solid dot) of the fluid element C (and D):

a. state whether the shear deformation (or velocity gradient) dydu / at the point on that surface is positive or negative,

b. state whether yxτ at the point on that surface is positive or negative, and

c. sketch the correct direction of the shear stress yxτ on that surface at that point. d. Also, state whether the fluid element C is being dragged forward (in x+ direction) or dragged

backward (in x− direction) by the fluid element adjacent to it at that surface.

x

y

z

1-, 2-, or 3-D?

.)..........(.........Vv

(a)

x

y

z

1-, 2-, or 3-D?

.)..........(.........Vv

(b)

x

y

z

1-, 2-, or 3-D?

.)..........(.........Vv

x

y

z

(c)


4

3.1. Flow in a channel with a moving plate in the middle. Assume linear velocity distribution across the gaps.

3.2. Wall jet. Assume smooth velocity profile.

C is a fluid element in the region where u is

increasing with increasing y

Marked surface on fluid element C

dydu / :

yxτ :

Forward/Backward (Being dragged forward or backward by the fluid element

adjacent to it at that surface?)

D is a fluid element in the region where u is

decreasing with increasing y

dydu / :

yxτ :



y

xA

B

D

Flow

Stationary wall

Stationary surrounding fluid far away above the wall

C


dydu / :

yxτ :



Marked surface on fluid element D

dydu / :

yxτ :



y

x

Flow

A

B

C

D

Flow

Stationary plate

Stationary plate

Moving plate, oV


5

Problem 4. Velocity Field and Flow Lines [Adapted from Fox et al., 2010, Problem 2.17, p. 45; and from 2145-213 – 2010 HW#1]

Consider the flow described by the velocity field jCyiAtBxV ˆ)(ˆ)1( ++=

v

where A , B , and C are constants and all numerical values in the above expression are dimensionless. 4.1. Is the velocity field steady? 4.2. What are the dimensions of A , B , and C in the system of primary dimensions MLtT? Pathline 4.3. Find the pathline of the particle that is located at the point (xo , yo ) at time to . 4.4. Find the velocity of the above particle at any time t . 4.5. Find the acceleration of the above particle at any time t . Streamline 4.6. Find the streamline that passes through the point (xo , yo ) at time to .

Problem 5. Newton’s Viscosity Law and Shear Stress [Çengel and Cimbala, 2010, Problem 2-75, p. 66.] A 50-cm× 30-cm× 20-cm block weighing 150 N is to be moved at a constant velocity of 0.8 m/s on an inclined

surface with a friction coefficient of 0.27. a) Determine the force F that needs to be applied in the horizontal direction. b) If a 0.4-mm thick oil film with a dynamic viscosity of 0.012 Pa-s is applied between the block and inclined

surface, determine the percent reduction in the required force.

50 cm

30 cm

W = 150 N

F

V = 0.8 m/s

θ = 20o


6

Solution


7

2141-365 Fluid Mechanics for International Engineers HW #1: Fundamental concepts: dimensions, field concepts, velocity field.

Pathlines and streamlines. Newton’s Viscosity Law.

Problem 1. The Principle of Dimensional Homogeneity

A valid physical relation for the change in thermal/internal energy of a substance ( 12 UUU −=Δ ) and the specific heat (at constant volume, c ) can be expressed as

TmcU Δ=Δ where m is the mass of the substance and 12 TTT −=Δ is the change in temperature. 1.1. What are the dimensions of the specific heat c in terms of the primary dimensions

• EAtT (E = Energy, M = Mass, t = time, T = Temperature), and • MLtT (M = Mass, L = Length, t = time, T = Temperature)?

1.2. What should be the physical interpretation of the specific heat c ? Solution

1.1. Tm

UcΔ

Δ= → [ ]

eTemperaturMassEnergy

MTE

TmUc

−==⎥⎦

⎤⎢⎣⎡

ΔΔ

= ANS

→ [ ] 12222

−−−

===⎥⎦⎤

⎢⎣⎡

ΔΔ

= TtLMT

tMLMTE

TmUc ANS

1.2. From [ ]eTemperaturMass

EnergyMTE

TmUc

−==⎥⎦

⎤⎢⎣⎡

ΔΔ

= , c can be interpreted as the quantity that is related to

energy per unit mass per unit change in temperature. ANS


8

Problem 2. Field Concept. Classification of Velocity Field. 2.1. Steadiness and Uniformity of the Property Field φ

Let φ be a property whose field representation is ),( txEvφφ = .

We are interested in • the uniformity of the field φ over a region of interest (it can be line, area, or volume) at any one instant, • the steadiness of the field φ over a period of time of interest (and over the region of interest).

Let the shading represent the value of the property φ , say light = high value of φ , dark = low value of φ . Answer the following questions.

(b) Sketch a similar diagram for a uniform but unsteady field. Solution

ANS

ANS

(b)

t t+dt

Uniform but unsteady field

(a)

t t+dt

Over the region of interest, is the property field φ steady over the period of time of interest? Yes At the instant t, is the property field φ uniform over the region of interest? No At the instant t+dt, is the property field φ uniform over the region of interest? No

(a)

t t+dt

Over the region of interest, is the property field φ steady over the period of time of interest? At the instant t, is the property field φ uniform over the region of interest? At the instant t+dt, is the property field φ uniform over the region of interest?


9

2.2. Classification of Velocity Field

(a) Fill in the table below. ba, , and c are non-zero, finite constants.

tV ∂∂ /v

Unsteady or Steady ..)(...,...,.VV

vv=

1-, 2-, or 3-dim.

Vvv

×∇=:ω Rotational

or Irrotational

Vv

⋅∇ [a] [b] [c] [d]

A kctjbyiaxV ˆ)(ˆ)(ˆ)( ++=v

kctV ˆ/ =∂∂v

Unsteady ),,( tyxVVvv

= 2 0vv

=×∇ V Irrotational baV +=⋅∇v

1/t 1/t L/t2 -

B ieayV bt ˆ)( 2 −=v

ieaby bt ˆ)( 2 −− Unsteady ),( tyVVvv

= 1 kaye bt ˆ)2( −− Irrotational 0 1/(Lt) 1/t - -

C kcjbyiaxyV ˆ)(ˆ)(ˆ)( +−=v

0v

Steady

),( yxVVvv

= 2 kax ˆ)(− Irrotational bay − 1/(Lt) 1/t L/t -

D ( ) jdicybtaV ˆ)(ˆsin +−=v

( )icybtabV ˆcos −=v

Unsteady

),( tyVV

vv= 1 kcybtac ˆ)cos( − Irrotational 0 L/t 1/t 1/L L/t

(c) Can you initially guess physically what the quantity b in Problem B should be related to? If so, what?

Because of its dimension ( [ b ] = 1/t = frequency), we can guess that b should be related to some kind of frequency. ANS


10

2.3. Dimensionality and Functionality of A Velocity Field Assuming that the following velocity fields are steady. State the dimensionality of the fields and write down its

functional relation (e.g., ),,,( tzyxVv

, ),,( tyxVv

, ),( zxVv

, or etc.)

(d) Similar to above, sketch an example of the velocity field in which iyVV x

ˆ)(=v

ANS

x

y

z

1-, 2-, or 3-D? 2-D

),( yxVv

(a)

x

y

z

1-, 2-, or 3-D? 2-D

),( zyVv

(b)

x

y

z

1-, 2-, or 3-D? 3-D

),,( zyxVv

x

y

z

(c)


11

Problem 3. Velocity Field and Its Boundary Conditions. Shear deformation and shear stress.

1. Qualitatively sketch the velocity profile along the transverse AB. Be mindful about physical constraints at the solid and free boundaries. Assume for simplicity first that there is only one dominant velocity component.

2. For the marked surface (marked by solid dot) of the fluid element C (and D):

a. state whether the shear deformation (or velocity gradient) dydu / at the point on that surface is positive or negative,

b. state whether yxτ at the point on that surface is positive or negative, and

c. sketch the correct direction of the shear stress yxτ on that surface at that point. d. Also, state whether the fluid element C is being dragged forward (in x+ direction) or dragged backward

(in x− direction) by the fluid element adjacent to it at that surface. 3.1. Flow in a channel with a moving plate in the middle. Assume linear velocity distribution across the gaps.

ANS


dydu / : negative

yxτ : negative

Forward/Backward backward (Being dragged forward or backward by the fluid element


Marked surface on fluid element D

dydu / : negative

yxτ : negative

Forward/Backward forward (Being dragged forward or backward by the fluid element


y

x

Flow

A

B

C

D

Flow

Stationary plate

Stationary plate

Moving plate, oV oV

C

D


12

3.2. Wall jet. Assume smooth velocity profile.

ANS

C is a fluid element in the region where u is

increasing with increasing y


dydu / : positive

yxτ : positive

Forward/Backward backward (Being dragged forward or backward by the fluid element


D is a fluid element in the region where u is

decreasing with increasing y

dydu / : negative

yxτ : negative

Forward/Backward forward (Being dragged forward or backward by the fluid element


y

xA

B

D

Flow

Stationary wall

Stationary surrounding fluid far away above the wall

C

C

D


13

Problem 4. Velocity Field and Flow Lines [Adapted from Fox et al., 2010, Problem 2.17, p. 45; and from 2145-213 – 2010 HW#1]

Consider the flow described by the velocity field jCyiAtBxV ˆ)(ˆ)1( ++=

v

where A , B , and C are constants and all numerical values in the above expression are dimensionless. 4.1. Is the velocity field steady? 4.2. What are the dimensions of A , B , and C in the system of primary dimensions MLtT? Pathline 4.3. Find the pathline of the particle that is located at the point (xo , yo ) at time to . 4.4. Find the velocity of the above particle at any time t . 4.5. Find the acceleration of the above particle at any time t . Streamline 4.6. Find the streamline that passes through the point (xo , yo ) at time to . Solution 4.1. Since ),,( tyxVV

vv= [ 0/

vv≠∂∂ tV ], the velocity field is not steady. ANS

4.2. ][A = 1/t, [ B ] = 1/t, and [ C ] = 1/t. ANS Pathline 4.3. Find the pathline of the particle that is located at the point (xo , yo ) at time to.

⎥⎦⎤

⎢⎣⎡ −+−

=

⎥⎦⎤

⎢⎣⎡ −+−=

+=

+=

+==

∫∫

)(2

)(

22

2

)(

22

)(

)(2

)(

2)(ln

)1(

)1(

oo

o

oo

ttAttB

o

oo

t

to

t

t

tx

x

x

extx

ttAttB

tAtBx

tx

dtAtBx

dx

dtAtBxdtVdx

)(

)(

)(

)(

)(ln

o

o

oo

ttCo

o

tt

o

t

t

ty

y

y

eyty

ttC

tCy

ty

Cdty

dy

Cydt

dtVdy

−=

−=

=

=

=

=

∫∫

Thus, the path line is given by

( ) LDimensionjeyiexjtyitxtr ooo ttC

o

ttAttB

o =+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=+= −⎥⎦

⎤⎢⎣⎡ −+−

,ˆˆˆ)(ˆ)()( )()(

2)( 22

v . ANS

4.4. Find the velocity of the above particle at any time t . The velocity of the particle in 4.3 can be found as follows. Approach 1: Directly time differentiate the position vector to the pathline.

( ) ( )

( ) ( )jeCyiexAtB

jCeyiAtBexjdt

tdyidt

tdxdt

trdtV

ooo

ooo

ttCo

ttAttB

o

ttCo

ttAttB

o

ˆˆ1

ˆˆ1ˆ)(ˆ)()()(

)()(

2)(

)()(

2)(

22

22

−⎥⎦⎤

⎢⎣⎡ −+−

−⎥⎦⎤

⎢⎣⎡ −+−

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=+==

vv

Approach 2: Recognize that 1. the velocity field ),( txV vv

gives the velocity at any point in space at any time t , and 2. the point/location of the particle at any time t is given by the pathline.

Hence, the velocity of the particle )(tVv

is the velocity at the point at which the particle

currently occupies, i.e., )(trx vv= . Thus,


14

( ) ( )[ ]

( )jeCyiAteBx

jtyitxtrxjCyiAtBx

txVtV

ooo ttC

o

ttAttB

o

trx

trx

ˆˆ)1(

pathlineˆ)(ˆ)()(,ˆˆ)1(

),()(

)()(

2)(

)(

)(

22

−⎥⎦⎤

⎢⎣⎡ −+−

=

=

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=

−+==++=

=

vv

vvv

vv

vv

Thus, ( ) VelocityDimensionjeCyiexAtBtV ooo ttC

o

ttAttB

o =+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+= −⎥⎦

⎤⎢⎣⎡ −+−

,ˆˆ)1()( )()(

2)( 22v

ANS

4.5. Find the acceleration of the above particle at any time t .

The acceleration of the particle in 2.3 can be found as follows.

( ) [ ]

( ) [ ] onAcceleratiDimensionjeyCiexAtBAB

jeyCieAtBAtAeBx

dttVdta

ooo

ooooo

ttCo

ttAttB

o

ttCo

ttAttBttAttB

o

=+⎥⎥

⎦

⎤

⎢⎢

⎣

⎡++=

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+++=

=

−⎥⎦⎤

⎢⎣⎡ −+−

−⎥⎦⎤

⎢⎣⎡ −+−⎥⎦

⎤⎢⎣⎡ −+−

,ˆˆ)1(

ˆˆ1)1()(

)()(

)(2)(

2)(

2

)(2)(

2)()(

2)(

22

2222

vv

ANS Streamline 4.6. Find the streamline that passes through the point (xo , yo ) at time to .

The streamline can be found as follows.

essDimensionlxx

yy

yy

Cxx

AtB

tCydy

AtBxdx

Vdy

Vdx

AtBC

oo

oo

y

y

x

x

yx

oo

)1(

ln1ln)1(

1

timefix,)1(

+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=+

=+

=

∫∫

ANS Example of a pathline and streamlines that pass through point (1, 1) for this velocity field is shown below.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

x (m)

y (m

)

Series1Series2Series3

Pathline

Streamline at t = 0 sStreamline at t = 1 s

Pathline

Streamline at t = 0 s

Streamline at t = 1 s


15

Problem 5. Newton’s Viscosity Law and Shear Stress [Çengel and Cimbala, 2010, Problem 2-75, p. 66.] A 50-cm× 30-cm× 20-cm block weighing 150 N is to be moved at a constant velocity of 0.8 m/s on an inclined

surface with a friction coefficient of 0.27. a) Determine the force F that needs to be applied in the horizontal direction. b) If a 0.4-mm thick oil film with a dynamic viscosity of 0.012 Pa-s is applied between the block and inclined

surface, determine the percent reduction in the required force.

Solution a)

System: Closed system (material volume MV). We define our system as the solid block only. The system is

shown by the free-body-diagram (FBD) above. Motion: Block moves with constant velocity, no acceleration. Governing Equation: 0

vv=∑F , Nf kμ=

∑ = 0xF : 0sincos =−− θθ WfF (A)

∑ = 0yF : 0cossin =+−− NWF θθ (B)

Nf kμ= (C) Equations (A)-(C) constitute a system of 3 equations in 3 unknonws: F , f , and N .

(B): θθ cossin WFN += (1) (C) → (A): 0sincos =−− θμθ WNF k (2)

(1) → (2):

NN

WF

WFWWFF

oo

ook

k

kk

k

5.105150)20sin27.020(cos)20cos27.020(sin

)sin(cos)cos(sin

)cos(sin)sin(cos0sin)cossin(cos

=××−

×+=

−+

=

+=−=−+−

θμθθμθ

θμθθμθθθθμθ

ANS

Body force W = 150 N

Surface (concentrated point) force F

V = 0.8 m/s

Surface force f , 27.0=kμ

Surface force N

x

y

θ

L = 50 cm

H = 30 cm

W = 150 N

F

V = 0.8 m/s

θ = 20o


16

(b)

Assumptions 1. Newtonian fluid 2. Velocity filed is steady. 3. Linear velocity distribution over the small gap. 4. μ = constant

System: Oil film

Newton’s viscosity law: (Below, we take Eq. (D) as a tensor equation, taking the signs of components into account.)

Pam

smsPa

hV

hV

dydu

erfaceoilblockyx

24104.0

/8.0012.0

(D)0

3

int

+=×

×⋅=

=−

==

−

−μμμτ

Fluid frictional force (Below, we take Eq. (E) as a tensor equation, taking the signs of components into account.)

NPamWL

Af

yx

yxy

4.2)24)(2.05.0()(

(E)2 +=+×+=+=

=′

τ

τ

Hence, on the oil film f ′ acts in the positive x direction. System: Solid Block

Due to Newton’s third law, f ′ acts on the block in the negative x direction. Since Eq. (A) – with f replaced by f ′ , and F by F ′ - is still applicable, we have

(A): 0sincos =−′−′ θθ WfF

→ NNNWfF o

o2.57

20cos20sin1504.2

cossin

=×+

=+′

=′θ

θ

Thus, the percent reduction in the required force is

%8.45%1005.105

2.575.105%100 =×−

=×′−

=N

NNF

FF ANS

NOTE: The required force is reduced considerably by using oil film.

Body force W = 150 N

Surface (concentrated point) force F ′

V = 0.8 m/s

Surface force f ′

Surface force N ′

x

y

θ

System is solid block

V

Oil film of thickness h

h Adv

System is oil film

yxτ

Surface force f ′

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2141-365 - 2010 - HW1 (Solution) - Fundamental Concepts, Pathlines and Streamlines

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