Problems in Fluid Mechanics Baranyi

Tutorial Problems in FLUID MECHANICS

with answers and specimen solutions

L. Baranyi

TEMPUS-JEP 1501 Project 9

University of Miskolc 1994

CONTENTS Preface 1 2 3 4 5 6 7

FLUID STATICS, RELATIVE EQUILIBRIUM OF MOVING FLUIDS .............. 1 KINEMATICS OF FLUIDS ......................................................................... 14 APPLICATION OF BERNOULLI'S EQUATION ... ... ..... .... .... .............. ... ...... 17 MOMENTUM EQUATION ............................................ ....... ...................... 24 PLANE POTENTIAL FLOW ...................................................................... 30 LAMINAR FLOW.. .... ............................. ................. ..... .. ........................... 34 TURBULENT FLOW. ... .................................................................. .... ....... 41

SOLUTIONS TO SOLUTIONS TO SOLUTIONS TO SOLUTIONS TO SOLUTIONS TO SOLUTIONS TO SOLUTIONS TO

CHAPTER 1 ........................... ............................................. 45 CHAPTER 2 ........................................................................ 62 CHAPTER 3 ........................................................................ 69 CHAPTER 4 .............. .... ........ ........................ .. .................... 75 CHAPTER 5 ........................................................................ 86 CHAPTER 6 .... ................ ......................... ..... ... ................... 99 CHAPTER 7 ........ ............ .. .. ..... ... ........................................ 11 4

List of References ..................................................... ... ........... .... ......... ... ...... 123 Appendixes A Notation ..... ... ...... ............. .... ........... .... .. ........................... .. ......... .. ... ..... . 124 B Moody diagram ......................................... ..................... ..... .... .... .......... 127

ii

Preface These tutorial problems are intended primarily for foreign students who study the subject FLUID MECHANICS in English medium at the Faculty of Mechanical Engineering, University of Miskolc. It is hoped, however, that it may also be of some service to other engineering students. The material is reasonably self-contained together with the unpublished lecture notes of the author but references are also given for further reading. The choice of notations largely follows the recommendations of the International Standards Organisation, and the international system of units (SI) is used throughout the material. The number of problems involved is relatively low, but more or less detailed specimen solutions are given to all problems. The author encourages students to try and solve a lot of problems by their own. I would like to emphasize physical understanding to make students aware of the variety of phenomena that occur in real fluid flow situations. The study aid contains seven chapters for wording the problems and further seven for the answers and specimen solutions. A list of symbols used and a copy of the Moody diagram are also included. The acceleration due to gravity should be taken g = 9.81 m/s 2 and the density of water is to be chosen to be 1OOO kg/m 3 if it is not stated otherwise. Figures are not to scale in this study aid. The author gratefully acknowledges the financial support of the TEMPUS JEP-1501. Laszlo Baranyi

1

1 FLUID STATICS, RELATIVE EQUILIBRIUM OF MOVING FLUIDS 1 .1 Determine the gravity force W that can be sustained by the force F acting on the piston of the Figure. 0= 240 mm diam

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1.6 The ratio of cistern diameter to tube diameter is 10. When the air in the tank is at atmospheric pressure, the free surface in the tube is at position 1 . When the cistern is pressurized, the liquid in the tube moves 20 mm Y= 200 mm} up the tube from position 1 to position 2. What is the cistern pressure that causes this deflection? The density of the liquid is p = 800 kg/mJ .

3

Cistern

1.7 Find the gauge difference N1 11 in the U-tube if the water level in the container is raised by ,1/-1 = 1.5 m (see the Figure).

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Water

=_-_-=-=-=-=--=--=-= -= -=-=- =- =-=- lqw=-1000 kg/~) _-_ - _-__- _-_- _- _- ~w-

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c.C> If)

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Mercury 3 {qm= 13600 kg/m ) t.h

11

A

A'

1.8 An aquarium at Marineland has a window as shown. Find: (a) The resultant force from seawater (p = 1015 kg/m 3) on the window, F (b) The line of action of the resultant force, in metre below the water surface, k.

4

d

y

a :0.6 m b = 1.Sm

c:0 .3m d ,,z.4m

1.9 Locate the pressure centre for the gate in the Figure. The gate is 1.3 m wide. (w

= 1.3 m)

r

1·

w

·1 l

h=2m

h

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le

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y

1.10 Determine the moment M at A required to hold the gate. (the gate is 1.2 m wide; w = 1.2 m)

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1.11 Find the resultant force due to water on both sides of the gate including its line of action.

H =-3m -GaTe_ -

~...:::.

-_-_- 9 w =-1. 3 m - (Gatewict!_h) ::_

h =2 m

6 1.12 Calculate the force exerted by water on one side of the vertical annular area shown in the Figure.

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Water

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(~

3

J

.. 10 kg /m)

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H= 2m

1.13

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Water (9 = 999 kg /m) _ -

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H =2m

7

Given: rectangular gate, hinged along A, width w = 5 m. Find: resultant force, F, of the water on the gate, and its line of action.

1.14

W=2m

- Water -



(1)

Writing equation (1) in componential form, and integrating, yields 4>=V0

x3 y2 z2 ) -+-+ - +z +const ( 3

2

2

2.8 dx dy -=-

(a)

x2

y2

a2

b2

- + - = 1·

Normally these are ellipses (a :1; where

a2

= 2ac ·,

They are circles if

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2 b2 = c f3 a

=f3

v = -