Basic Fluid Mechanics by Prof. V.Sundar BASIC FLUID MECHANICS Prof. V. Sundar, Ocean Engineering Centre, IIT Madras 1.1GENERAL Matter exists in three states namely solid, liquid and gas. Liquid and gas are commonly called fluids. Fluids can be defined as a substance, which undergoes continuous deformation under the action of shear forces regardless of their magnitude. The main distinction between a liquid and a gas lies in their rate of change in the density. The density of gas changes more readily than that of liquid. However, they can be treated in the same way without taking into account the change of density, provided that the speed of flow is low as compared with the speed of sound propagation in the fluid. The fluid is called incompressible if the change of the density is negligible. Ideal Fluids Fluids with no viscosity, no surface Tension and are incompressible are called as ideal fluids. That is, for ideal fluids no resistance is encountered as the fluid moves. Ideal fluids do not exist in nature. They are imaginary fluids. Fluids, which have, low viscosity such as air, water may however be treated as ideal fluid without much error. Practical or Real Fluids Exist in nature. Has Viscosity, surface tension and are compressible. 1.2 TYPES OF FLOW: 1.Steady and Unsteady flow 2.Uniform and Non-uniform 3.Rotational and Irrotational 4.Laminar and Turbulent. Steady flow:Fluid characteristics such as Velocity, u, pressure, p, density, ρ, temp, T, etc., at any point do not change with time. i.e. At(x,y,z), . 0 t , 0 t p , 0 t u = ∂ ρ ∂ = ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ Unsteady Flow:Fluid characteristics do change with time i.e. At(x,y,z), 0 t T , 0 t , 0 t p , 0 t u ≠ ∂ ∂ ≠ ∂ ρ ∂ ≠ ∂ ∂ ≠ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ Most of the practical problems of engineering involve only steady flow conditions and is simpler to solve than problems of unsteady flow. 1
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Prof. V. Sundar, Ocean Engineering Centre, IIT Madras
1.1 GENERAL
Matter exists in three states namely solid, liquid and gas. Liquid and gas are commonly
called fluids. Fluids can be defined as a substance, which undergoes continuous
deformation under the action of shear forces regardless of their magnitude. The maindistinction between a liquid and a gas lies in their rate of change in the density. The
density of gas changes more readily than that of liquid. However, they can be treated in
the same way without taking into account the change of density, provided that the speedof flow is low as compared with the speed of sound propagation in the fluid. The fluid is
called incompressible if the change of the density is negligible.
Ideal Fluids
Fluids with no viscosity, no surface Tension and are incompressible are called as ideal
fluids. That is, for ideal fluids no resistance is encountered as the fluid moves. Ideal fluidsdo not exist in nature. They are imaginary fluids. Fluids, which have, low viscosity such
as air, water may however be treated as ideal fluid without much error.
Practical or Real Fluids
Exist in nature. Has Viscosity, surface tension and are compressible.
1.2 TYPES OF FLOW:
1.
Steady and Unsteady flow
2. Uniform and Non-uniform
3. Rotational and Irrotational4. Laminar and Turbulent.
Steady flow: Fluid characteristics such as Velocity, u, pressure, p, density, ρ, temp, T,
etc., at any point do not change with time.
i.e. At(x,y,z), .0t
,0t
p,0
t
u=
∂ρ∂
=∂∂
=⎟ ⎠
⎞⎜⎝
⎛ ∂∂
Unsteady Flow: Fluid characteristics do change with time
i.e. At(x,y,z), 0t
T,0
t,0
t
p,0
t
u≠
∂∂
≠∂ρ∂
≠∂∂
≠⎟ ⎠
⎞⎜⎝
⎛ ∂∂
Most of the practical problems of engineering involve only steady flow conditions and is
When fluid properties does not change both in magnitude and direction, from point to
point in the fluid at any given instant of time, the flow is said to be uniform.
0
1tts
u=
=⎟ ⎠
⎞⎜⎝
⎛ ∂∂
Velocity with respect to distance
Eg. Flow of liquids under pressure through long pipelines of constant diameter is uniformflow.
Non-Uniform Flow:
If velocity of fluid changes from point to point at any instant, the flow is non-uniformEg. Flow of liquid under pressure through long pipelines of varying diameter.
Steady (or unsteady) and uniform (or nonuniform) flow can exist independently of eachother, so that any of four combinations is possible. Thus the flow of liquid at a constant
rate in a long straight pipe of constant diameter is steady uniform flow, the flow of liquid
at a constant rate through a conical pipe is steady nonuniform flow, while at a changingrate of flow these cases become unsteady uniform and unsteady nonuniform flow
respectively.
Rotational Flow:
A flow is said to be rotational if the fluid particles while moving in the direction of flow
rotate, about their mass centres.Eg. Liquid in a rotating tank.
Irrotational Flow:
The fluid particles while moving in the direction of flow do not rotate about their mass
centres. This type of flow exists only in the case of Ideal Fluid for which no tangential or
Shear Stresses occur. But the flow of Real fluids may also be assumed to be irrotational ifthe viscosity of the fluid has little significance. For a fluid flow to be irrotational the
following conditions are to be satisfied.
It can be proved that the rotation components about the axes parallel to x and y axes can
and if at every point in the flowing fluid the rotation components W x, Wy, and Wz areequal to zero then
y
u
x
v;0W
x
w
z
u;0W
z
v
y
w;0W
z
yx
∂∂
=∂∂
=
∂∂
=∂∂
=
∂
∂=
∂
∂=
(1.1)
where u, v and w are velocities in the x, y, and z directions respectively.
Laminar Flow:
A flow is said to be laminar if the fluid particles move along straight parallel paths in
layers, such that the path of the individual fluid particles do not cross those of theneighbouring particles. In other words, the fluids appear to move by the sliding oflaminations of infinitesimal thickness relative to adjacent layers. This type of flow occurs
when the viscous forces dominate the inertia forces at low velocities. Laminar flow canoccur in flow through pipes, open channels, and through porous media.
Turbulent Flow:
A fluid motion is said to be turbulent when the fluid particles move in an entirely randomor disorderly manner, that results in a rapid and continuous mixing of the fluid leading to
momentum transfer as flow occurs. A distinguishing characteristic of turbulence is its
irregularity, there being no definite frequency, as in wave motion, and no observable
pattern, as in the case of large eddies. Eddies or Vortices of different sizes and shapes are present moving over large distances in such a fluid flow. Flow in natural streams,
artificial channels, Sewers, etc. are a few examples of turbulent flow.
Therefore, the net mass of fluid that has remained in the fluid medium per unit time
through the pair of faces ABCD and A’B’C’D’ is obtained as = ( ) .xz.y.u.x
∆∆∆ρ∂∂−
= ( ) z.y.x.ux
∆∆∆ρ∂∂−
The area has been taken out of the parenthesis since it is not a function of x.( z.y∆∆ )By applying the same procedure the net mass of fluid that remains in the cube per unit
time through the other two pairs of faces of the cube may also be obtained as
( ) ( z.y.x.vy
∆∆∆ρ∂∂−
= ) through pair of faces A A’D’D and B B’C’C
( ) ( z.y.x.wZ
∆∆∆ρ∂ ∂−= )
( )
through pair of faces D D’C’C and A A’ B’B.
By adding all these expressions the net total mass of fluid that has remained in the cube
per unit time is
( ) ( )z.y.x
z
w
y
v
x
u∆∆∆⎥
⎦
⎤⎢⎣
⎡∂ρ∂
+∂ρ∂
+∂ρ∂
− (1.2)
Since the fluid is neither created nor destroyed in the cube, any increase in the mass of
the fluid contained in this space per unit time, is equal to the net total mass of fluid,
that has remained in the cube per unit time, which is expressed by the aboveexpression.
( )z.y.x ∆∆∆ρThe mass of the fluid in the cube is and its rate of increase with time is
( ) ( )z.y.x.t
z.y.x.t
∆∆∆∂ρ∂
=∆∆∆ρ∂∂
(1.3)
Equating the two expressions Eq. (1.2) = Eq.(1.3)
( ) ( ) ( ) ( )z.y.x.t
z.y.xzw
yv
xu ∆∆∆
∂ρ∂=∆∆∆⎥
⎦⎤⎢
⎣⎡
∂ρ∂+
∂ρ∂+
∂ρ∂−
Dividing both sides of above expression by the volume of the parallelopiped ( )z.y.x ∆∆∆
And taking the limit so that the fluid medium shrinks to the point P(x,y,z) the continuity
This equation represents the continuity eqn. In its most general form and is applicable forsteady as well as unsteady flow, Uniform and Non-Uniform flow and compressible as
well as incompressible fluids.
For Steady flow 0t=
∂ρ∂
i.e, the above eq. Becomes
( ) ( ) ( ).0
z
w
y
v
x
u=
∂ρ∂
+∂ρ∂
+∂ρ∂
Further for an incompressible fluid the mass density ‘ρ’ does not change with x,y,z and
t, hence, the above equation simplifies to
0z
w
y
v
x
u=
∂∂
+∂∂
+∂∂
(1.5)
1.4 FORCES ACTING ON FLUIDS IN MOTION
The different forces influencing the fluid motion are due to gravity, pressure, viscosity,turbulence, surface tension and compressibility and are listed below.
Fg (Gravity Force) Due to Wt. of Fluid
= (Mass * gravitational constant)
F p (Pressure Force) Due to pressure gradient
Fv (Viscous Force) Due to Viscosity
Ft (Turbulent Force) Due to Turbulence
Fs (Surface Tension Force) Due to Surface Tension
Fe (Compressibility Force) Due to elastic property of the fluid
If a certain mass of fluid in motion is influenced by all the above mentioned forces then
according to Newton’s Second law of motion the following equation of motion may bewritten
Where M is the mass is of fluid and ax , ay , az are fluid acceleration in the x, y and z
directions respectively.In most fluid problems Fe and Fs may be neglected
Hence Ma = Fg + F p +Fv + Ft (1.7)
Eq.(1.7) is known as Reynold’s Equation of Motion
For laminar Flows Ft is negligible
Hence Ma = Fg + F p + Fv (1.8)
Eq.(1.8) known as Navier Stokes equation
In case of Ideal Fluids Fv = o
Hence Ma = Fg + F p (1.9)
Eq.(1.9) is known as the Euler’s Equation of Motion
1.5 EULER’S EQUATION OF MOTION
Only pressure forces and the fluid weight or in general, the body force, are assumed to be
acting on the mass of fluid in motion. Consider a point P(x,y,z) in a flowing mass of fluid
at which let u, v and w be the velocity components in directions x,y and z respectively. ρ
mass density; p be pressure intensity. Let X, Y, and Z be the components of the body
force per unit mass at the same point.
Mass of fluid in the fluid medium considered in Fig. 1.2 is ( )z.y.x ∆∆∆ρ . Therefore, total
component of the body force acting on the cube in x direction = ( )z.y.xX ∆∆∆ρ . Similarly
in y and z direction are ( )z.y.xY ∆∆∆ρ and ( )z.y.xZ ∆∆∆ρ ’p’ is pressure intensity at
point ‘P’. Since the lengths of the edges of the fluid medium are extremely small, it may be assumed that the ‘p’ on the face PQR’S is uniform and equal to p.
In these derivations no assumptions has been made that ‘ρ ’ is a constant. Hence, these
equations are applicable to compressible or incompressible, nonviscous fluid in steady or
unsteady state of flow.
1.6 PATHLINES AND STREAMLINES:
A Pathlines is the trace made by a single particle over a period of time. The pathlineshows the direction of the velocity of the fluid particle at successive instants of time.
Streamlines show the mean direction of a number of particles at the same instant of time.
If a camera were to take a short time exposure of a flow in which there were a large
number of particles, each particle would trace a short path, which would indicate its
velocity during that brief interval. A series of curves drawn tangent to the means of thevelocity vectors are Streamlines. Pathlines and Streamlines are identical in the Steady
flow of a fluid in which there are no fluctuating velocity components, in other words, for
truly steady flow. The equation of a Streamline is represented as
dz
w
dy
v
dx
u== (1.12)
1.7 VELOCITY POTENTIAL :
Velocity Potential, is defined as a scalar function of space and time such that its
derivative with respect to any direction yields velocity in that direction. Hence, for anydirection S, in which the velocity is V
φ
s
sVs=
∂φ∂
yv,
x ∂φ∂
=∂φ∂
u =
when substituted in continuity eq. (1.5) we get
∇ 02 =φ
1.8 STREAM FUNCTION:
Stream function, ψ is defined as a scalar function of space and time, such that its partial
derivative with respect to any direction gives the velocity component at right angles (inthe counter clockwise direction) to this direction.
1. Daugherty, R.L., Franzini, J.B. and Finnemore, E.J. “Fluid Mechanics with
Engineering Applications”, McGraw Hill, Inc. 1985.
2.
Garde, R.J. “Fluid Mechanics through Problems”, Wiley Eastern Limited, 1989.
3. Modi, P.N. and Seth, S.M. “Hydraulics and Fluid Mechanics”, published by StandardBook House, 1985.
Problem1.
The rate at which water flows through a horizontal 25cm pipe is increased linearly from30to150 liters/sec in 3.5secs. What pressure gradient must exist to produce this
acceleration? What difference in pressure intensity will prevail between sections 8m
apart? Take ρ = 102m slug/m3 (1000 kg/m
3 in SI unit).
Solution
Euler’s Equation along the pipe axis may be written as
x
uu
t
u
x
p1X
∂∂
+∂∂
=∂∂
ρ−
since the pipe has a constant diameter 0x
u=
∂∂
and since it is horizontal, the body force per unit volume, X along the flow direction is
also zero. The above equation of motion, therefore, reduces to
x.
1
t
u
∂ρ∂
ρ−=
∂∂
The changes in velocity as the flow changes from 30 to 150 liters/sec in
= -71.25 Kg/m2 /m in MKS= -1000 * 0.698 in SI= -698 N/m
2/m
Difference in pressure between sections 8m apart
= 8*x∂ρ∂
= -71.25*8
= -570.0 Kg/m2 (or)
= -698* 8N/m2
= -5.58 KN/m2
Problem2.
The wind velocity in a cyclone may be assumed to vary according to free Vortex law. Ifthe velocity is 16 Km/ hr, 50 km from the centre of the cyclone, what pressure gradient
should obtain at this point? What reduction in barometric pressure should occur over a
radial distance of 10 km from this point towards the centre of the storm? Take massdensity of air as 1.208 Kg (mass ) per m
3.
Solution
In a cyclone the wind velocity is given to vary according to the free Vortex law. Thevelocity distribution in a free Vortex is given by
A conical tube is fixed vertically with its smaller end upwards. The velocity of flow down
the tube is 4.5 m/sec at the upper end and 1.5 m/sec at the lower end. The tube is 1.5mlong and the pressure head at the upper end is 3.1 m of the liquid. The loss in the tube
expressed as a head is 0.3g2
2)VV( 21− where V1 and V2 are the velocities at the upper
and lower ends respectively. What is the pressure head at the lower end?
Solution
Applying Bernoulli’s equation between the upper and the lower ends,