Fluid kinematics

MODULE -2
Fluid Kinematics and Dynamics:
Fluid Kinematics: Types of Flow-steady, unsteady, uniform, non-uniform, laminar, turbulent,
one, two and three dimensional, compressible, incompressible, rotational, irrotational,
stream lines, path lines, streak lines, velocity components, convective and local
acceleration, velocity potential, stream function, continuity equation in Cartesian co-
ordinates. Rotation, vorticity and circulation, Laplace equation in velocity potential and
Poisson equation in stream function, flow net, Problems.
Fluid Dynamics:
Momentum equation, Impacts of jets- force on fixed and moving vanes, flat and curved.
Numericals. Euler’s equation, Integration of Euler’s equation to obtain Bernoulli’s equation,
Bernoulli’s theorem, Application of Bernoulli’s theorem such as venture meter, orifice
meter, rectangular and triangular notch, pitot tube, orifices etc., related numericals.
12 Hours

Deals with motion of fluid particles without considering the
forces causing the motion
Focus on – velocity and acceleration of fluid flow

Methods of describing Fluid motion
 Lagrangian method
 Eulerian method
Lagrangian Method
 Single fluid particle is followed
 The path followed by the particle and its changes can be focused
X = f(x.y,z,t) y = f(x,y,z,t) Z = f(x,y,z,t)
Then,
u =dx/dt; v =dy/dt; w = dz/dt
Resultant velocity, V = sq.root( u2+v2+w2)
ax = d2x/dt2; ay = dy2/dt2; az = dz2/dt2
Resultant acceleration, a = sq.root(ax2+ay2+az2)

Eulerian Method
 At a point in a flow system is followed
 Fluid flow properties at that point is studied
 State of motion at various points in the fluid system is focused
u = f1(x,y,z,t); v =f2(x,y,z,t); w = f3(x,y,z,t)
du = du/dx .dx+dv/dy .dy+dw/dz .dz+ du/dt .dt
ax = du/dt = (du/dx .dx/dt + du/dy .dy/dt + du/dz ) + du/dt.dt/dt
= (u.du/dx + v.du/dy + w.du/dz) + du/dt

Types of fluid flows
 Steady and unsteady flows
 Uniform and non-uniform flows
 Laminar and turbulent flows
 One, two, three dimensional flows
 Compressible and incompressible flows
 Rotational and irrotational flows

Steady Flow:
The flow in which the fluid characteristics like velocity, pressure,
density etc. at a point do not change with time
(du/dt) = 0; (dv/dt) = 0 ; (dw/dt) = 0; (dp/dt) = 0;
(d/dt) = 0 ( At Some Point x0, y0, z0)
e.g. : Flow through a conduit with constant flow rate,
u = ax2+bx+c
Unsteady Flow:
The flow in which the fluid characteristics like velocity, pressure,
density etc. at a point changes with time
(du/dt)  0; (dv/dt)  0 ; (dw/dt)  0;
(dp/dt)  0; (d/dt)  0 ( at some point x0, y0, z0)
e.g.: Flow through a pipe whose valve is gradually opened or closed
u = ax2+bxt
-------Change w.r.t time is very important------

Uniform Flow:
The flow in which velocity at any given time does not change with
space
(dV/ds) = 0; ( t = constant)
e.g. : Flow through a straight conduit with constant constant diameter
Non-uniform Flow:
The flow in which velocity at any given time changes with space
(dV/ds)  0; ( t = constant)
e.g.: Flow through a non prismatic conduit
-------Change w.r.t space is very important------

Laminar Flow: ( Viscous Flow) or (Stream line Flow)
The flow in which the fluid particles move along a well defined
paths. i.e. each layers are straight and parallel
e.g. : flow through capillary tube, flow of blood in veins
Turbulent Flow:
The flow in which the fluid particles move in zig-zag way
i.e. each layers are not straight and not parallel
e.g.: High velocity flow in large conduit
Reynold’s Number – non-dimensional number
Re  2000 ----laminar
Re  4000---- turbulent

1D Flow:
The flow in which the flow parameter, velocity is a function of time
and one space co-ordinate only.
u= f(x,t) here v =0 and w =0
2D Flow:
The flow in which the flow parameter, velocity is a function of time
and two space co-ordinates.
u = f1(x,y,t) and v =f2(x,y,t) but w = 0
e.g.: flow between parallel plates
3D Flow:
The flow in which the flow parameter, velocity is a function of
time and all the three space co-ordinates.
u = f1(x,y,z,t) and v =f2(x,y,z,t) w = f3(x,y,z,t)
e.g.: flow in a converging or diverging pipe

Compressible Flow:
The flow in which the density changes from point to point
i.e.   0
e.g. : flow of gases through orifices, nozzles gas turbines
Incompressible Flow:
The flow in which the density do not changes from point to point
i.e.  = 0
e.g.: liquids are generally considered flowing incompressibly

Rotational Flow:
The flow in which the fluid particles rotate about their own axis while
flowing
e.g. : motion of liquid in a rotating tank
flow near the solid boundaries is rotational
Irrotational Flow:
The flow in which the fluid particles do not rotate about their own
axis while flowing
e.g.: flow above the drain hole of a stationary tank or wash basin
Flow outside the boundary layer

Types flow lines
 Stream lines
 Path lines
 Streak lines

Stream Lines
It is an imaginary line within the flow such that the tangent at any point on
it indicates the velocity at that point
Equation of stream line 3D
dx/u = dy/v = dz/w
 A stream line can not intersect or two stream lines never crosses
 Stream line spacing varies inversely as the velocity
 converging of stream lines in any direction shows accelerated
flow
 Stream line indicates the direction of number of particles at a
time
 stream line represent the flow pattern

Path Line
 It is the path followed by a fluid particle in motion
 it shows the direction of motion of flow
Streak Lines
It is curve that gives instantaneous picture of the location of the fluid
particles which have passed through the given point

Velocity Components and Acceleration  
Lagrangian Analysis : ( single fluid particle )
If the position of fluid particle in a space are given at time t
x = f1 (a, b, c, t )
y = f2 ( a, b, c, t )
z = f3 (a, b, c, t )
then velocity components can be found
u =  x/  t ; v =  y/  t ; w =  z/  t ;
and the resultant velocity is given by
V =  u2+v2+w2

Similarly the acceleration components are found
ax = 2x/  t2 ; ay = 2y/  t2 ; az = 2z/  t2
and resultant acceleration is given by
a =  ax2 +ay2 + az2

Eulerian Analysis : ( at a point )
If the velocity components at a point are given
u = f1 ( x, y, z, t )
v = f2 ( x, y, z, t )
w = f3 ( x, y, z, t ) ----- these are the velocity components
Resultant velocity
V = u2 +v2 +w2
Then the increment in these components in their respective directions are
du = u/ x .dx + u/ y .dy + u/ z .dz + u/ t.dt
dv = v/ x .dx + v/ y .dy + v/ z .dz + v/ t .dt
dw = w/ x .dx + w/ y .dy + w/ z .dz + w/ t.dt

The acceleration components are obtained as follows
ax =du/dt = u/ x.dx/dt +u/ y.dy/dt +u/ z.dz/dt +u/ t.dt/dt
= (u/ x.u +u/ y.v +u/ z.w) +u/ t.
= (u.u/ x + v.u/ y + w.u/ z) + u/ t
Similarly in other two directions
ay = (u.v/ x + v.v/ y + w.v/ z) + v/ t
az = (u.w/ x + v.w/ y + w.w/ z) + w/ t
Resultant acceleration is
a =  ax2 + ay2 + az2

 
ay = (u.v/ x + v.v/ y + w.v/ z) + v/ t
Convective acceleration Local acceleration
Local acceleration– rate of increase of velocity with respect to time
at a given point in a fluid flow
dV/dt, u/ t, v/ t , w/ t
Convective acceleration – rate of change of velocity due to change of
position of fluid particles in a flow
V.dV/ds

Problem 1
The velocity vector in a fluid flow is given by
V = 4x3i – 10x2yj + 2tk
find the velocity and acceleration of fluid particle at (2,1,3) at time
t = 1
u = 32 units
v = -40 units
w =2 units
V = 51.26 units
ax = 1536 units
ay = 320 units
az = 2 units
a = 1568.9 units

Problem 2
Find the velocity and acceleration for the velocity field
V = x2yi + y2zj – (2xyz+yz2)k, at (2,1,3)
Ans: V =21.587 units a = 108.71 units
Problem 3
For the velocity field V = (3x+2y)i + (2z+3x2)j + (2t-3z)k
Find the i) speed at (1,1,1)
ii) speed at (0,0,2) and t = 2s
Also state type of flow
Ans: V1 = 4t2 -12t +59 V2 = 4.472
unsteady, non uniform and 3D

Problem 4
find the velocity and acceleration at a point (1,2,3) after 1 sec for a three
dimensional flow given by
u = yz+t, v = xz-t, w = xy, m/s
Ans: V = 7.55 m/s
a = 29.34 m/s2
Problem 5
the velocity along a central line of a nozzle of length ‘l’ is given by
V = 2t ( 1- x/2l)^2
V in m/s, t in sec and x in m. Calculate the convective , local and total
acceleration when t =6 sec, x =1 m, l = 1.6m
Ans: dV/dt = 0.945 m/s2
V.(dV/dx) = -29.24 m/s2
a = -28.295 m/s2

Rate of flow or discharge (Q)
 Continuity Equation
“ If no fluid is added or removed from the pipe in any length, then the
mass of fluid passing across different sections shall be same ”
1.A1.V1 = 2.A2.V2
For incompressible fluid flow
A1.V1 = A2.V2

Continuity Equation in Cartesian coordinates
dy
dz

Rate of mass of fluid entering the face ABCD= . u. dy.dz
Rate of mass of fluid leaving the face EFGH= .u.dy.dz+/x (. u.dy.dz).dx
The rate of gain of mass of fluid in X-direction
= . u. dy.dz – [ . u. dy.dz + /x (. u. dy.dz).dx]
= - /x (u) dx.dy.dz
Similarly in Y-direction, = - /y (v) dx.dy.dz
in Z-direction = - /z (w) dx.dy.dz
Net gain of fluid mass per unit time
= - [/x (u) + /y (v) + /z (w)]dx.dy.dz-----(1)
Rate of change of mass of control volume
= /t ( dx.dy.dz) ------(2)

(1) = (2)
- [/x (u) + /y (v) + /z (w)]dx.dy.dz = /t ( dx.dy.dz)
[/x (u) + /y (v) + /z (w)] + /t () = 0
Problem 1
Determine which of the following satisfy the continuity equation
i) u =A sinxy, v = -A sinxy
ii) u = x+y; v =x-y
iii) u = 2x2 +3y; v = -2xy +3y3 +3zy ; w = -(3/2)z2 -2xz -6yz

Problem 2
Calculate the unknown velocity component so that te continuity equation is
Satisfied
u =? ; v = Axy
Problem 3
Given that u =xy and v = 2yz Examine whetherthese velocity components
represent two, three dimensional incompressible flow, if three dimensional,
find the third component.
Problem 4
In an incompressible flow the velocity vector is given by
V = (6xt+yz2) i + (3t + xy2) j + (xy -2xyz -6tz) k
Verify whether the continuity equation satisfied or not
Also find the acceleration vector at (2,2,2) at t = 2

Circulation and Vorticity
Circulation ( )
It is defined as the line integral of the tangential velocity about the closed
path
 =  V cos

Circulation around the regular curves can be obtained by integration
d = u. x + (v + v/x . x).y – (u + u/y . y).x - v. y
= (v/x - u/y ). x y

Vorticity :
It is defined as the circulation per unit enclosed area
 =  / A
= d / x y
= (v/x - u/y )
Worthy points
 If a flow possesses vorticity then it is rotational
 Rotation is defined as one-half of the vorticity and rotation is denoted
by 
 If rotation  is zero, then the flow is irrotational
  = (1/2) (v/x - u/y )

 For 3D flow the rotation components along all the 3 directions are
z = (1/2) (v/x - u/y )
x = (1/2) (w/y - v/z )
y = (1/2) (w/y - v/z )
 Then the vorticity is given by twice the rotation
 The flow is said to be irrotational if the components of rotation or
vorticity are zero

Velocity potential ( )
It is a scalar function of space and time such that its
negative derivative w.r.t any direction gives the fluid velocity in that
direction
 = f (x, y, z, t ) ------for unsteady
 = f(x, y, z ) ------for steady
According to definition,
u = -( /x )
v = -( /y )
w = -( /z )
-ve sign indicates that the velocity potential always decreasing in a given
direction

continuity equation as applied to velocity potential
The continuity equation is
u /x + v /y + w /z = 0
/x (- /x ) + /y (- /y ) + /z (- /z ) = 0
2 /x2 + 2 /y2 + 2 /z2 = 0
This equation is known as “Laplace Equation”
If the velocity potential satisfies the Laplace equation, then it is a case of
possible steady, incompressible, irrotational flow.

Stream Function ( )
Is defined as the function of space and time such that its partial derivative
w.r.t. any direction gives the velocity component at right angles to this
direction
 = f(x, y, t ) -------- for unsteady flow
 = f( x, y ) ----------for steady flow
then, u =  / y
v = - / x

For two dimensional flow
u /x + v /y = 0
/x ( / y) + /y (- / x) = 0
2 /xy - 2 /yx = 0
The existence of stream function means a possible case of flow
For rotational or irritational,
z = (1/2) (v/x - u/y )
= - (1/2) (2 /x2 + 2 /y2)
This equation is known as Poisson’s equation

For the flow to be irrotational, z = 0
2 /x2 + 2 /y2 = 0
Cauchy Riemann Equations
u = - /x =  / y
v = - /y = - / x
• Velocity potential function exists only for irrotational flow
• Stream function applies to both rotational and irritational flows
• For irritational flow both stream function and velocity potential
function satisfy the Laplace equation

Equipotential Line
A line along which the velocity potential is constant
i.e.  = constant
therefore, d = 0
consider,  = f( x, y )------ for steady flow
d = /x.dx + /y.dy
= -u.dx- v.dy =0
Therefore, dy/dx = - u /v = slope of equipotential line

Line of constant stream function
 = constant
Therefore , d = 0
Consider,  = f ( x, y ) -------for steady flow
d = /x.dx + /y.dy
= -v.dx + u.dy =0
Therefore, dy/dx = v/u
The velocity potential and the stream lines are orthogonal

Problem 1
Verify whether the following given function  = A( x2 – y2 ) is a valid
potential function
Ans: Yes
Problem 2
The velocity components in fluid flow are u = 2xy, v = a2 + x2 – y2
Find the stream function.
Ans :  = xy2 – a2x – x3/3 + C

Problem 3
If u = 2cx and v = -2cy, find the stream function
Ans :  = 2cxy +c1
Problem 4
A two dimensional flow field is given by,  = 3xy determine
i) The stream function
ii) The velocity at L(2,6) and M(6,6)
iii) The pressure difference between the points L and M
iv) The discharge between the streamlines passing through L and M
Ans :  = 3/2 (x2-y2); V1= 18.97; V2 = 25.45
pL –pM = 14.68; Q = 48 =

Problem 5
For the stream function,  = 3x2y –y3, Determine the velocity at a point
(1,2)
Ans: V = 15 units
Problem 6
In a two dimensional incompressible flow the fluid velocity components
are given by u= x-4y and v= -y-4x. Show that the velocity potential
exists. And determine its form as well as stream function.
Ans : rotation or vorticity should be zero;
 = ½( y2 –x2 ) 8xy;  = 2( x2-y2) + 2xy

Flow Nets:
These are the gridlines obtained by drawing a series of stream lines and
equipotential lines
Flow nets provides information regarding two dimensional irrotational
flow where it is difficult to go by mathematical solution
Methods of drawing Flow nets
• Analytical Method
• Graphical method
• Hydraulic Models
• Electrical Analogy

Applications
• To determine the stream lines and equipotential lines
• To determine the quantity of seepage below the hydraulic structure
• To determine the design of the outlets for their streamlining
• To determine the velocity and pressure distribution for a given
boundary flow

Fluid kinematics

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