last updated on: 10-02-2020€¦ · dynamics of fluid flow: module-3 fluid kinematics and dynamics...
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Last Updated on:
Module-3
Fluid Kinematics and Dynamics
98795 10743
Fluid Mechanics & Hydraulics (3140611)
10-02-2020
Prof. Mehul Pujara
2Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Fluid:
• The term fluid refers to both aliquid and a gas; it is generallydefined as a state of matter inwhich the material flows freelyunder the action of a shearstress.
3Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
A branch of fluid mechanics whichdeals with the study of velocity andacceleration of the particles offluids in motion and theirdistribution in space withoutconsidering any forces or energyinvolved.
It provides
(i) an idea about the rate offlow
(II) an idea about differenttypes of velocities of flow
Fluid Kinematics???
• Equation of Continuity is the first and fundamental equation of flow.
4Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
1. Mass flow rate ( 𝑚)
2. Volume flow rate( 𝑄)
Rate of Flow:
5Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Volume Flow Rate:
6Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Mass Flow Rate:
7Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Control Volume Approach:
• A system is whatever the engineer selects for study.• Open System (Control Volume system)• Closed System (Control Mass system)
• The closed system involves selection and analysisof a specific collection of matter, the closedsystem is a Lagrangian concept.
• The control volume involves selection and analysisof a region in space, the CV is an Eulerian concept.
8Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Observer concentrates on themovement of single particle.
Observer has to move with thefluid particle to observe itsmovement
The path and changes in velocity,acceleration, pressure and densityof a single particle are described.
Not commonly use.
Langrangian Concept:
P
PP
P
𝑢 =𝜕𝑥
𝜕𝑡𝑣 =
𝜕𝑦
𝜕𝑡𝑤 =
𝜕𝑧
𝜕𝑡
𝑎𝑥 =𝜕2𝑥
𝜕𝑡2𝑎𝑦 =
𝜕2𝑦
𝜕𝑡2𝑎𝑧 =
𝜕2𝑧
𝜕𝑡2
9Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Observer concentrates on various fixed point particles.
Observer remains steady and observe the change in the fluidparameters at the fixed point only
Method describes the overall flow characteristics at various points as fluid particle pass.
Commonly used.
Eulerian Concept:
u = f1 (x,y,z,t), v = f2(x,y,z,t), W = f3(x,y,z,t)
10Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
1. Path line
2. Stream line
3. Stream tube
4. Streak line or filament line
5. Potential line
Lines of flow to describe the motion of the fluid:
11Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The path traced by single fluid particle inmotion over a period of time is called itspath line.
The path line shows the direction ofvelocity of the particle as it moves ahead.
During steady flow, path line coincide withstream line.
However path line fluctuates betweendifferent stream lines during an unsteadyflow.
It is real line showing successive position ofone particle.
Particle may cross its path line.
Path line:
12Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
A stream line is imaginary line drawnthrough a flowing fluid such that the tangentat each point on the line indicates thedirection of the velocity of the fluid particleat that point.
It shows positions of various fluid particles.
Stream line can not intersect each other,they are always parallel.
No flow across stream line.
Stream line:
13Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Stream Tube:
A circular space formed by the collection ofstream lines passing through the perimeterof a closed curve in a steady flow.
Stream tube behaves as a solid surfacetube.
It may be of regular or irregular shape.
14Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Real line showing an instantaneouspicture of the position of all fluidparticles in the flow which havepassed from a given point.
The line of smoke from chimney isnothing but a streak line.
It may change instant to instant.
Flow across streak line is possible.
Streak line or Filament line:
15Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The lines of equal velocity potential are called potential line.
Stream line and potential line are at right angle to each other.
Potential Line:
16Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Steady or Unsteady
Uniform or Non-uniform
Laminar or Turbulent
Compressible or Incompressible
Rotational or Irrotational
One, Two or Three Dimensional
Types of fluid flow:
17Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Fluid characteristics like velocity,acceleration, pressure and densitydo not change with time at anypoint in the fluid is called steadyflow.
𝜕𝑣
𝜕𝑡=0,
𝜕𝑃
𝜕𝑡=0,
𝜕𝜌
𝜕𝑡=0 ,
𝜕2𝑣
𝜕𝑡2=0
Flow of water with constantdischarge through pipeline.
If fluid properties changes with timeits called unsteady flow.
Steady vs. Unsteady Flow:
18Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Velocity of the flow does not changealong its direction of flow at any point oftime is called Uniform flow.
Flow through a constant diameter pipeline is uniform flow.
𝜕𝑣
𝜕𝑠=0
Velocity of flow changes with its directionof flow any instant of time is called Non-uniform flow.
Uniform vs. Non-uniform flow:
19Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Fluid particles move in layers witheach layer sliding over the other, theflow is called laminar Flow.
Velocity of flow is very small.
Reynolds number<2100
The flow in which adjacent layerscross each other and the layers donot move along the well definedpath is called turbulent Flow.
Reynolds number>4000
Laminar vs. Turbulent Flow:
20Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The volume and hence density of thefluid does not remain constant duringthe flow is called compressible flow.
Gases are compressible.
The flow in which the changes involume and thereby density of the fluidare insignificant, the flow isincompressible.
Liquids are incompressible.
For gases in subsonic aerodynamiccondition air flow can be considerincompressible.
Compressible vs. Incompressible Flow:
21Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Ex.
Fluid particles rotate about their own axiswhile flowing along stream lines, is calledrotational flow.
Motion of a liquid in a rotating cylinder is arotational flow.
The flow in which fluid particles do notrotate about their own axis while flowingalong stream lines is called Irrotationalflow.
Flow of water in emptying wash basing isIrrotational flow.
Rotational vs. Irrotational Flow:
22Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
One, Two and Three Dimensional Flows:
Flow parameter such as velocity is afunction of time and one space co-ordinate only, say x is called OneDimensional flow.
Ex. u=f(x), v=0 and w=0
Flow parameter such as velocity is afunction of time and two space co-ordinate only, say x and y is called TwoDimensional flow.
Ex. u=f1(x,y), v=f2(x,y) and w=0
Flow parameter such as velocity is afunction of time and three space co-ordinate only, say x,y and z is calledThree Dimensional flow.
Ex. u=f1(x,y,z), v=f2(x,y,z) and w=f3(x,y,z)
Ex.
Ex.
Ex.
23Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Critical depth (yc) is defined as the depth of flow where energy is at aminimum for a particular discharge.
For a given value of specific energy, the critical depth gives the greatestdischarge, or conversely, for a given discharge, the specific energy is aminimum for the critical depth.
Critical Depth:
24Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Subcritical flow occurs when the actual water depth is greater than criticaldepth.
Subcritical flow is dominated by gravitational forces and behaves in a slowor stable way.
It is defined as having a Froude number less than one.
Supercritical flow is dominated by inertia forces and behaves as rapid orunstable flow.
When the actual depth is less than critical depth it is classified assupercritical.
Supercritical flow has a Froude number greater than one.
Critical flow is the transition or control flow that possesses the minimumpossible energy for that flow rate. Critical flow has a Froude number equalto one.
Sub Critical, Critical and Super Critical flow:
25Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
CONTINUITY EQUATION:
It follows “Principle of conservationof mass”
“Mass neither be created nor bedestroyed, total mass of systemremains constant.”
Mass offluidenteringunit time
Mass offluidmovingout perunit time
Rate ofincreaseof massper unittime
- =
26Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
CONTINUITY EQUATION:
Considering flow though the stream tube.
Mass of fluid entering into stream tube perunit time is
= Density * Discharge
= Density * Area * velocity
= 𝜌 𝐴 𝑣 (1)
Mass of fluid coming out from stream tubeper unit time is
= 𝜌 𝐴 𝑣 +𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠ds (2)
Net mass of fluid reaming in stream tubeper unit time is
= -𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠ds (3)
𝜌 𝐴 𝑣
𝜌 𝐴 𝑣 +𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠ds
27Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The mass of fluid in stream tube is
= density * volume of fluid in tube
= 𝜌 𝐴 ds (4)
Rate of increase of mass of fluid with timein stream tube is
=𝜕 (𝜌 𝐴𝑑𝑠 )
𝜕𝑡
=𝜕 (𝜌 𝐴 )
𝜕𝑡𝑑𝑠 (5)
Net mass remained in the stream tube per unittime is equal to rate of increase of mass withthe time.
Equation (3) = Equation (5)
-𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠𝑑𝑠 =
𝜕 (𝜌 𝐴 )
𝜕𝑡𝑑𝑠
CONTINUITY EQUATION:
-𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠= 𝜕 (𝜌 𝐴 )
𝜕𝑡
𝜕 (𝜌 𝐴 )
𝜕𝑡+ 𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠= 0 (6)
Equation (6) is known as continuityequation for one dimensional flow, it isapplicable to all cases of flow.
28Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
For Steady flow𝜕 (𝜌 𝐴 )
𝜕𝑡= 0
Therefore𝜕 (𝜌 𝐴 𝑣)
𝜕𝑠= 0
𝜌 𝐴 𝑣 = Constant
At section 1 and 2 we can write continuityequation
𝜌1 𝐴1 𝑣1 = 𝜌2 𝐴2 𝑣2
In case of incompressible flow
𝜌1 = 𝜌2
𝐴1 𝑣1 = 𝐴2 𝑣2
CONTINUITY EQUATION:
1
2
29Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Hydrodynamics
Aerodynamics
Electromagnetism
Quantum mechanics
Flow of fluid through pipe, ducts or tubes
Rivers
Process plants
Power plants
Dairies
Logistics in general
CONTINUITY EQUATION APPLICATION:
1
2
30Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Consider a fluid element of lengths dx, dy anddz in the direction of x, y and z.
Let inlet velocity components are u, v and w inthe direction of x, y and z respectively
Mass of fluid entering the face ABCD persecond is
= density * velocity in x-direction * area of ABCD
= ρ * u * (dy*dz)
Mass of fluid leaving the face EFGH per secondis
= ρu dydz +𝜕 (ρu dydz)
𝜕𝑥dx
CONTINUITY EQUATION IN THREE DIMENSTIONS:
31Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Net gain of mass in x direction is
= ρu dydz - [ ρu dydz +𝜕 (ρu dydz)
𝜕𝑥dx]
= -𝜕 (ρu dydz)
𝜕𝑥dx
= -𝜕 (ρu)𝜕𝑥
dx dy dz
Similarly net gain of mass in y direction is
= -𝜕 (ρv)𝜕𝑦
dx dy dz
Net gain of mass in z direction is
= -𝜕 (ρw)
𝜕𝑧dx dy dz
CONTINUITY EQUATION IN THREE DIMENSTIONS:
32Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Total Net gain of mass per unit time is
= - [𝜕 (ρu)𝜕𝑥
+𝜕 (ρv)𝜕𝑦
+𝜕 (ρw)
𝜕𝑧] dx dy dz
The net increase of mass per unit time in thefluid element must be equal to the rate ofincrease of mass of fluid in the element.
The mass of fluid is
= density * volume
= ρ * dx dy dz
The rate of increase of mass of fluid in theelement
=𝑑ρ
𝑑𝑡dx dy dz
CONTINUITY EQUATION IN THREE DIMENSTIONS:
33Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Equation two expression
- [𝜕 (ρu)𝜕𝑥
+𝜕 (ρv)𝜕𝑦
+𝜕 (ρw)
𝜕𝑧] dx dy dz =
𝑑ρ
𝑑𝑡dx dy dz
𝜕𝜌
𝜕𝑡+𝜕 (ρu)𝜕𝑥
+𝜕 (ρv)𝜕𝑦
+𝜕 (ρw)
𝜕𝑧= 0
The above equation is continuity equation incartesian coordinates in general form.
It is applicable to Steady and unsteady flow
Uniform and non-uniform flow
Compressible and incompressible flow
CONTINUITY EQUATION IN THREE DIMENSTIONS:
34Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
General Continuity equation is
𝜕𝜌
𝜕𝑡+𝜕 (ρu)𝜕𝑥
+𝜕 (ρv)𝜕𝑦
+𝜕 (ρw)
𝜕𝑧= 0
For Steady flow, the equation is
𝜕 (ρu)𝜕𝑥
+𝜕 (ρv)𝜕𝑥
+𝜕 (ρw)
𝜕𝑥= 0
For incompressible flow, the equation is𝜕u𝜕𝑥
+𝜕v𝜕𝑦
+𝜕w𝜕𝑧
= 0
For One dimensional flow, the equation is
𝜕u𝜕𝑥
= 0
CONTINUITY EQUATION IN THREE DIMENSTIONS:
35Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
It is the study of fluid motion with the forces causing flow.
The dynamic bhaviour of the fluid flow is analyzed by the Newton’s second law of motion.
F = m a
In fluid flow, the following forces are present:
Fg, gravity force
Fp, pressure force
Fv, viscous force
Ft, turbulent force
Fc, force due to compressibility
F = Fg + Fp + Fv + Ft + Fc
If Fc is neglected
F = Fg + Fp + Fv + Ft
Equation is known as Reynold’s Equation of motion.
DYNAMICS OF FLUID FLOW:
36Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
If Ft is neglected
F = Fg + Fp + Fv
Equation is known as Navier-Stokes Equation.
If is Fv neglected
F = Fg + Fp
Equation is known as Euler’s equation of motion.
DYNAMICS OF FLUID FLOW:
37Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The below equation which consider effect of gravitational and pressure force is known as Euler’s equation of motion.
F = Fg + Fp
Consider a stream-line in which flow is taking place.
Consider cylindrical element of cross section dA and length ds. The forces acting on it is: Pressure force pdA in direction of flow
pressure force (p + 𝜕𝑝
𝜕𝑠ds) dA opposite to
the direction of flow
Weight of element ρ g dA ds
EULER’S EQUATION OF MOTION:
38Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Let θ is the angle between the direction of flow and the line of action of the weight of element.
pdA - (p + 𝝏𝒑
𝝏𝒔ds) dA - ρ g dA ds cos θ = m * as
pdA - (p + 𝜕𝑝
𝜕𝑠ds) dA - ρ g dA ds cos θ = ρ dA ds * as
Now,
as = 𝒅𝒗
𝒅𝒕, where v is a function of s and t.
= 𝜕𝑣
𝜕𝑠
𝑑𝑠
𝑑𝑡+
𝜕𝑣
𝜕𝑡
= v 𝜕𝑣
𝜕𝑠+ 𝜕𝑣
𝜕𝑡
If Flow is steady then 𝜕𝑣
𝜕𝑡= 0
as = v 𝝏𝒗
𝝏𝒔
EULER’S EQUATION OF MOTION:
39Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Substituting value of as in equation,
-𝝏𝒑
𝝏𝒔ds dA - ρ g dA ds cos θ = ρ dA ds * v
𝝏𝒗
𝝏𝒔
Dividing by ρ ds dA above equation,
-𝝏𝒑
ρ𝝏𝒔- g cos θ = v
𝝏𝒗
𝝏𝒔
𝝏𝒑
ρ𝝏𝒔+ g cos θ + v
𝝏𝒗
𝝏𝒔= 0
Now,
cos θ =𝒅𝒛
𝒅𝒔
The above equation can be rewrite
𝟏
ρ𝝏𝒑
𝝏𝒔+ g
𝒅𝒛
𝒅𝒔+ v
𝝏𝒗
𝝏𝒔= 0
EULER’S EQUATION OF MOTION:
𝒅𝒑
ρ+ g dz + v dv= 0
The above equation is known as Euler’s Equation of Motion.
40Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
It States, “The sum of pressure energy(pressure head), the kinetic energy (velocityhead) and the potential energy (potentialhead) is constant along a stream line.”
Integrating Euler’s equation of motion
𝑑𝑝
𝜌+ 𝑔𝑑𝑧 + 𝑣𝑑𝑣 = 0
Or p/ 𝜌 + gZ + v2/2 = Constant
Or p/ 𝜌g + v2/2g + Z = Constant
Substituting 𝜌g = 𝛾
p/𝛾 + v2/2g + Z = Constant
The above equation known as Bernoulli’sequation.
BERNOULLI’S EQUATION:
p/𝛾 = Pressure Headv2/2g = Velocity HeadZ = Potential Head
p/𝛾 + v2/2g + Z = Total Head or Hydrodynamic Head
41Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Assumption made in Bernoulli’s equation:
Flow is steady: No variation with respect to time
Fluid is ideal: No frictional effect due to viscosity
Flow is incompressible: density constant
One-dimensional flow
Velocity of flow is uniform.
Bernoulli’s equation at section 1&2
p1/𝛾 + v12/2g + Z1 = p2/𝛾 + v2
2/2g + Z2
BERNOULLI’S EQUATION:
1 2
42Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
It is defined as a scalar function of space and time such that its negativederivative with respect to any direction gives the fluid velocity in thatdirection.
Mathematically, the velocity potential is defined as ∅ = f (x , y, z) for steadyflow such that,
u = -𝜕∅
𝜕𝑥
v = -𝜕∅
𝜕𝑦
w = -𝜕∅
𝜕𝑧where u, v and w are the components of velocity in x, y and z directionsrespectively.
VELOCITY POTENTIAL FUNCTION:
43Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The continuity equation for an incompressible steady flow is𝜕u𝜕𝑥
+𝜕v𝜕𝑦
+𝜕w𝜕𝑧
= 0
Substituting the values of u, v and w from previous equation, we get𝜕
𝜕𝑥(-𝜕∅
𝜕𝑥) +
𝜕
𝜕𝑦(-𝜕∅
𝜕𝑦) +
𝜕
𝜕𝑦(-𝜕∅
𝜕𝑧) = 0
𝜕2∅
𝜕𝑥2+𝜕2∅
𝜕𝑦2 +𝜕2∅
𝜕𝑧2= 0
The equation is known as Laplace equation.
The rotational component is
ωz =1
2(𝜕v𝜕𝑥
−𝜕u𝜕𝑦
), ωy =1
2(𝜕u𝜕𝑧
−𝜕w𝜕𝑥
), ωx =1
2(𝜕w𝜕𝑦
−𝜕v𝜕𝑧
)
or
ωz =1
2(−
𝜕2∅
𝜕𝑥𝜕𝑦+
𝜕2∅
𝜕𝑦𝜕𝑥), ωy =
1
2(−
𝜕2∅
𝜕𝑧𝜕𝑥+
𝜕2∅
𝜕𝑥𝜕𝑧), ωx =
1
2(−
𝜕2∅
𝜕𝑦𝜕𝑧+
𝜕2∅
𝜕𝑧𝜕𝑦),
VELOCITY POTENTIAL FUNCTION:
44Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The rotational component is
ωz =1
2(−
𝜕2∅
𝜕𝑥𝜕𝑦+
𝜕2∅
𝜕𝑦𝜕𝑥), ωy =
1
2(−
𝜕2∅
𝜕𝑧𝜕𝑥+
𝜕2∅
𝜕𝑥𝜕𝑧), ωx =
1
2(−
𝜕2∅
𝜕𝑦𝜕𝑧+
𝜕2∅
𝜕𝑧𝜕𝑦)
If ∅ is continuous function then,
𝜕2∅
𝜕𝑥𝜕𝑦=
𝜕2∅
𝜕𝑦𝜕𝑥,𝜕2∅
𝜕𝑧𝜕𝑥=
𝜕2∅
𝜕𝑥𝜕𝑧,𝜕2∅
𝜕𝑦𝜕𝑧=
𝜕2∅
𝜕𝑧𝜕𝑦
ωz = ωy = ωx = 0
When rotational components are zero, the flow is called Irrotational.
The properties of potential function are:
1. If velocity potential (∅) is exist, the flow should be Irrotational.
2. If velocity potential (∅ ) is satisfies the Laplace equation, itrepresents steady incompressible Irrotational flow.
VELOCITY POTENTIAL FUNCTION:
45Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
It is defined as the scalar function of space and time, such that its partialderivative with respect to any direction gives the velocity component at rightangles to that direction.
It is defined only for two dimension flow.
Mathematically, for steady flow it is defined as 𝜑= f (x, y) such that
𝜕𝜑
𝜕𝑥= v
𝜕𝜑
𝜕𝑦= -u
where u and v are the components of velocity in x and y directionsrespectively.
STREAM FUNCTION:
46Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The continuity equation for two dimensional flow is,
𝜕u𝜕𝑥
+𝜕v𝜕𝑦
= 0
𝜕
𝜕𝑥−
𝜕𝜑
𝜕𝑦+
𝜕
𝜕𝑦(𝜕𝜑
𝜕𝑥) = 0
−𝜕2𝜑
𝜕𝑥𝜕𝑦+
𝜕2𝜑
𝜕𝑦𝜕𝑥= 0
The rotational component is
ωz =1
2(𝜕v𝜕𝑥
−𝜕u𝜕𝑦
) =1
2(𝜕2𝜑
𝜕𝑥2+𝜕2𝜑
𝜕𝑦2)
For Irrotational flow ωz = 0 so,
𝜕2𝜑
𝜕𝑥2+𝜕2𝜑
𝜕𝑦2 = 0 , it is Laplace equation for 𝜑.
STREAM FUNCTION:
47Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
The velocity potential is,
u = -𝜕∅
𝜕𝑥
v = -𝜕∅
𝜕𝑦
The stream function is,
𝜕𝜑
𝜕𝑥= v
𝜕𝜑
𝜕𝑦= -u
So
u = -𝜕∅
𝜕𝑥= -
𝜕𝜑
𝜕𝑦& v = -
𝜕∅
𝜕𝑦=𝜕𝜑
𝜕𝑥
𝜕∅
𝜕𝑥=
𝜕𝜑
𝜕𝑦&
𝜕∅
𝜕𝑦= -
𝜕𝜑
𝜕𝑥
Relation between Stream Function and Velocity Potential Function:
48Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
A grid obtained by drawing a series of equipotential lines and stream linesis called a flow net.
The flow net is an important tool in analyzing two-dimensional irrotationalflow problems.
FLOW NET:
49Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
Example:
50Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot
References:
1. Fluid Mechanics and Fluid Power Engineering by D.S. Kumar, S.K.Kataria & Sons
2. Fluid Mechanics and Hydraulic Machines by R.K. Bansal, LaxmiPublications
3. Fluid Mechanics and Hydraulic Machines by R.K. Rajput, S.Chand & Co
4. Fluid Mechanics; Fundamentals and Applications by John. M. CimbalaYunus A. Cengel, McGraw-Hill Publication