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Phy 352: Fluid Dynamics, Spring 2013
Prasad [email protected]
Indian Institute of Science Education and Research (IISER), Pune
Subramanian Fluid Dynamics
Introduction
Course webpage:http://www.iiserpune.ac.in/∼p.subramanian/FluidsSpring2013.html
Subramanian Fluid Dynamics
Introduction
Course webpage:http://www.iiserpune.ac.in/∼p.subramanian/FluidsSpring2013.html
3 credit course offered to students of the 6th and 8thsemesters at IISER Pune. Also offered over the NationalKnowledge Network to students at IISER Kolkata
Subramanian Fluid Dynamics
Introduction
Course webpage:http://www.iiserpune.ac.in/∼p.subramanian/FluidsSpring2013.html
3 credit course offered to students of the 6th and 8thsemesters at IISER Pune. Also offered over the NationalKnowledge Network to students at IISER Kolkata
11:30 am - 12:25 pm, Tuesdays and Fridays
Subramanian Fluid Dynamics
Introduction
Course webpage:http://www.iiserpune.ac.in/∼p.subramanian/FluidsSpring2013.html
3 credit course offered to students of the 6th and 8thsemesters at IISER Pune. Also offered over the NationalKnowledge Network to students at IISER Kolkata
11:30 am - 12:25 pm, Tuesdays and Fridays
Useful for students wishing to gain an overview of the vastfield of fluid dynamics. Applications: astrophysics,aerodynamics, biofluid dynamics, computational fluiddynamics, etc.
Subramanian Fluid Dynamics
Introduction
Course webpage:http://www.iiserpune.ac.in/∼p.subramanian/FluidsSpring2013.html
3 credit course offered to students of the 6th and 8thsemesters at IISER Pune. Also offered over the NationalKnowledge Network to students at IISER Kolkata
11:30 am - 12:25 pm, Tuesdays and Fridays
Useful for students wishing to gain an overview of the vastfield of fluid dynamics. Applications: astrophysics,aerodynamics, biofluid dynamics, computational fluiddynamics, etc.
Pre-requisites: Classical mechanics, a sound knowledge of(basic) vector calculus (e.g., gradient, divergence, curl, etc.)and some familiarity with tensors
Subramanian Fluid Dynamics
Course organization - I
The continuum hypothesis, kinematics, conservation laws:continuity equation, Euler and Navier-Stokes equations(chapters 1, 3 and 4, Kundu & Cohen, 3 weeks)
Subramanian Fluid Dynamics
Course organization - I
The continuum hypothesis, kinematics, conservation laws:continuity equation, Euler and Navier-Stokes equations(chapters 1, 3 and 4, Kundu & Cohen, 3 weeks)
Dimensionless numbers, dynamic similarity, aerodynamics(chapters 8 and 15, Kundu and Cohen, 2 weeks)
Subramanian Fluid Dynamics
Course organization - I
The continuum hypothesis, kinematics, conservation laws:continuity equation, Euler and Navier-Stokes equations(chapters 1, 3 and 4, Kundu & Cohen, 3 weeks)
Dimensionless numbers, dynamic similarity, aerodynamics(chapters 8 and 15, Kundu and Cohen, 2 weeks)
Quiz 1 (15 %)
Subramanian Fluid Dynamics
Course organization - I
The continuum hypothesis, kinematics, conservation laws:continuity equation, Euler and Navier-Stokes equations(chapters 1, 3 and 4, Kundu & Cohen, 3 weeks)
Dimensionless numbers, dynamic similarity, aerodynamics(chapters 8 and 15, Kundu and Cohen, 2 weeks)
Quiz 1 (15 %)
Compressible flows, speed of sound, shocks (2 weeks, chapter16, Kundu and Cohen, also relevant parts from Physics ofFluids and Plasmas, Arnab Rai Choudhuri)
Subramanian Fluid Dynamics
Course organization - I
The continuum hypothesis, kinematics, conservation laws:continuity equation, Euler and Navier-Stokes equations(chapters 1, 3 and 4, Kundu & Cohen, 3 weeks)
Dimensionless numbers, dynamic similarity, aerodynamics(chapters 8 and 15, Kundu and Cohen, 2 weeks)
Quiz 1 (15 %)
Compressible flows, speed of sound, shocks (2 weeks, chapter16, Kundu and Cohen, also relevant parts from Physics ofFluids and Plasmas, Arnab Rai Choudhuri)
Mid-term exam (30 %)
Subramanian Fluid Dynamics
Course organization - II
Fluid instabilities and turbulence (chapters 12 and 13, Kunduand Cohen, also relevant parts from Physics of Fluids andPlasmas, Arnab Rai Choudhuri, 2 weeks)
Subramanian Fluid Dynamics
Course organization - II
Fluid instabilities and turbulence (chapters 12 and 13, Kunduand Cohen, also relevant parts from Physics of Fluids andPlasmas, Arnab Rai Choudhuri, 2 weeks)
A quick introduction to computational fluid dynamics(chapter 11, Kundu and Cohen, 1 week)
Subramanian Fluid Dynamics
Course organization - II
Fluid instabilities and turbulence (chapters 12 and 13, Kunduand Cohen, also relevant parts from Physics of Fluids andPlasmas, Arnab Rai Choudhuri, 2 weeks)
A quick introduction to computational fluid dynamics(chapter 11, Kundu and Cohen, 1 week)
Quiz 2 (15 %)
Subramanian Fluid Dynamics
Course organization - II
Fluid instabilities and turbulence (chapters 12 and 13, Kunduand Cohen, also relevant parts from Physics of Fluids andPlasmas, Arnab Rai Choudhuri, 2 weeks)
A quick introduction to computational fluid dynamics(chapter 11, Kundu and Cohen, 1 week)
Quiz 2 (15 %)
Applications of fluid dynamics in Astrophysics: e.g.,Astrophysical jets, the de Laval nozzle, spherical accretiononto compact objects, the solar wind (2 weeks)
Subramanian Fluid Dynamics
Course organization - II
Fluid instabilities and turbulence (chapters 12 and 13, Kunduand Cohen, also relevant parts from Physics of Fluids andPlasmas, Arnab Rai Choudhuri, 2 weeks)
A quick introduction to computational fluid dynamics(chapter 11, Kundu and Cohen, 1 week)
Quiz 2 (15 %)
Applications of fluid dynamics in Astrophysics: e.g.,Astrophysical jets, the de Laval nozzle, spherical accretiononto compact objects, the solar wind (2 weeks)
Applications of fluid dynamics in geophysics (chapter 14,Kundu and Cohen, 1 week)
Subramanian Fluid Dynamics
Course organization - II
Fluid instabilities and turbulence (chapters 12 and 13, Kunduand Cohen, also relevant parts from Physics of Fluids andPlasmas, Arnab Rai Choudhuri, 2 weeks)
A quick introduction to computational fluid dynamics(chapter 11, Kundu and Cohen, 1 week)
Quiz 2 (15 %)
Applications of fluid dynamics in Astrophysics: e.g.,Astrophysical jets, the de Laval nozzle, spherical accretiononto compact objects, the solar wind (2 weeks)
Applications of fluid dynamics in geophysics (chapter 14,Kundu and Cohen, 1 week)
Final exam (40 %)
Subramanian Fluid Dynamics
Reference Material
Fluid Mechanics by Kundu and Cohen, 4th edition, 2008,Elsevier (Indian edition): primary text
Subramanian Fluid Dynamics
Reference Material
Fluid Mechanics by Kundu and Cohen, 4th edition, 2008,Elsevier (Indian edition): primary text
The Physics of Fluids and Plasmas, Arnab Rai Choudhuri,Cambridge University Press (Indian edition)
Subramanian Fluid Dynamics
Reference Material
Fluid Mechanics by Kundu and Cohen, 4th edition, 2008,Elsevier (Indian edition): primary text
The Physics of Fluids and Plasmas, Arnab Rai Choudhuri,Cambridge University Press (Indian edition)
Fluid Mechanics, Landau & Lifshitz, 2nd edition, PergamonPress (Indian edition)
Subramanian Fluid Dynamics
The continuum hypothesis
Sub-branch of continuum mechanics (as opposed to themechanics of point bodies)
Subramanian Fluid Dynamics
The continuum hypothesis
Sub-branch of continuum mechanics (as opposed to themechanics of point bodies)
We don’t concern ourselves with microscopics (like collisionsbetween atoms/molecules); materials are only characterizedby bulk properties like conductivity, elasticity (for solids),viscosity (for fluids)
Subramanian Fluid Dynamics
The continuum hypothesis
Sub-branch of continuum mechanics (as opposed to themechanics of point bodies)
We don’t concern ourselves with microscopics (like collisionsbetween atoms/molecules); materials are only characterizedby bulk properties like conductivity, elasticity (for solids),viscosity (for fluids)
These bulk properties are characterized by the relevanttransport coefficients, which can be derived frommicroscopic properties
Subramanian Fluid Dynamics
The continuum hypothesis
Sub-branch of continuum mechanics (as opposed to themechanics of point bodies)
We don’t concern ourselves with microscopics (like collisionsbetween atoms/molecules); materials are only characterizedby bulk properties like conductivity, elasticity (for solids),viscosity (for fluids)
These bulk properties are characterized by the relevanttransport coefficients, which can be derived frommicroscopic properties
To operate in the continuum limit, we need (roughly speaking)N (number of particles in some macroscopic volume) ≫ 1.
Subramanian Fluid Dynamics
Bulk properties from kinetics: examples
Consider a gas whose molecules are in thermodynamicequilibrium. It has a definite distribution f (say,Maxwell-Boltzmann distribution)
Subramanian Fluid Dynamics
Bulk properties from kinetics: examples
Consider a gas whose molecules are in thermodynamicequilibrium. It has a definite distribution f (say,Maxwell-Boltzmann distribution)
Number density n (cm−3) =∫
f d3xd3v
Subramanian Fluid Dynamics
Bulk properties from kinetics: examples
Consider a gas whose molecules are in thermodynamicequilibrium. It has a definite distribution f (say,Maxwell-Boltzmann distribution)
Number density n (cm−3) =∫
f d3xd3v
The average velocity (if any) would be 〈v〉 =∫
v f d3xd3v
Subramanian Fluid Dynamics
Bulk properties from kinetics: examples
Consider a gas whose molecules are in thermodynamicequilibrium. It has a definite distribution f (say,Maxwell-Boltzmann distribution)
Number density n (cm−3) =∫
f d3xd3v
The average velocity (if any) would be 〈v〉 =∫
v f d3xd3v
The average kinetic energy would be〈(1/2)mv2〉 =
∫
(1/2)mv2 f d3xd3v . For a thermaldistribution, (i.e., a Maxwell-Boltzmann f ) we know that thisis equal to (3/2)kT
Subramanian Fluid Dynamics
Bulk properties from kinetics: examples
Consider a gas whose molecules are in thermodynamicequilibrium. It has a definite distribution f (say,Maxwell-Boltzmann distribution)
Number density n (cm−3) =∫
f d3xd3v
The average velocity (if any) would be 〈v〉 =∫
v f d3xd3v
The average kinetic energy would be〈(1/2)mv2〉 =
∫
(1/2)mv2 f d3xd3v . For a thermaldistribution, (i.e., a Maxwell-Boltzmann f ) we know that thisis equal to (3/2)kT
The pressure of the gas is thus a macroscopic (fluid) concept,representing a statistical average of the force per unit areadue to molecules striking the walls of a container.
Subramanian Fluid Dynamics
Solids vs fluids
Fluids are characterized mainly by their response to shear forces;they cannot resist shear; they deform/flow continuously forarbitrarily small shear. Solids, however, can be elastic.Complications: viscoelesticity, etc.
Subramanian Fluid Dynamics
How do fluids respond to shear?
They flow. Viscosity tends to reduce the velocity gradient (thesolid line velocity profile tends towards the dashed line)
Subramanian Fluid Dynamics
Stress
Given an area element dA, one can define a
normal stress τn ≡ dFn/dA (the scalar pressure we are familiarwith) and a
Subramanian Fluid Dynamics
Stress
Given an area element dA, one can define a
normal stress τn ≡ dFn/dA (the scalar pressure we are familiarwith) and a
shear stress τs ≡ dFs/dA
Subramanian Fluid Dynamics
Shear stress, viscosity
For a wide class of fluids called Newtonian fluids, the shear stressis experimentally observed to be linearly proportional to thevelocity gradient:
τs = µdu
dy
where µ (g cm−1 s−1) is called the coefficient of dynamic viscosity.In fact, the components of stress form a tensor
Subramanian Fluid Dynamics
Shear stress, viscosity
For a wide class of fluids called Newtonian fluids, the shear stressis experimentally observed to be linearly proportional to thevelocity gradient:
τs = µdu
dy
where µ (g cm−1 s−1) is called the coefficient of dynamic viscosity.In fact, the components of stress form a tensor
Subramanian Fluid Dynamics
Fluids: kinematics
A study of the appearance of motion (e.g., displacement, velocity,acceleration, rotation) of fluid elements without explicitlyconsidering the forces acting on them. There are two alternativedescriptions possible:
Subramanian Fluid Dynamics
Fluids: kinematics
A study of the appearance of motion (e.g., displacement, velocity,acceleration, rotation) of fluid elements without explicitlyconsidering the forces acting on them. There are two alternativedescriptions possible:
The Eulerian description: fluid flow as seen by an observer inthe lab frame (the “field” picture): A fluid variable F (likedensity, velocity) is expressed in terms of (lab) position x andtime t: F (x, t)
Subramanian Fluid Dynamics
Fluids: kinematics
A study of the appearance of motion (e.g., displacement, velocity,acceleration, rotation) of fluid elements without explicitlyconsidering the forces acting on them. There are two alternativedescriptions possible:
The Eulerian description: fluid flow as seen by an observer inthe lab frame (the “field” picture): A fluid variable F (likedensity, velocity) is expressed in terms of (lab) position x andtime t: F (x, t)
The Lagrangian description: fluid flow as seen by an observersitting on a fluid parcel (the “particle” picture): F is afunction of the position of the fluid parcel x0 at a referencetime t = t0 and time t: F (x0, t)
Subramanian Fluid Dynamics
Fluids: kinematics
A study of the appearance of motion (e.g., displacement, velocity,acceleration, rotation) of fluid elements without explicitlyconsidering the forces acting on them. There are two alternativedescriptions possible:
The Eulerian description: fluid flow as seen by an observer inthe lab frame (the “field” picture): A fluid variable F (likedensity, velocity) is expressed in terms of (lab) position x andtime t: F (x, t)
The Lagrangian description: fluid flow as seen by an observersitting on a fluid parcel (the “particle” picture): F is afunction of the position of the fluid parcel x0 at a referencetime t = t0 and time t: F (x0, t)
the two pictures are related via a (typically Galilean) frametransformation involving the bulk fluid velocity
Subramanian Fluid Dynamics
The “material/substantive/particle” derivative
What is the rate of change of F experienced by a Lagrangian
observer (i.e.; one sitting on a fluid parcel) expressed in regular lab(Eulerian) variables?
Subramanian Fluid Dynamics
The “material/substantive/particle” derivative
What is the rate of change of F experienced by a Lagrangian
observer (i.e.; one sitting on a fluid parcel) expressed in regular lab(Eulerian) variables?
dF =∂F
∂tdt +
∂F
∂xidxi
: ∂/∂t gives the local (time) rate of change at a point x
Subramanian Fluid Dynamics
The “material/substantive/particle” derivative
What is the rate of change of F experienced by a Lagrangian
observer (i.e.; one sitting on a fluid parcel) expressed in regular lab(Eulerian) variables?
dF =∂F
∂tdt +
∂F
∂xidxi
: ∂/∂t gives the local (time) rate of change at a point x
dF
dt=∂F
∂t+ ui
∂F
∂xi
Subramanian Fluid Dynamics
The “material/substantive/particle” derivative
What is the rate of change of F experienced by a Lagrangian
observer (i.e.; one sitting on a fluid parcel) expressed in regular lab(Eulerian) variables?
dF =∂F
∂tdt +
∂F
∂xidxi
: ∂/∂t gives the local (time) rate of change at a point x
dF
dt=∂F
∂t+ ui
∂F
∂xi=∂F
∂t+ u .∇F
Subramanian Fluid Dynamics
The “material/substantive/particle” derivative
What is the rate of change of F experienced by a Lagrangian
observer (i.e.; one sitting on a fluid parcel) expressed in regular lab(Eulerian) variables?
dF =∂F
∂tdt +
∂F
∂xidxi
: ∂/∂t gives the local (time) rate of change at a point x
dF
dt=∂F
∂t+ ui
∂F
∂xi=∂F
∂t+ u .∇F
dF/dt (often written as DF/Dt is the material derivative; thetotal rate of change in quantity F ) felt by a Lagrangianobserver
Subramanian Fluid Dynamics
Streamlines
If |u| = q, and we define a “streamline coordinate” s that pointsalong the local direction of u, the material derivative dF/dt can bewritten as
Subramanian Fluid Dynamics
Streamlines
If |u| = q, and we define a “streamline coordinate” s that pointsalong the local direction of u, the material derivative dF/dt can bewritten as
dF
dt=∂F
∂t+ u .∇F
Subramanian Fluid Dynamics
Streamlines
If |u| = q, and we define a “streamline coordinate” s that pointsalong the local direction of u, the material derivative dF/dt can bewritten as
dF
dt=∂F
∂t+ u .∇F =
∂F
∂t+ q
∂F
∂s
Alternatively, if ds = (dx , dy , dz) and u = (ux , uy , uz), astreamline can be defined by
Subramanian Fluid Dynamics
Streamlines
If |u| = q, and we define a “streamline coordinate” s that pointsalong the local direction of u, the material derivative dF/dt can bewritten as
dF
dt=∂F
∂t+ u .∇F =
∂F
∂t+ q
∂F
∂s
Alternatively, if ds = (dx , dy , dz) and u = (ux , uy , uz), astreamline can be defined by
dx
ux=
dy
uy=
dz
uz
Subramanian Fluid Dynamics
Streamlines
If |u| = q, and we define a “streamline coordinate” s that pointsalong the local direction of u, the material derivative dF/dt can bewritten as
dF
dt=∂F
∂t+ u .∇F =
∂F
∂t+ q
∂F
∂s
Alternatively, if ds = (dx , dy , dz) and u = (ux , uy , uz), astreamline can be defined by
dx
ux=
dy
uy=
dz
uz
Equivalently, ds× u = 0.
Subramanian Fluid Dynamics
Laminar flow around a sphere: streamlines
Subramanian Fluid Dynamics
Turbulent flow around a sphere: streamlines
Subramanian Fluid Dynamics
Streamlines, pathlines and streaklines
Pathline: trajectory of a fluid parcel of fixed identity
Subramanian Fluid Dynamics
Streamlines, pathlines and streaklines
Pathline: trajectory of a fluid parcel of fixed identity
Streakline: Current location of all fluid parcels that havepassed through a fixed spatial point (e.g., injectdye/smoke/some kind of tracer)
Subramanian Fluid Dynamics
Streamlines, pathlines and streaklines
Pathline: trajectory of a fluid parcel of fixed identity
Streakline: Current location of all fluid parcels that havepassed through a fixed spatial point (e.g., injectdye/smoke/some kind of tracer)
Streamline, pathline, streakline all equivalent for steady flow(what precisely does “steady” mean?)
Subramanian Fluid Dynamics
The streamfunction
For 2-dimensional flows (i.e., no z-dependence), its often useful todefine a streamfunction ψ:
Subramanian Fluid Dynamics
The streamfunction
For 2-dimensional flows (i.e., no z-dependence), its often useful todefine a streamfunction ψ:
u = −∇× [zψ(x , y)]
Subramanian Fluid Dynamics
The streamfunction
For 2-dimensional flows (i.e., no z-dependence), its often useful todefine a streamfunction ψ:
u = −∇× [zψ(x , y)]
Using this definition, and the definition of streamlines, one canshow that (work it out!)
Subramanian Fluid Dynamics
The streamfunction
For 2-dimensional flows (i.e., no z-dependence), its often useful todefine a streamfunction ψ:
u = −∇× [zψ(x , y)]
Using this definition, and the definition of streamlines, one canshow that (work it out!)
δψ =∂ψ
∂xdx +
∂ψ
∂ydy
Subramanian Fluid Dynamics
The streamfunction
For 2-dimensional flows (i.e., no z-dependence), its often useful todefine a streamfunction ψ:
u = −∇× [zψ(x , y)]
Using this definition, and the definition of streamlines, one canshow that (work it out!)
δψ =∂ψ
∂xdx +
∂ψ
∂ydy = 0 ,
where dx and dy are along a streamline.
Subramanian Fluid Dynamics
The streamfunction
For 2-dimensional flows (i.e., no z-dependence), its often useful todefine a streamfunction ψ:
u = −∇× [zψ(x , y)]
Using this definition, and the definition of streamlines, one canshow that (work it out!)
δψ =∂ψ
∂xdx +
∂ψ
∂ydy = 0 ,
where dx and dy are along a streamline. In other words thestreamfunction remains constant along a streamline.
Subramanian Fluid Dynamics
Vorticity
The vorticity ω of a flow is defined as
ω = ∇× u
Subramanian Fluid Dynamics
Vorticity
The vorticity ω of a flow is defined as
ω = ∇× u
If the vorticity of a flow equals zero, the flow is irrotational
Subramanian Fluid Dynamics
Vorticity
The vorticity ω of a flow is defined as
ω = ∇× u
If the vorticity of a flow equals zero, the flow is irrotational
For irrotational flows, one can define a potential φ, such thatu = −∇φ (why?)
Subramanian Fluid Dynamics
Vorticity
The vorticity ω of a flow is defined as
ω = ∇× u
If the vorticity of a flow equals zero, the flow is irrotational
For irrotational flows, one can define a potential φ, such thatu = −∇φ (why?)
and then ∇φ .∇ψ = 0
Subramanian Fluid Dynamics
Vorticity
The vorticity ω of a flow is defined as
ω = ∇× u
If the vorticity of a flow equals zero, the flow is irrotational
For irrotational flows, one can define a potential φ, such thatu = −∇φ (why?)
and then ∇φ .∇ψ = 0
Why, and can you think of parallels in electrostatics? What
about the Cauchy-Riemann conditions in complex algebra?
Subramanian Fluid Dynamics
Mass conservation: the equation of continuity
First, define the mass flux across a bounding surface: theamount of mass per unit area per unit time flowing out of (orinto) the surface:
Subramanian Fluid Dynamics
Mass conservation: the equation of continuity
First, define the mass flux across a bounding surface: theamount of mass per unit area per unit time flowing out of (orinto) the surface:
Mass flux = ρu, where ρ is the matter density and u is theflow velocity
Subramanian Fluid Dynamics
Mass conservation: the equation of continuity
First, define the mass flux across a bounding surface: theamount of mass per unit area per unit time flowing out of (orinto) the surface:
Mass flux = ρu, where ρ is the matter density and u is theflow velocity
The mass contained in a volume∫
ρ dV can change only dueto mass flux through the bounding surface (mass can’t becreated/destroyed within the volume);
Subramanian Fluid Dynamics
Mass conservation: the equation of continuity
First, define the mass flux across a bounding surface: theamount of mass per unit area per unit time flowing out of (orinto) the surface:
Mass flux = ρu, where ρ is the matter density and u is theflow velocity
The mass contained in a volume∫
ρ dV can change only dueto mass flux through the bounding surface (mass can’t becreated/destroyed within the volume); in other words,
Subramanian Fluid Dynamics
Mass conservation: the equation of continuity
First, define the mass flux across a bounding surface: theamount of mass per unit area per unit time flowing out of (orinto) the surface:
Mass flux = ρu, where ρ is the matter density and u is theflow velocity
The mass contained in a volume∫
ρ dV can change only dueto mass flux through the bounding surface (mass can’t becreated/destroyed within the volume); in other words,
∂
∂t
∫
ρ dV = −∫
ρu . dS
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get∫[
∂ρ
∂t+∇ . (ρu)
]
dV = 0
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get∫[
∂ρ
∂t+∇ . (ρu)
]
dV = 0
Since this is true for an arbitrary volume,
∂ρ
∂t+∇ . (ρu) = 0
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get∫[
∂ρ
∂t+∇ . (ρu)
]
dV = 0
Since this is true for an arbitrary volume,
∂ρ
∂t+∇ . (ρu) = 0
This is the conservative form of the mass continuity equation
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get∫[
∂ρ
∂t+∇ . (ρu)
]
dV = 0
Since this is true for an arbitrary volume,
∂ρ
∂t+∇ . (ρu) = 0
This is the conservative form of the mass continuity equationIn general, the conservative form of any quantity goes as
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get∫[
∂ρ
∂t+∇ . (ρu)
]
dV = 0
Since this is true for an arbitrary volume,
∂ρ
∂t+∇ . (ρu) = 0
This is the conservative form of the mass continuity equationIn general, the conservative form of any quantity goes as(Partial) time derivative of quantity + divergence of flux ofthat quantity = 0
Subramanian Fluid Dynamics
Equation of continuity (Mass conservation)
∂
∂t
∫
ρ dV = −∫
ρu . dS
Use Gauss’s law on RHS to get∫[
∂ρ
∂t+∇ . (ρu)
]
dV = 0
Since this is true for an arbitrary volume,
∂ρ
∂t+∇ . (ρu) = 0
This is the conservative form of the mass continuity equationIn general, the conservative form of any quantity goes as(Partial) time derivative of quantity + divergence of flux ofthat quantity = 0Alternatively, using the Lagrangian derivative d/dt, (show!)
dρ
dt+ ρ∇ .u = 0
Subramanian Fluid Dynamics
Incompressibility
The mass continuity equation can be rewritten as
1
ρ
dρ
dt+∇ .u = 0
Subramanian Fluid Dynamics
Incompressibility
The mass continuity equation can be rewritten as
1
ρ
dρ
dt+∇ .u = 0
One frequently encounters situations where (to a fairapproximation) the first term is negligible;
Subramanian Fluid Dynamics
Incompressibility
The mass continuity equation can be rewritten as
1
ρ
dρ
dt+∇ .u = 0
One frequently encounters situations where (to a fairapproximation) the first term is negligible; i.e.,
1
ρ
dρ
dt≈ 0
Subramanian Fluid Dynamics
Incompressibility
The mass continuity equation can be rewritten as
1
ρ
dρ
dt+∇ .u = 0
One frequently encounters situations where (to a fairapproximation) the first term is negligible; i.e.,
1
ρ
dρ
dt≈ 0
From the continuity equation, the condition forincompressibility then becomes
∇ .u = 0
Subramanian Fluid Dynamics
Incompressibility
The mass continuity equation can be rewritten as
1
ρ
dρ
dt+∇ .u = 0
One frequently encounters situations where (to a fairapproximation) the first term is negligible; i.e.,
1
ρ
dρ
dt≈ 0
From the continuity equation, the condition forincompressibility then becomes
∇ .u = 0
Applicable when flow speeds ≪ sound speed
Subramanian Fluid Dynamics
Incompressibility...some more
In the Boussinesq approximation, density variations in thefluid can be neglected everywhere except where the density ismultiplied by g
Subramanian Fluid Dynamics
Incompressibility...some more
In the Boussinesq approximation, density variations in thefluid can be neglected everywhere except where the density ismultiplied by g
Consider a (gravitationally) vertically stratified atmosphere. Ithas a vertical scale height L ≈ c2s /g ; this is the e-folding scaleheight for the density (show that this is so, at least for
isothermal flows)
Subramanian Fluid Dynamics
Incompressibility...some more
In the Boussinesq approximation, density variations in thefluid can be neglected everywhere except where the density ismultiplied by g
Consider a (gravitationally) vertically stratified atmosphere. Ithas a vertical scale height L ≈ c2s /g ; this is the e-folding scaleheight for the density (show that this is so, at least for
isothermal flows)
For the Boussinesq approximation to be valid, the scale of theflow should be ≪ L
Subramanian Fluid Dynamics
Incompressibility...some more
In the Boussinesq approximation, density variations in thefluid can be neglected everywhere except where the density ismultiplied by g
Consider a (gravitationally) vertically stratified atmosphere. Ithas a vertical scale height L ≈ c2s /g ; this is the e-folding scaleheight for the density (show that this is so, at least for
isothermal flows)
For the Boussinesq approximation to be valid, the scale of theflow should be ≪ L
The Boussinseq approximation often just reduces to ∇ .u = 0,but not always
Subramanian Fluid Dynamics
Momentum conservation
Newton’s second law of motion: F = m a
Subramanian Fluid Dynamics
Momentum conservation
Newton’s second law of motion: F = m a
The RHS (m a) is simply the rate of change of momentum
Subramanian Fluid Dynamics
Momentum conservation
Newton’s second law of motion: F = m a
The RHS (m a) is simply the rate of change of momentum
The momentum can change because of “intrinsic” (∂/∂t)change inside the volume:
∫
∂(ρu)
∂tdV
Subramanian Fluid Dynamics
Momentum conservation
Newton’s second law of motion: F = m a
The RHS (m a) is simply the rate of change of momentum
The momentum can change because of “intrinsic” (∂/∂t)change inside the volume:
∫
∂(ρu)
∂tdV
..and also due to the flux of momentum through the boundingsurface
∫
(ρu)u . dA
Subramanian Fluid Dynamics
Momentum conservation - momentum flux
Using Gauss’s divergence theorem (for tensors, since uu is asecond rank tensor), the surface integral
∫
(ρu)u . dS
becomes
Subramanian Fluid Dynamics
Momentum conservation - momentum flux
Using Gauss’s divergence theorem (for tensors, since uu is asecond rank tensor), the surface integral
∫
(ρu)u . dS
becomes
∫
∇ .(ρuu) dV
Subramanian Fluid Dynamics
Momentum conservation - momentum flux
Using Gauss’s divergence theorem (for tensors, since uu is asecond rank tensor), the surface integral
∫
(ρu)u . dS
becomes
∫
∇ .(ρuu) dV
So that the rate of change of momentum is
Subramanian Fluid Dynamics
Momentum conservation - momentum flux
Using Gauss’s divergence theorem (for tensors, since uu is asecond rank tensor), the surface integral
∫
(ρu)u . dS
becomes
∫
∇ .(ρuu) dV
So that the rate of change of momentum is
∫[
∂(ρu)
∂t+∇ .(ρuu)
]
dV
Subramanian Fluid Dynamics
Momentum conservation - forces
The rate of change of momentum is equal to the force, ofcourse
Subramanian Fluid Dynamics
Momentum conservation - forces
The rate of change of momentum is equal to the force, ofcourseThe force on the fluid element too can have a volume
component (due to body forces),∫
ρg dV
Subramanian Fluid Dynamics
Momentum conservation - forces
The rate of change of momentum is equal to the force, ofcourseThe force on the fluid element too can have a volume
component (due to body forces),∫
ρg dV..note, g represents the acceleration due to any kind of bodyforce (like gravity, for instance)
Subramanian Fluid Dynamics
Momentum conservation - forces
The rate of change of momentum is equal to the force, ofcourseThe force on the fluid element too can have a volume
component (due to body forces),∫
ρg dV..note, g represents the acceleration due to any kind of bodyforce (like gravity, for instance)Surface forces are a little trickier. Recall, one can have forceson a surface area element that are normal to it, as well astangential to it:
Subramanian Fluid Dynamics
Momentum conservation - forces
The rate of change of momentum is equal to the force, ofcourseThe force on the fluid element too can have a volume
component (due to body forces),∫
ρg dV..note, g represents the acceleration due to any kind of bodyforce (like gravity, for instance)Surface forces are a little trickier. Recall, one can have forceson a surface area element that are normal to it, as well astangential to it:
Subramanian Fluid Dynamics
Momentum conservation - the pressure tensor
The ijth component of the pressure tensor represents the forcein the ith direction felt by an area element whose outwardnormal points in the jth direction:
(d Fsurface)i = −Pij d Aj
So that
Subramanian Fluid Dynamics
Momentum conservation - the pressure tensor
The ijth component of the pressure tensor represents the forcein the ith direction felt by an area element whose outwardnormal points in the jth direction:
(d Fsurface)i = −Pij d Aj
So that
(Fsurface)i = −∫
Pij d Aj
Subramanian Fluid Dynamics
Momentum conservation - the pressure tensor
The ijth component of the pressure tensor represents the forcein the ith direction felt by an area element whose outwardnormal points in the jth direction:
(d Fsurface)i = −Pij d Aj
So that
(Fsurface)i = −∫
Pij d Aj = −∫
∂Pij
∂xjdV
So the total force on a fluid element is∫
ρg dV −∫
p dA =
∫[
ρg −∇ .P
]
dV
Subramanian Fluid Dynamics
Momentum conservation - the pressure tensor
The ijth component of the pressure tensor represents the forcein the ith direction felt by an area element whose outwardnormal points in the jth direction:
(d Fsurface)i = −Pij d Aj
So that
(Fsurface)i = −∫
Pij d Aj = −∫
∂Pij
∂xjdV
So the total force on a fluid element is∫
ρg dV −∫
p dA =
∫[
ρg −∇ .P
]
dV
where
(∇ .P)i =∂Pij
∂xj
Subramanian Fluid Dynamics
Momentum conservation - the pressure tensor
The ijth component of the pressure tensor represents the forcein the ith direction felt by an area element whose outwardnormal points in the jth direction:
(d Fsurface)i = −Pij d Aj
So that
(Fsurface)i = −∫
Pij d Aj = −∫
∂Pij
∂xjdV
So the total force on a fluid element is∫
ρg dV −∫
p dA =
∫[
ρg −∇ .P
]
dV
where
(∇ .P)i =∂Pij
∂xj
Subramanian Fluid Dynamics
Momentum conservation - putting it together
Taken together, the momentum conservation equation is
Subramanian Fluid Dynamics
Momentum conservation - putting it together
Taken together, the momentum conservation equation is
∂
∂t(ρu) +∇ . (ρuu) = −∇ .P+ ρ g
Subramanian Fluid Dynamics
Momentum conservation - putting it together
Taken together, the momentum conservation equation is
∂
∂t(ρu) +∇ . (ρuu) = −∇ .P+ ρ g
Note, uu is a second order tensor, in the sense of
a b =
a1 b1 a1 b2 a1 b3a2 b1 a2 b2 a2 b3a3 b1 a3 b2 a3 b3
.
Subramanian Fluid Dynamics
Momentum conservation: “conservative” form
Neglect body forces (g)
Subramanian Fluid Dynamics
Momentum conservation: “conservative” form
Neglect body forces (g)
The “conservative” form of the momentum conservationequation is
∂
∂t(ρu) +∇ . (ρuu+ P) = 0
Subramanian Fluid Dynamics
Momentum conservation: “conservative” form
Neglect body forces (g)
The “conservative” form of the momentum conservationequation is
∂
∂t(ρu) +∇ . (ρuu+ P) = 0
which is of the form: partial time derivative of quantity +
divergence of flux of that quantity = 0
Subramanian Fluid Dynamics
Momentum conservation: “conservative” form
Neglect body forces (g)
The “conservative” form of the momentum conservationequation is
∂
∂t(ρu) +∇ . (ρuu+ P) = 0
which is of the form: partial time derivative of quantity +
divergence of flux of that quantity = 0
Subramanian Fluid Dynamics
Momentum conservation for inviscid fluids: The Eulerequation
For an inviscid fluid, its enough to consider the scalar pressure
Pij = p δij
Subramanian Fluid Dynamics
Momentum conservation for inviscid fluids: The Eulerequation
For an inviscid fluid, its enough to consider the scalar pressure
Pij = p δij
; i.e., only the diagonal elements of the pressure tensor arenon-zero
Subramanian Fluid Dynamics
Momentum conservation for inviscid fluids: The Eulerequation
For an inviscid fluid, its enough to consider the scalar pressure
Pij = p δij
; i.e., only the diagonal elements of the pressure tensor arenon-zero
Furthermore, we expand
∂
∂t(ρu) +∇ . (ρuu) = 0
Subramanian Fluid Dynamics
Momentum conservation for inviscid fluids: The Eulerequation
For an inviscid fluid, its enough to consider the scalar pressure
Pij = p δij
; i.e., only the diagonal elements of the pressure tensor arenon-zero
Furthermore, we expand
∂
∂t(ρu) +∇ . (ρuu) = 0
and use the mass continuity equation
Subramanian Fluid Dynamics
Momentum conservation for inviscid fluids: The Eulerequation
For an inviscid fluid, its enough to consider the scalar pressure
Pij = p δij
; i.e., only the diagonal elements of the pressure tensor arenon-zero
Furthermore, we expand
∂
∂t(ρu) +∇ . (ρuu) = 0
and use the mass continuity equation
∂ρ
∂t+∇ . (ρu) = 0
Subramanian Fluid Dynamics
Momentum conservation for inviscid fluids: The Eulerequation
For an inviscid fluid, its enough to consider the scalar pressure
Pij = p δij
; i.e., only the diagonal elements of the pressure tensor arenon-zero
Furthermore, we expand
∂
∂t(ρu) +∇ . (ρuu) = 0
and use the mass continuity equation
∂ρ
∂t+∇ . (ρu) = 0
to get the Euler equation (work it out!)
ρ∂u
∂t+ ρ (u .∇)u = −∇ p + ρ g
Subramanian Fluid Dynamics
Some alternatives
If this treatment doesn’t appeal to you, consider following thethe Lagrangian treatment (sit on top of a fluid parcel andwork out momentum conservation); e.g., Arnab RaiChoudhuri’s book
Subramanian Fluid Dynamics
Some alternatives
If this treatment doesn’t appeal to you, consider following thethe Lagrangian treatment (sit on top of a fluid parcel andwork out momentum conservation); e.g., Arnab RaiChoudhuri’s book
Keep in mind, the Euler equation
ρ∂u
∂t+ ρ (u .∇)u = −∇ p + ρ g
is valid only for inviscid fluids (ones for which Pij = p δij).
Subramanian Fluid Dynamics
Some alternatives
If this treatment doesn’t appeal to you, consider following thethe Lagrangian treatment (sit on top of a fluid parcel andwork out momentum conservation); e.g., Arnab RaiChoudhuri’s book
Keep in mind, the Euler equation
ρ∂u
∂t+ ρ (u .∇)u = −∇ p + ρ g
is valid only for inviscid fluids (ones for which Pij = p δij).
Else, there are additional terms in the pressure tensor(involving viscosity), and we get the Navier-Stokes equation
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
By virtue of irrotationality, we can define a (scalar) velocitypotential φ defined by
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
By virtue of irrotationality, we can define a (scalar) velocitypotential φ defined by
u = ∇φ
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
By virtue of irrotationality, we can define a (scalar) velocitypotential φ defined by
u = ∇φ
This also has to with inviscid flows, as we’ll see later.
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
By virtue of irrotationality, we can define a (scalar) velocitypotential φ defined by
u = ∇φ
This also has to with inviscid flows, as we’ll see later.
By virtue of incompressibility, the velocity potential satsifiesLaplace’s equation
∇2 φ = 0
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
By virtue of irrotationality, we can define a (scalar) velocitypotential φ defined by
u = ∇φ
This also has to with inviscid flows, as we’ll see later.
By virtue of incompressibility, the velocity potential satsifiesLaplace’s equation
∇2 φ = 0
and we can use the solutions familiar to us from electrostatics
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..I
Lets consider flows that are incompressible (∇ .u = 0) andirrotational (ω = ∇ × u = 0)
By virtue of irrotationality, we can define a (scalar) velocitypotential φ defined by
u = ∇φ
This also has to with inviscid flows, as we’ll see later.
By virtue of incompressibility, the velocity potential satsifiesLaplace’s equation
∇2 φ = 0
and we can use the solutions familiar to us from electrostatics
Recall, the scalar potential has to be featureless
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..II
By virtue of incompressibility, we can define a (scalar)streamfunction ψ
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..II
By virtue of incompressibility, we can define a (scalar)streamfunction ψ (for 2D velocity fields, at any rate)
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..II
By virtue of incompressibility, we can define a (scalar)streamfunction ψ (for 2D velocity fields, at any rate) ..andrecall, streamlines are lines along which the streamfunction isconserved
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..II
By virtue of incompressibility, we can define a (scalar)streamfunction ψ (for 2D velocity fields, at any rate) ..andrecall, streamlines are lines along which the streamfunction isconserved
u(x , y) = −∇× (zψ)
Subramanian Fluid Dynamics
Before we move onto Navier-Stokes..II
By virtue of incompressibility, we can define a (scalar)streamfunction ψ (for 2D velocity fields, at any rate) ..andrecall, streamlines are lines along which the streamfunction isconserved
u(x , y) = −∇× (zψ)
Taken together, ∇φ .∇ψ = 0; i.e., equipotential lines andstreamlines are orthogonal.
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
..and this can be related to the stream function as
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
..and this can be related to the stream function as
ux ≡ ∂ψ
∂yuy ≡ ∂ψ
∂x
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
..and this can be related to the stream function as
ux ≡ ∂ψ
∂yuy ≡ ∂ψ
∂x
From irrotationality we get
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
..and this can be related to the stream function as
ux ≡ ∂ψ
∂yuy ≡ ∂ψ
∂x
From irrotationality we get
ux ≡ −∂φ∂x
uy ≡ −∂φ∂y
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
..and this can be related to the stream function as
ux ≡ ∂ψ
∂yuy ≡ ∂ψ
∂x
From irrotationality we get
ux ≡ −∂φ∂x
uy ≡ −∂φ∂y
Which of course gives the Cauchy-Riemann (like) condition
Subramanian Fluid Dynamics
Application of complex variables - I
For 2D flows, from incompressibility (∇ .u = 0) we get
∂ux∂x
+∂uy∂y
= 0
..and this can be related to the stream function as
ux ≡ ∂ψ
∂yuy ≡ ∂ψ
∂x
From irrotationality we get
ux ≡ −∂φ∂x
uy ≡ −∂φ∂y
Which of course gives the Cauchy-Riemann (like) condition
∂φ
∂x=∂ψ
∂yand
∂φ
∂y= −∂ψ
∂x
Subramanian Fluid Dynamics
Application of complex variables - II
Its as though ux and uy are components of a single complexvariable;
Subramanian Fluid Dynamics
Application of complex variables - II
Its as though ux and uy are components of a single complexvariable; one can therefore apply powerful techniques fromcomplex analysis to solve for the flow field
Subramanian Fluid Dynamics
Application of complex variables - II
Its as though ux and uy are components of a single complexvariable; one can therefore apply powerful techniques fromcomplex analysis to solve for the flow field
Note, you can also easily verify ∇ψ .∇φ = 0 from theCauchy-Riemann condition, and
Subramanian Fluid Dynamics
Application of complex variables - II
Its as though ux and uy are components of a single complexvariable; one can therefore apply powerful techniques fromcomplex analysis to solve for the flow field
Note, you can also easily verify ∇ψ .∇φ = 0 from theCauchy-Riemann condition, and
Also show that ∇2 ψ (in addition to ∇2 φ) = 0
Subramanian Fluid Dynamics
Application of complex variables - II
Its as though ux and uy are components of a single complexvariable; one can therefore apply powerful techniques fromcomplex analysis to solve for the flow field
Note, you can also easily verify ∇ψ .∇φ = 0 from theCauchy-Riemann condition, and
Also show that ∇2 ψ (in addition to ∇2 φ) = 0
So both the streamfunction ψ and the velocity potential φ aresolutions to Laplace’s equation
Subramanian Fluid Dynamics
Application of complex variables - II
Its as though ux and uy are components of a single complexvariable; one can therefore apply powerful techniques fromcomplex analysis to solve for the flow field
Note, you can also easily verify ∇ψ .∇φ = 0 from theCauchy-Riemann condition, and
Also show that ∇2 ψ (in addition to ∇2 φ) = 0
So both the streamfunction ψ and the velocity potential φ aresolutions to Laplace’s equation
In practice, though, one usually identifies an already knownsolution to the Laplace’s equation and then hunts for aphysical situation where its applicable!
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
i.e., normal component of velocity vanishes on the surface of thesphere (tangential slip allowed);
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
i.e., normal component of velocity vanishes on the surface of thesphere (tangential slip allowed); how do you think the boundary
condition at the sphere’s surface would look like for a viscous fluid?
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
i.e., normal component of velocity vanishes on the surface of thesphere (tangential slip allowed); how do you think the boundary
condition at the sphere’s surface would look like for a viscous fluid?
and
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
i.e., normal component of velocity vanishes on the surface of thesphere (tangential slip allowed); how do you think the boundary
condition at the sphere’s surface would look like for a viscous fluid?
andu∞ = −U x
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
i.e., normal component of velocity vanishes on the surface of thesphere (tangential slip allowed); how do you think the boundary
condition at the sphere’s surface would look like for a viscous fluid?
andu∞ = −U x or φ = U r cos θ
Subramanian Fluid Dynamics
Inviscid, incompressible, irrotational flow around a smoothsphere
The general solution for ∇2 φ = 0 in 2D polar coordinates is
φ = (A0+B0 lnr) (C0+D0 θ)+Σ
(
An rn+
Bn
rn
)
(Cn cos n θ+Dn sin n θ)
For a smooth sphere of radius a, the boundary conditions are
∂φ
∂r= 0 at r = a
i.e., normal component of velocity vanishes on the surface of thesphere (tangential slip allowed); how do you think the boundary
condition at the sphere’s surface would look like for a viscous fluid?
andu∞ = −U x or φ = U r cos θ
i.e., the velocity is “unchanged” at large distances
Subramanian Fluid Dynamics
..and the particular solution is..
φ = U cos θ
(
r +a2
r
)
Subramanian Fluid Dynamics
..and the particular solution is..
φ = U cos θ
(
r +a2
r
)
This incorporates both the boundary conditions mentioned earlier.
Subramanian Fluid Dynamics
..and the particular solution is..
φ = U cos θ
(
r +a2
r
)
This incorporates both the boundary conditions mentioned earlier.The velocity field is (from u = −∇φ)
u = −U x+ Ua2
r2(cos θ r + sin θ θ)
Subramanian Fluid Dynamics
Laminar flow around a sphere: streamlines
Subramanian Fluid Dynamics
Laplace’s eq: numerical solution
Finite-difference representation of ∇2φ = 0 (assuming equal stepsin x and y)
Subramanian Fluid Dynamics
Laplace’s eq: numerical solution
Finite-difference representation of ∇2φ = 0 (assuming equal stepsin x and y)
φi , j =1
4[φi−1 , j + φi+1 , j + φi , j−1 + φi , j+1]
Subramanian Fluid Dynamics
The Bernoulli “constant” - I
Start with the Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p + g ,
Subramanian Fluid Dynamics
The Bernoulli “constant” - I
Start with the Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p + g ,
characterize the body acceleration g using a potential g = −∇Φ,and
Subramanian Fluid Dynamics
The Bernoulli “constant” - I
Start with the Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p + g ,
characterize the body acceleration g using a potential g = −∇Φ,and recognize that (u .∇)u = 1/2∇(u .u)− u× (∇× u)
Subramanian Fluid Dynamics
The Bernoulli “constant” - I
Start with the Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p + g ,
characterize the body acceleration g using a potential g = −∇Φ,and recognize that (u .∇)u = 1/2∇(u .u)− u× (∇× u) to write(for steady flows)
∇(1
2u2)− u× (∇× u) = −1
ρ∇ p −∇Φ
Subramanian Fluid Dynamics
The Bernoulli “constant” - I
Start with the Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p + g ,
characterize the body acceleration g using a potential g = −∇Φ,and recognize that (u .∇)u = 1/2∇(u .u)− u× (∇× u) to write(for steady flows)
∇(1
2u2)− u× (∇× u) = −1
ρ∇ p −∇Φ
Lets us now (line) integrate this whole equation along astreamline element d l
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
∫
d l .
[
∇(1
2u2)− u× (∇× u) +
1
ρ∇ p +∇Φ
]
yields
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
∫
d l .
[
∇(1
2u2)− u× (∇× u) +
1
ρ∇ p +∇Φ
]
yields
1
2u2 +
∫
d p
ρ+Φ = Constant
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
∫
d l .
[
∇(1
2u2)− u× (∇× u) +
1
ρ∇ p +∇Φ
]
yields
1
2u2 +
∫
d p
ρ+Φ = Constant
This is essentially a statement of energy conservation along astreamline
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
∫
d l .
[
∇(1
2u2)− u× (∇× u) +
1
ρ∇ p +∇Φ
]
yields
1
2u2 +
∫
d p
ρ+Φ = Constant
This is essentially a statement of energy conservation along astreamline (no surprise, since we integrated force along a lineelement)
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
∫
d l .
[
∇(1
2u2)− u× (∇× u) +
1
ρ∇ p +∇Φ
]
yields
1
2u2 +
∫
d p
ρ+Φ = Constant
This is essentially a statement of energy conservation along astreamline (no surprise, since we integrated force along a lineelement)
The Bernoulli constant is yet another label for a streamline
Subramanian Fluid Dynamics
The Bernoulli “constant” - II
Since d l is along a streamline, (i.e., along u) d l is always ⊥u× (∇× u)
∫
d l .
[
∇(1
2u2)− u× (∇× u) +
1
ρ∇ p +∇Φ
]
yields
1
2u2 +
∫
d p
ρ+Φ = Constant
This is essentially a statement of energy conservation along astreamline (no surprise, since we integrated force along a lineelement)
The Bernoulli constant is yet another label for a streamline
Subramanian Fluid Dynamics
Application of Bernoulli constant: flow from an orifice
Speed of water out of a hole in a water tank at depth h:uout =
√2 g h
Subramanian Fluid Dynamics
Application of Bernoulli constant: flow over airfoil
Pbelow > Pabove, hence airfoil experiences a lift
Subramanian Fluid Dynamics
Vorticity: some applications
Recap: Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p −∇Φ ,
Subramanian Fluid Dynamics
Vorticity: some applications
Recap: Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p −∇Φ ,
and use(u .∇)u = 1/2∇(u .u)− u× (∇× u)
to write it as
Subramanian Fluid Dynamics
Vorticity: some applications
Recap: Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p −∇Φ ,
and use(u .∇)u = 1/2∇(u .u)− u× (∇× u)
to write it as
∂u
∂t+ 1/2∇(u .u)− u× (∇× u) = −1
ρ∇ p −∇Φ
Subramanian Fluid Dynamics
Vorticity: some applications
Recap: Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p −∇Φ ,
and use(u .∇)u = 1/2∇(u .u)− u× (∇× u)
to write it as
∂u
∂t+ 1/2∇(u .u)− u× (∇× u) = −1
ρ∇ p −∇Φ
Take the curl of the whole equation to get the evolution ofvorticity ω = ∇× u:
Subramanian Fluid Dynamics
Vorticity: some applications
Recap: Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p −∇Φ ,
and use(u .∇)u = 1/2∇(u .u)− u× (∇× u)
to write it as
∂u
∂t+ 1/2∇(u .u)− u× (∇× u) = −1
ρ∇ p −∇Φ
Take the curl of the whole equation to get the evolution ofvorticity ω = ∇× u:
∂ ω
∂t= ∇× (u× ω) +
1
ρ2∇ ρ×∇ p
Subramanian Fluid Dynamics
Vorticity: some applications
Recap: Euler equation
∂u
∂t+ (u .∇)u = −1
ρ∇ p −∇Φ ,
and use(u .∇)u = 1/2∇(u .u)− u× (∇× u)
to write it as
∂u
∂t+ 1/2∇(u .u)− u× (∇× u) = −1
ρ∇ p −∇Φ
Take the curl of the whole equation to get the evolution ofvorticity ω = ∇× u:
∂ ω
∂t= ∇× (u× ω) +
1
ρ2∇ ρ×∇ p
show!Subramanian Fluid Dynamics
Vorticity equation for incompressible, barotropic fluids
If p = f (ρ), (barotropic fluid) then the vorticity equation becomessimpler (because ∇ ρ and ∇ p are parallel):
Subramanian Fluid Dynamics
Vorticity equation for incompressible, barotropic fluids
If p = f (ρ), (barotropic fluid) then the vorticity equation becomessimpler (because ∇ ρ and ∇ p are parallel):
∂ ω
∂t= ∇× (u× ω)
Subramanian Fluid Dynamics
Vorticity equation for incompressible, barotropic fluids
If p = f (ρ), (barotropic fluid) then the vorticity equation becomessimpler (because ∇ ρ and ∇ p are parallel):
∂ ω
∂t= ∇× (u× ω)
In principle, this is a dynamical equation for vorticity, sinceω = ∇× u
Subramanian Fluid Dynamics
Vorticity equation for incompressible, barotropic fluids
If p = f (ρ), (barotropic fluid) then the vorticity equation becomessimpler (because ∇ ρ and ∇ p are parallel):
∂ ω
∂t= ∇× (u× ω)
In principle, this is a dynamical equation for vorticity, sinceω = ∇× u ..and specifying both the curl and divergence(∇ .u = 0) of a vector field specifies it uniquely, (caveats?) so wehave the entire velocity dynamics specified.
Subramanian Fluid Dynamics
Kelvin’s vorticity theorem
Define the circulation K =∮
u . d l =∫
(∇× u) . d A
Subramanian Fluid Dynamics
Kelvin’s vorticity theorem
Define the circulation K =∮
u . d l =∫
(∇× u) . d A
Write the Euler equation using the material derivative:
Subramanian Fluid Dynamics
Kelvin’s vorticity theorem
Define the circulation K =∮
u . d l =∫
(∇× u) . d A
Write the Euler equation using the material derivative:
d u
dt=
1
ρ∇p +∇Φ
Subramanian Fluid Dynamics
Kelvin’s vorticity theorem
Define the circulation K =∮
u . d l =∫
(∇× u) . d A
Write the Euler equation using the material derivative:
d u
dt=
1
ρ∇p +∇Φ
From the definition of K , we can write its material derivativeas
d K
dt=
∮
d u
dt. d l+
∮
u .d
dtd l
Subramanian Fluid Dynamics
Kelvin’s vorticity theorem
Define the circulation K =∮
u . d l =∫
(∇× u) . d A
Write the Euler equation using the material derivative:
d u
dt=
1
ρ∇p +∇Φ
From the definition of K , we can write its material derivativeas
d K
dt=
∮
d u
dt. d l+
∮
u .d
dtd l
Subramanian Fluid Dynamics
When is the circulation conserved?
Using the Euler equation,
d K
dt=
∮
∇Φ . d l+
∮
dp
ρ+
∮
u .d l
dt
Subramanian Fluid Dynamics
When is the circulation conserved?
Using the Euler equation,
d K
dt=
∮
∇Φ . d l+
∮
dp
ρ+
∮
u .d l
dt
Each term is a perfect differential (show!), so the integral overa closed path is zero.
Subramanian Fluid Dynamics
When is the circulation conserved?
Using the Euler equation,
d K
dt=
∮
∇Φ . d l+
∮
dp
ρ+
∮
u .d l
dt
Each term is a perfect differential (show!), so the integral overa closed path is zero.
So circulation is conserved in an inviscid, barotropic fluid. Ifcirculation = 0 to begin with, it will always remain that way.
Subramanian Fluid Dynamics
Irrotational flow around a cylinder
Subramanian Fluid Dynamics
Irrotational flow + circulation around a cylinder: themagnus effect
Subramanian Fluid Dynamics
Irrotational flow + circulation around a cylinder: themagnus effect
Pbelow > Pabove:
Subramanian Fluid Dynamics
Irrotational flow + circulation around a cylinder: themagnus effect
Pbelow > Pabove: so spinning ball experiences a “lift” (why?).
Subramanian Fluid Dynamics
Irrotational flow + circulation around a cylinder: themagnus effect
Pbelow > Pabove: so spinning ball experiences a “lift” (why?). Thedirection of the force is ⊥ the rotation axis as well as the (local)flow. Remind you of Lorentz forces?
Subramanian Fluid Dynamics
Viscosity
Subramanian Fluid Dynamics
Viscosity
Consider viscous flow between two (unbounded) parallel plates.
Subramanian Fluid Dynamics
Viscosity
Consider viscous flow between two (unbounded) parallel plates.Due to viscosity, the flow doesn’t slip at the boundaries;
Subramanian Fluid Dynamics
Viscosity
Consider viscous flow between two (unbounded) parallel plates.Due to viscosity, the flow doesn’t slip at the boundaries; it sticks.
Subramanian Fluid Dynamics
Momentum equation with viscosity
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
But there is also a “frictional” opposing force (in steadystate), else the fluid would accelerate
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
But there is also a “frictional” opposing force (in steadystate), else the fluid would accelerate
Recall, for Newtonian fluids, the viscous stress is proportionalto the velocity gradient (strain): τ = µ du/dy
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
But there is also a “frictional” opposing force (in steadystate), else the fluid would accelerate
Recall, for Newtonian fluids, the viscous stress is proportionalto the velocity gradient (strain): τ = µ du/dy
So, per unit length in the z-direction (i.e., per dz), the forcebalance reads
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
But there is also a “frictional” opposing force (in steadystate), else the fluid would accelerate
Recall, for Newtonian fluids, the viscous stress is proportionalto the velocity gradient (strain): τ = µ du/dy
So, per unit length in the z-direction (i.e., per dz), the forcebalance reads
[
µ
(
du
dy
)
y+dy
− µ
(
du
dy
)
y
]
=
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
But there is also a “frictional” opposing force (in steadystate), else the fluid would accelerate
Recall, for Newtonian fluids, the viscous stress is proportionalto the velocity gradient (strain): τ = µ du/dy
So, per unit length in the z-direction (i.e., per dz), the forcebalance reads
[
µ
(
du
dy
)
y+dy
− µ
(
du
dy
)
y
]
= µd2u
dy2dy =
Subramanian Fluid Dynamics
Momentum equation with viscosity
Fluid flows along the x-direction due to the pressure gradientdp/dx
But there is also a “frictional” opposing force (in steadystate), else the fluid would accelerate
Recall, for Newtonian fluids, the viscous stress is proportionalto the velocity gradient (strain): τ = µ du/dy
So, per unit length in the z-direction (i.e., per dz), the forcebalance reads
[
µ
(
du
dy
)
y+dy
− µ
(
du
dy
)
y
]
= µd2u
dy2dy =
dp
dxdy
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
More generally,
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
More generally,
µ∇2u = ∇ p
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
More generally,
µ∇2u = ∇ p
With this additional term due to viscous stresses, the fullmomentum equation reads
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
More generally,
µ∇2u = ∇ p
With this additional term due to viscous stresses, the fullmomentum equation reads (Euler equation plus viscous stressterm)
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
More generally,
µ∇2u = ∇ p
With this additional term due to viscous stresses, the fullmomentum equation reads (Euler equation plus viscous stressterm)
ρ∂u
∂t+ ρ (u .∇)u = −∇ p + µ∇2u+ ρ g
Subramanian Fluid Dynamics
The Navier-Stokes equation
So
µd2u
dy2=
dp
dx
More generally,
µ∇2u = ∇ p
With this additional term due to viscous stresses, the fullmomentum equation reads (Euler equation plus viscous stressterm)
ρ∂u
∂t+ ρ (u .∇)u = −∇ p + µ∇2u+ ρ g
This is the Navier-Stokes equation.
Subramanian Fluid Dynamics
Navier-Stokes ...a little more formally
Remember, we can have normal, as well as tangential stresses oneach face in a fluid volume element:
Subramanian Fluid Dynamics
Navier-Stokes ...a little more formally
Remember, we can have normal, as well as tangential stresses oneach face in a fluid volume element:
Subramanian Fluid Dynamics
Navier-Stokes ...a little more formally
Remember, we can have normal, as well as tangential stresses oneach face in a fluid volume element:
...so instead of Pij = p δij , which is okay for inviscid fluids, we’dhave Pij = p δij + σij for viscous fluids.
Subramanian Fluid Dynamics
Navier-Stokes ...a little more formally
Remember, we can have normal, as well as tangential stresses oneach face in a fluid volume element:
...so instead of Pij = p δij , which is okay for inviscid fluids, we’dhave Pij = p δij + σij for viscous fluids. As always, Pij is the forcein the ith direction on the face whose outward normal is in the jthdirection.
Subramanian Fluid Dynamics
Navier-Stokes ...a little more formally
Remember, we can have normal, as well as tangential stresses oneach face in a fluid volume element:
...so instead of Pij = p δij , which is okay for inviscid fluids, we’dhave Pij = p δij + σij for viscous fluids. As always, Pij is the forcein the ith direction on the face whose outward normal is in the jthdirection. p is still the thermodynamic pressure, but there are somecaveats
Subramanian Fluid Dynamics
So what would σij look like?
Subramanian Fluid Dynamics
So what would σij look like?
Recall, for Newtonian fluids,
τ = µdu
dy
Subramanian Fluid Dynamics
So what would σij look like?
Recall, for Newtonian fluids,
τ = µdu
dy
..so σij will involve velocity derivatives like dui/dxj
Subramanian Fluid Dynamics
..σij ..some more..
∂ui∂xj
=1
2
(
∂ui∂xj
+∂uj∂xi
)
+1
2
(
∂ui∂xj
− ∂uj∂xi
)
Subramanian Fluid Dynamics
..σij ..some more..
∂ui∂xj
=1
2
(
∂ui∂xj
+∂uj∂xi
)
+1
2
(
∂ui∂xj
− ∂uj∂xi
)
The second term represents rigid body rotation (note, u = Ω× ris rigid body rotation) and therefore doesn’t involve shear stresses(show)
Subramanian Fluid Dynamics
..σij ..some more..
∂ui∂xj
=1
2
(
∂ui∂xj
+∂uj∂xi
)
+1
2
(
∂ui∂xj
− ∂uj∂xi
)
The second term represents rigid body rotation (note, u = Ω× ris rigid body rotation) and therefore doesn’t involve shear stresses(show) If we insist on Newtonian fluids, the most general secondrank tensor involving velocity gradients is of the form
σij = a
(
∂ui∂xj
+∂uj∂xi
)
+ b δij ∇ .u
Subramanian Fluid Dynamics
...furthermore..
Recall Pij = pδij + σij .
Subramanian Fluid Dynamics
...furthermore..
Recall Pij = pδij + σij . If we take p to be the (scalar)thermodynamic pressure, it contains the trace of Pij ; i.e.,
Subramanian Fluid Dynamics
...furthermore..
Recall Pij = pδij + σij . If we take p to be the (scalar)thermodynamic pressure, it contains the trace of Pij ; i.e.,
p =1
3Pii
Subramanian Fluid Dynamics
...furthermore..
Recall Pij = pδij + σij . If we take p to be the (scalar)thermodynamic pressure, it contains the trace of Pij ; i.e.,
p =1
3Pii
...in which case σij had better be traceless.
Subramanian Fluid Dynamics
...furthermore..
Recall Pij = pδij + σij . If we take p to be the (scalar)thermodynamic pressure, it contains the trace of Pij ; i.e.,
p =1
3Pii
...in which case σij had better be traceless. The only way this canhappen is if b = −(2/3)a, by which
σij = −µ(
∂ui∂xj
+∂uj∂xi
− 2
3δij∇ .u
)
Subramanian Fluid Dynamics
..putting it all together
ρdui
dt= ρgi −
∂p
∂xi+
∂
∂xj
[
µ
(
∂ui∂xj
+∂uj∂xi
− 2
3δij ∇ .u
)]
Subramanian Fluid Dynamics
..putting it all together
ρdui
dt= ρgi −
∂p
∂xi+
∂
∂xj
[
µ
(
∂ui∂xj
+∂uj∂xi
− 2
3δij ∇ .u
)]
..if µ is isotropic, it can be considered to be a scalar, and we get
Subramanian Fluid Dynamics
..putting it all together
ρdui
dt= ρgi −
∂p
∂xi+
∂
∂xj
[
µ
(
∂ui∂xj
+∂uj∂xi
− 2
3δij ∇ .u
)]
..if µ is isotropic, it can be considered to be a scalar, and we get
ρdu
dt= ρg −∇p + µ
[
∇2u+1
3∇(∇ .u)
]
Subramanian Fluid Dynamics
..putting it all together
ρdui
dt= ρgi −
∂p
∂xi+
∂
∂xj
[
µ
(
∂ui∂xj
+∂uj∂xi
− 2
3δij ∇ .u
)]
..if µ is isotropic, it can be considered to be a scalar, and we get
ρdu
dt= ρg −∇p + µ
[
∇2u+1
3∇(∇ .u)
]
..and for incompressible flows we recover
Subramanian Fluid Dynamics
..putting it all together
ρdui
dt= ρgi −
∂p
∂xi+
∂
∂xj
[
µ
(
∂ui∂xj
+∂uj∂xi
− 2
3δij ∇ .u
)]
..if µ is isotropic, it can be considered to be a scalar, and we get
ρdu
dt= ρg −∇p + µ
[
∇2u+1
3∇(∇ .u)
]
..and for incompressible flows we recover
du
dt= g − 1
ρ∇p +
µ
ρ∇2u
Subramanian Fluid Dynamics
Finally, the energy equation
We have already seen energy conservation in the guise of theBernoulli constant, but now lets include viscosity (we followLandau & Lifshitz here)
Subramanian Fluid Dynamics
Finally, the energy equation
We have already seen energy conservation in the guise of theBernoulli constant, but now lets include viscosity (we followLandau & Lifshitz here)
Recall, Pij = pδij + σij , and for incompressible fluids, theviscous stress tensor is
Subramanian Fluid Dynamics
Finally, the energy equation
We have already seen energy conservation in the guise of theBernoulli constant, but now lets include viscosity (we followLandau & Lifshitz here)
Recall, Pij = pδij + σij , and for incompressible fluids, theviscous stress tensor is
σij = −µ(
∂ui∂xj
+∂uj∂xi
)
Subramanian Fluid Dynamics
Finally, the energy equation
We have already seen energy conservation in the guise of theBernoulli constant, but now lets include viscosity (we followLandau & Lifshitz here)
Recall, Pij = pδij + σij , and for incompressible fluids, theviscous stress tensor is
σij = −µ(
∂ui∂xj
+∂uj∂xi
)
The Navier-Stokes equation can be equivalently be written as
Subramanian Fluid Dynamics
Finally, the energy equation
We have already seen energy conservation in the guise of theBernoulli constant, but now lets include viscosity (we followLandau & Lifshitz here)
Recall, Pij = pδij + σij , and for incompressible fluids, theviscous stress tensor is
σij = −µ(
∂ui∂xj
+∂uj∂xi
)
The Navier-Stokes equation can be equivalently be written as
∂ui∂t
= −uk∂ui∂xk
− 1
ρ
∂p
∂xi+
1
ρ
∂σik∂xk
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time.
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time. In other words, evaluate
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time. In other words, evaluate
∂(1/2ρu2)
∂t= ρui
∂ui∂t
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time. In other words, evaluate
∂(1/2ρu2)
∂t= ρui
∂ui∂t
Using the form of Navier-Stokes
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time. In other words, evaluate
∂(1/2ρu2)
∂t= ρui
∂ui∂t
Using the form of Navier-Stokes
∂ui∂t
= −uk∂ui∂xk
− 1
ρ
∂p
∂xi+
1
ρ
∂σik∂xk
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time. In other words, evaluate
∂(1/2ρu2)
∂t= ρui
∂ui∂t
Using the form of Navier-Stokes
∂ui∂t
= −uk∂ui∂xk
− 1
ρ
∂p
∂xi+
1
ρ
∂σik∂xk
we get
∂(1/2ρu2)
∂t= −ρu . (u .∇)u− u .∇p + ui
∂σik∂xk
Subramanian Fluid Dynamics
Energy equation ..cont’d
We’re interested in knowing how the kinetic energy 1/2ρu2 of aparcel of fluid evolves with time. In other words, evaluate
∂(1/2ρu2)
∂t= ρui
∂ui∂t
Using the form of Navier-Stokes
∂ui∂t
= −uk∂ui∂xk
− 1
ρ
∂p
∂xi+
1
ρ
∂σik∂xk
we get
∂(1/2ρu2)
∂t= −ρu . (u .∇)u− u .∇p + ui
∂σik∂xk
=
−ρ(u .∇)
(
1
2u2 +
p
ρ
)
+ div(u . σ)− σik∂ui∂xk
Subramanian Fluid Dynamics
Energy equation..cont’d
Since we’re considering incompressible fluids (∇ .u = 0),
Subramanian Fluid Dynamics
Energy equation..cont’d
Since we’re considering incompressible fluids (∇ .u = 0),
∂(1/2ρu2)
∂t= −∇ .
[
ρu
(
1
2u2 +
p
ρ
)
− u . σ
]
− σik∂ui∂xk
Subramanian Fluid Dynamics
Energy equation..cont’d
Since we’re considering incompressible fluids (∇ .u = 0),
∂(1/2ρu2)
∂t= −∇ .
[
ρu
(
1
2u2 +
p
ρ
)
− u . σ
]
− σik∂ui∂xk
Integrating over a macroscopic volume (and bounding area),
Subramanian Fluid Dynamics
Energy equation..cont’d
Since we’re considering incompressible fluids (∇ .u = 0),
∂(1/2ρu2)
∂t= −∇ .
[
ρu
(
1
2u2 +
p
ρ
)
− u . σ
]
− σik∂ui∂xk
Integrating over a macroscopic volume (and bounding area),
∫
∂
∂t(1/2ρu2) dV = −
∮[
ρu
(
1
2u2+
p
ρ
)
−u . σ
]
. dA−∫
σik∂ui∂xk
dV
Subramanian Fluid Dynamics
Energy equation..cont’d
Since we’re considering incompressible fluids (∇ .u = 0),
∂(1/2ρu2)
∂t= −∇ .
[
ρu
(
1
2u2 +
p
ρ
)
− u . σ
]
− σik∂ui∂xk
Integrating over a macroscopic volume (and bounding area),
∫
∂
∂t(1/2ρu2) dV = −
∮[
ρu
(
1
2u2+
p
ρ
)
−u . σ
]
. dA−∫
σik∂ui∂xk
dV
The first term on the RHS vanishes (why?)
Subramanian Fluid Dynamics
..leaving, finally
Ekin = −∫
σik∂ui∂xk
dV =
Subramanian Fluid Dynamics
..leaving, finally
Ekin = −∫
σik∂ui∂xk
dV = − 1
2
∫
σik
(
∂ui∂xk
+∂uk∂xi
)
dV
Subramanian Fluid Dynamics
..leaving, finally
Ekin = −∫
σik∂ui∂xk
dV = − 1
2
∫
σik
(
∂ui∂xk
+∂uk∂xi
)
dV
and substituting for σik ,
σik = −µ(
∂ui∂xk
+∂uk∂xi
)
Subramanian Fluid Dynamics
..leaving, finally
Ekin = −∫
σik∂ui∂xk
dV = − 1
2
∫
σik
(
∂ui∂xk
+∂uk∂xi
)
dV
and substituting for σik ,
σik = −µ(
∂ui∂xk
+∂uk∂xi
)
we get
Ekin = −1
2µ
∫(
∂ui∂xk
+∂uk∂xi
)2
dV
Subramanian Fluid Dynamics