Equations of Continuity
description
Transcript of Equations of Continuity
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Equations of Continuity
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Outline
1.Time Derivatives & Vector Notation
2.Differential Equations of Continuity
3.Momentum Transfer Equations
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Introduction
FLUID
In order to calculate forces exerted by a moving fluid as well as the consequent transport effects, the dynamics of flow must be described mathematically (kinematics).
Continuousmedium
Infinitesimal pieces of fluid
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Eulerian Perspective – the flow as seen at fixed locations in space, or over fixed volumes of space (the perspective of most analysis)
Lagrangian Perspective – the flow as seen by the fluid material (the perspective of the laws of motion)Control volume: finite fixed region of space (Eulerian)
Coordinate: fixed point in space (Eulerian)
Fluid system: finite piece of the fluid material (Lagrangian)
Fluid particle: differentially small finite piece of the fluid material (Lagrangian)
Perspectives of Fluid Motion
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Lagrangian Perspective
The motion of a fluid particle is relative to a specific initial position in space at an initial time.
1 , 2 , 3 ,
position: ( , , ) ( , , ) ( , , ) p p p p p p p p px f x y z t y f x y z t z f x y z t
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Lagrangian Perspective
x
y
zLagrangian coordinate system
pathline
position vector p p px y zr i j k
1 ,
2 ,
3 ,
( , , )velocity:
( , , )
( , , )
p p px
p p py
p p pz
f x y z tv
tf x y z t
vt
f x y z tv
t
partial (local) time derivatives
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Lagrangian Perspective
x
y
z
t = t1 t = t2y
xz
1 1 1, 1
2 1 1, 1
3 1 1, 1
position 1: ( , , )
( , , )
( , , )
p p p
p p p
p p p
x f x y z t
y f x y z t
z f x y z t
1 2 2, 2
2 2 2, 2
3 2 2, 2
position 2: ( , , )
( , , )
( , , )
p p p
p p p
p p p
x f x y z t
y f x y z t
z f x y z t
Consider a small fluid element with a mass concentration moving through Cartesian space:
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Lagrangian Perspective
x
y
z
t = t1 t = t2y
xz
Consider a small fluid element with a mass concentration moving through Cartesian space:
1 1 1, 1
concentration 1: ( , , ) p p px y z t 2 2 2, 2
concentration 2: ( , , ) p p px y z t
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Lagrangian Perspective
Total change in the mass concentration with respect to time:
2 1 2 1 2 1
2 1 2 1 2 1
2 1 2 1
2 1 2 1
t t x xt t t t t x t t
y y z zy t t z t t
If the timeframe is infinitesimally small:
2 1 2 1
2 1 2 1
2 1 2 1
2 1 2 1
2 1 2 1
2 1 2 1
lim lim
lim lim
t t t t
t t t t
x xt t t x t t
y y z zy t t z t t
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Lagrangian Perspective
Total Time Derivative
d x y zdt t x t y t z t
x y z
D v v vDt t x y z
local derivative convective derivative
Substantial Time Derivative
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Lagrangian Perspective
x y z
D v v vDt t x y z
DDt t
v
x y zv v v
x y z
v i j k
i j k
vector notation
stream velocity
gradient
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Lagrangian Perspective
Problem with the Lagrangian Perspective
The concept is pretty straightforward but very difficult to implement (since to describe the whole fluid motion, kinematics must be applied to ALL of the moving particles), often would produce more information than necessary, and is not often applicable to systems defined in fluid mechanics.
DDt t
v
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Eulerian Perspective
flow
x
y
z
Motion of a fluid as a continuum
Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z).
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Eulerian Perspective
flow
x
y
z
Motion of a fluid as a continuum
Velocity expressed as a function of time t and spatial position (x, y, z)
Eulerian coordinate system
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Eulerian Perspective
Difference from the Lagrangian approach:
Eulerian Lagrangian
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Eulerian Perspective
Difference from the Lagrangian approach:
Eulerian
Lagrangian
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Outline
1.Time Derivatives & Vector Notation
2.Differential Equations of Continuity
3.Momentum Transfer Equations
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Equation of Continuity
differential control volume:
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Differential Mass Balance
Rate of Rate of Rate ofaccumulation mass in mass out
mass balance:
Rate ofmass in
x y zx zyv y z v x z v x y
Rate ofmass out
x y zx x z zy yv y z v x z v x y
Rate of mass accumulation
x y zt
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Differential Mass Balance
Substituting:
x y zx zy
x y zx x z zy y
x y z v y z v x z v x yt
v y z v x z v x y
Rearranging:
x xx x x
y yy y y
z zz z z
x y z v v y zt
v v x z
v v x y
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Differential Equation of Continuity
Dividing everything by ΔV:
Taking the limit as ∆x, ∆y and ∆z 0:
y yx x z zy y yx x x z z zv vv v v v
t x y z
yx zvv vt x y z
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Differential Equation of Continuity
yx zvv v
t x y z
v
divergence of mass velocity vector (v)
Partial differentiation:
yx zx y z
vv v v v vt x y z x y z
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Differential Equation of Continuity
Rearranging:
yx zx y z
vv vv v vt x y z x y z
substantial time derivative
yx zvv vD
Dt x y zv
If fluid is incompressible: 0 v
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Differential Equation of Continuity
In cylindrical coordinates:
1 1 0
r zrv v vd
dt r r r z
2 2 1where , tan yr x yx
If fluid is incompressible:
1 0
r r zvv v vr r r z
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Outline
1.Time Derivatives & Vector Notation
2.Differential Equations of Continuity
3.Momentum Transfer Equations
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Differential Equations of Motion
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Control Volume
For 1D fluid flow, momentum transport occurs in 3 directions
Fluid is flowing in 3 directions
Momentum transport is fully defined by 3 equations of motion
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Momentum Balance
Sum of forcesRate of Rate of Rate of
acting in accumulation momentum in momentum out
the systemx x xx
convective
Rate of Rate ofmomentum in momentum out
Rate of Rate of
momentum in momentum out
Rate of
momentum in
x x
x x
molecular
Rate ofmomentum outx x
Consider the x-component of the momentum transport:
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Momentum Balance
convective
Rate of Rate ofmomentum in momentum out
x x x xx x
x x
xv v v v y z
y x y xy y y
z x z xz z z
v v v v x z
v v v v x y
Due to convective transport:
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Momentum Balance
molecular
Rate of Rate ofmomentum in momentum out
xx xxx x x
yx yx y
x
y
x
y z
y
zx zxz z z
x z
x y
Due to molecular transport:
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Momentum Balance
Sum of forcesRate of Rate of Rate of
acting in accumulation momentum in momentum out
the systemx x xx
Sum of forces
acting in the system
x x x x
x
p p y z g x y z
Consider the x-component of the momentum transport:
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Momentum Balance
Sum of forcesRate of Rate of Rate of
acting in accumulation momentum in momentum out
the systemx x xx
Rate ofaccumulation
x
x
vx y z
t
Consider the x-component of the momentum transport:
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Differential Momentum Balance
Substituting:
x x x xx x x
y x y xy y y
z x
xx xxx x
x
z
x
xz z z
y z
vx y z
tv v v v y z
v v v v x z
v v v v x y
yx yxy y y
z
x x x x
x zxz z z
x z
x
p p y z g x y z
y
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Differential Momentum Balance
Dividing everything by ΔV:
y x y xx x x x y y yx x x
z x xx xxx x xz
yx yx zx zxy y y z z z
x x
xz z
x
z
xx
p pg
x
x
y
v v v vv v v v
x y
v v
z
t
v v
z
v
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Differential Equation of Motion
Taking the limit as ∆x, ∆y and ∆z 0:
y xx x
yxxx z
z x
xx
x
x y zp gx
v vv v v vx y z
vt
Rearranging:
y xx x
xyxxx x
z
z
xxv vv v v vv
t
x yg
x y
p
z
z x
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Differential Momentum Balance
For the convective terms:
y xx x z x
yx zx x xx y z x
v vv v v vx y z
vv vv v vv v v vx y z x y z
For the accumulation term:
x xx
yx x zx x y z
v v vt t t
vv v vv v v vt x y z x y z
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Differential Equation of Motion
Substituting:
x x x
yx x z
x
x x
y z
yx z
yxxx z
y
x
z
xx
vv v
v v vv v v
x y z
vv vv
x y z
vv v v vt x y z x
x y z
z
g
y
px
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Differential Equation of Motion
x xx xx y
yxxx zxx
zv v vv v vx y z
v
gx y x
t
zp
EQUATION OF MOTION FOR THE x-COMPONENT
Substituting:
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Differential Equation of Motion
Substituting:
EQUATION OF MOTION FOR THE y-COMPONENT
y yy yx y
xy yy zyy
z
v v vv v v
x y z
xg
v
y z
t
py
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Differential Equation of Motion
Substituting:
EQUATION OF MOTION FOR THE z-COMPONENT
z zz zx y
yzxz zzz
zv v vv v vx y z
v
gx y z
t
zp
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Differential Equation of Motion
Substantial time derivatives:
yxxx zxxx
xy yy zyyy
yzxz zzzz
Dv p gDt x y z x
Dv p gDt x y z y
Dv p gDt x y z z
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Differential Equation of Motion
In vector-matrix notation:
yxxx zx
x xxy yy zy
y y
z z
yzxz zz
px y z xv g
D pv gDt x y z y
v gpzx y z
D pDt
v g
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Differential Equation of Motion
Cauchy momentum equation• Equation of motion for a pure fluid• Valid for any continuous medium (Eulerian)• In order to determine velocity distributions, shear
stress must be expressed in terms of velocity gradients and fluid properties (e.g. Newton’s law)
D pDt
v τ g
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Cauchy Stress Tensor
Stress distribution: normal stresses
shear stresses
xx
yy
zz
xy yx
xz zx
yz zy
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Cauchy Stress Tensor
Stokes relations (based on Stokes’ hypothesis)
2232232= 23
xxx
yyy
zzz
vxvyvz
v
v
v
yxxy yx
x zxz zx
y zyz zy
vvy x
v vz xv vz y
where yx zvv v
x y z
v
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Navier-Stokes Equations
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Assumptions
1. Newtonian fluid2. Obeys Stokes’ hypothesis3. Continuum4. Isotropic viscosity5. Constant density
Divergence of the stream velocity is zero
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Navier-Stokes Equations
Applying the Stokes relations per component:
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
yxxx zx x x x
xy yy zy y y y
yzxz zz z z z
v v vx y z x y z
v v vx y z x y z
v v vx y z x y z
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Navier-Stokes Equations
Navier-Stokes equations in rectangular coordinates2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
x x x xx
y y y yy
z z z zz
Dv v v v p gDt x y z x
Dv v v v p gDt x y z y
Dv v v v p gDt x y z z
2D pDt
v g v
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Cylindrical Coordinates
2
2 2
2 2 2 2
2
1 1 2
1
r r r rr z
r r r rr
r r r rr z
v vv v v vv vt r r r z
rv v v vp gr r r r r r z
v vv v v vv vt r r r z
pr
2 2
2 2 2 2
2
2 2
2 2 2
1 1 2
1 1
r r r rr z
z z zz
rv v v vg
r r r r r z
v vv v v vv vt r r r z
v v vp g rz r r r r z
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Applications of Navier-Stokes Equations
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
1. Steady state flow
0
0 all components
tvt
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
2. Unidirectional flow
2 2 2 2
2 2 2 2
flow along -direction only
x x x xv v v vx y z x
x
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
2. Unidirectional flow
2 2
2 2 2 2
2
2
1 1 2
flow along -direction only
rv v v vr r r r r z
vz
z
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
3. Constant fluid properties • Isotropy (independent of position/direction)• Independent with temperature and pressure
incompressible fluid: 0 from the equation of continuity v
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
4. No viscous dissipation (INVISCID FLOW)
0 0D pDt
τv g Euler’s Equation
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
5. No external forces acting on the system
0 no external pressure gradient
0 no gravity effects
p
g
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Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
6. Laminar flow
x x x x xx y z
Dv v v v vv v vDt t x y z
x xDv vDt t
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Example
Derive the equation giving the velocity distribution at steady state for laminar flow of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates. The velocity profile desired is at a point far from the inlet or outlet of the channel. The two plates will be considered to be fixed and of infinite width, with flow driven by the pressure gradient in the x-direction.
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Example
Derive the equation giving the velocity distribution at steady state for laminar, downward flow in a vertical pipe of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates.