TrajectoriesTrajectories

Eulerian ViewEulerian View

In the Lagrangian view each In the Lagrangian view each body is described at each body is described at each point in space.point in space.• Difficult for a fluid with many Difficult for a fluid with many

particles.particles.

In the Eulerian view the In the Eulerian view the points in space are points in space are described.described.• Bulk properties of density Bulk properties of density

and velocityand velocity

),( 0 trr

),( tr

),( trv

Fluid ChangeFluid Change

A change in a property like A change in a property like pressure depends on the pressure depends on the view.view.

In the Lagrangian view the In the Lagrangian view the total time derivative depends total time derivative depends on position and time.on position and time.

The Eulerian view uses just The Eulerian view uses just the partial derivative with the partial derivative with time.time.• Points in space are fixedPoints in space are fixed

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p

dt

dp

dt

dz

z

p

dt

dy

y

p

dt

dx

x

p

t

p

dt

dp

vtdt

d

t

p

dt

dp

constr

CompressibilityCompressibility

A change in pressure on a A change in pressure on a fluid can cause deformation.fluid can cause deformation.

Compressibility measures Compressibility measures the relationship between the relationship between volume change and volume change and pressure.pressure.• Usually expressed as a bulk Usually expressed as a bulk

modulus modulus BB

Ideal liquids are Ideal liquids are incompressible.incompressible.

V

p

pV

V

1

V

pVB

Volume ChangeVolume Change

Consider a fixed amount of Consider a fixed amount of fluid in a volume fluid in a volume VV..• Cubic, Cartesian geometryCubic, Cartesian geometry

• Dimensions Dimensions xx, , yy, , zz..

The change in The change in VV is related is related to the divergence.to the divergence.• Incompressible fluids must Incompressible fluids must

have no velocity divergencehave no velocity divergence

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vz

dt

d

yy

vy

dt

d

xx

vx

dt

d

z

y

x

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y

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x

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dt

d zyx

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d

Jacobian TensorJacobian Tensor

A general coordinate A general coordinate transformation can be transformation can be expressed as a tensor.expressed as a tensor.• Partial derivatives between Partial derivatives between

two systemstwo systems

• JacobianJacobian NNNN real matrix real matrix

• Inverse for nonsingular Inverse for nonsingular Jacobians.Jacobians.

Cartesian coordinate Cartesian coordinate transformations have an transformations have an additional symmetry.additional symmetry.• Not generally true for other Not generally true for other

transformationstransformations

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q

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i

q

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j

iij x

x

x

xJ

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x cos

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i

q

xi

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x

qq

Transformation GradientTransformation Gradient

The components of a The components of a gradient of a scalar do not gradient of a scalar do not transform like a position transform like a position vector.vector.• Inverse transformationInverse transformation

• CovariantCovariant behavior behavior

• Position is Position is contravariantcontravariant

Gradients use a shorthand Gradients use a shorthand index notation.index notation. ii ,

ii xe

i

m

mii x

q

qx

mm qe

mm

ii q

q

xx

i

jij x

vv

,

Volume ElementVolume Element

An infinitessimal volume An infinitessimal volume element is defined by element is defined by coordinates.coordinates.• dVdV = = dxdx11dxdx22dxdx33

Transform a volume element Transform a volume element from other coordinates.from other coordinates.• components from the components from the

transformationtransformation

The Jacobian determinant is The Jacobian determinant is the ratio of the volume the ratio of the volume elements.elements.

x1

x2

x3

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xq

q

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Continuity EquationContinuity Equation

A mass element must A mass element must remain constant in time.remain constant in time.• Conservation of massConservation of mass

Combine with divergence Combine with divergence relationship.relationship.

Write in terms of a point in Write in terms of a point in space.space.

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0 Vdt

dm

dt

d

0

VvVdt

ddt

VdV

dt

d

0 vdt

d

0

vvt

0)(

vt

StreamlinesStreamlines

A A streamlinestreamline follows the follows the tangents to fluid velocity.tangents to fluid velocity.• Lagrangian viewLagrangian view

• Dashed lines at leftDashed lines at left

• Stream tube follows an areaStream tube follows an area

A A streaklinestreakline (blue) shows the (blue) shows the current position of a particle current position of a particle starting at a fixed point.starting at a fixed point.

A pathline (red) tracks an A pathline (red) tracks an individual particle.individual particle.

Wikimedia image

Rotational FlowRotational Flow

The curl of velocity The curl of velocity measures rotation per unit measures rotation per unit area.area.• Stokes’ theoremStokes’ theorem

Fluid with zero curl is Fluid with zero curl is irrotational.irrotational.• Transform to rotating Transform to rotating

system with zero curlsystem with zero curl

• Defines angular velocityDefines angular velocity

next

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2

3

v

v

rrv

rvv

rvv

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