Theory and modeling of multiphase flows

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Theory and modeling of multiphase flows
Payman Jalali Department of Energy and
Environmental Technology Lappeenranta University
of Technology Lappeenranta, Finland Fall 2006
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Vectors in Cartesian coordinate system
Exercise 1 The temperature near the surface of a
bubble is given by T(R/R0)T0, where R0 is the
radius of the bubble and R is the radial position
of a point near the surface of the bubble. T0 is
the temperature at the surface of the bubble.
Assume that the bubble can be considered
two-dimensional. What is the heat flux at the
bubble surface?
Solution The heat flux at the surface is
The systematic way of solution
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Vectors in Cartesian coordinate system
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Tensor algebra, summation convention and tensor
notations
Exercise 2 Which statement for the substantial
derivative is correct?
x
x
x
Time derivative is not written by index
convention. Twice repeated index represents the
convective derivative because it is a summation
over all three directions.
x
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Tensor algebra, summation convention and tensor
notations
Exercise 3 Which one describes the continuity
equation for single phase flow?
x
x
x
x
There is only one equation of continuity. No free
index should exist. The second term represents
the divergence of velocity.
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Tensor algebra, summation convention and tensor
notations
Exercise 4 Which one describes the momentum
equation for single phase flow?
i is the free index for each component of
velocity. Tji is the stress tensor.
x
x
x
x
x
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Tensor algebra, summation convention and tensor
notations
Exercise 5 How the momentum equation will change
in an inviscid flow? If there is no viscous
forces in the flow (inviscid flow) the stress
tensor will only have the hydrostatic term
Only 1 term remains, that is, if j i.
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Tensor algebra, summation convention and tensor
notations
Exercise 6 What is dyadic product of a second
order tensor and a vector?
k1
k2
k3
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Tensor algebra, summation convention and tensor
notations
Exercise 7 What is the contraction product of a
vector and a second order tensor?
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Tensor algebra, summation convention and tensor
notations
Exercise 8 Expand different components of the
curl of the tensor T.
For example, if iq1
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Tensor algebra, summation convention and tensor
notations
Verify this result is working if T is a vector.
In case of a vector, there will be only 1 index
for T
From the last answer for A11 we can write
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Tensor algebra, summation convention and tensor
notations
Exercise 9 Write tensorial form of this
expression