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Diffusion+Advection+Reaction
(solution)
Pure diffusion in a 1D infinite pipe:
�� – radius of the tube
• A mass of tracer �� is injected at the point �� = 0 at time
�� = 0, uniformly across
the cross-section area �� = ����2
• How does the tracer spreads in time, due to molecular
diffusion alone?
Boundary conditions:
Initial condition: ��(��) = �
0, �� > 0
+∞, �� = 0
0, �� < 0
Conservation of mass:
Dirac delta function
�
−∞
+∞
�� �� ���� = 1
Recall dimensional analysis:
• Buckingham π-theorem
– Consider a process that can be described by �� dimensional
variables (parameters)
– This full set of variables contains �� different physical
dimensions
(e.g. ��, ��, ��)
– Then there are (��− ��) independent non-dimensional
groups that
can be formed from these variables…
– …and the variables can be related according to
��1 = ��(��2, … ,����−��)
One-dimensional pure diffusion in an infinite pipe:
Number of parameters: �� = 5
Number of dimensions: �� = 3
Number of (independent) dimensionless groups: �� − �� = 2
��1 = ��(��2)
��
⁄�� (�� ����)
= ��(
��
����
)
�� = ⁄�� (�� ����)��(
��
����
)
One-dimensional pure diffusion in an infinite pipe:
�� ��, �� =
��
�� ����
��(
��
����
)
�� =
��
����
New variable:
����
����
=
1
����
����
����
= �� −
1
2
���� −
1
2−1�� = −
��
2��
�� ��, �� =
��
�� ����
��(��)
����
���� =
��
����
��
�� ����
��(��) =
��
����
��
�� ����
�� �� +
��
�� ����
����
����
����
���� = −
��
2���� ����
�� + ��
����
����
��2��
����2 =
��
������ ����
��2��
����2
(diffusion equation in 1D)��
��2��
����2 =
����
����
One-dimensional pure diffusion in an infinite pipe:
��
��
������ ����
��2��
����2 = −
��
2���� ����
�� + ��
����
����
��2��
����2 +
1
2 �� + ��
����
����
= 0
��2��
����2 +
1
2 �� + ��
����
���� =
��
����
����
���� +
1
2 ���� = 0 ���
����
���� +
1
2 ���� = �����
����
���� +
1
2 ���� = ��0
= 0
1
�� ���� = −
1
2 ������ �� = ��1������ −
��2
4
(it can be shown that ��0 = 0
satisfies initial and boundary
conditions)
One-dimensional pure diffusion in an infinite pipe:
�� ��, �� =
��
�� ����
��(��)
�� = �
��
�� ��, �� ���� = �
−∞
∞
�
0
�� ��
�� ����
��(��) 2������������
���� = ���� ����
(balance of mass)
1 = �
−∞
∞
�� �� ����
��(��) = ��1������ −
��2
4
�
−∞
∞
��1������ −
��2
4 ���� = 1 ��1 =
1
∫−∞
∞ ������ −��
2
4 ����
=
1
2 ��
(solution from the
integral tables)
One-dimensional pure diffusion in an infinite pipe:
��1 =
1
2 ��
��(��) = ��1������ −
��2
4
�� ��, �� =
��
�� ����
�� ��
�� =
��
����
�� ��, �� =
��
�� 4������
������ −
��2
4����
Gaussian distribution, with a standard
deviation, ��2 = 2����
Solution
for Diffusion+Advection in 1D:
(translating with u)

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Diffusion+Advection+Reaction(solution)Pure diffusion.docx

  • 1. Diffusion+Advection+Reaction (solution) Pure diffusion in a 1D infinite pipe: �� – radius of the tube • A mass of tracer �� is injected at the point �� = 0 at time �� = 0, uniformly across the cross-section area �� = ����2 • How does the tracer spreads in time, due to molecular diffusion alone? Boundary conditions: Initial condition: ��(��) = � 0, �� > 0 +∞, �� = 0 0, �� < 0 Conservation of mass: Dirac delta function � −∞ +∞
  • 2. �� �� ���� = 1 Recall dimensional analysis: • Buckingham π-theorem – Consider a process that can be described by �� dimensional variables (parameters) – This full set of variables contains �� different physical dimensions (e.g. ��, ��, ��) – Then there are (��− ��) independent non-dimensional groups that can be formed from these variables… – …and the variables can be related according to ��1 = ��(��2, … ,����−��) One-dimensional pure diffusion in an infinite pipe: Number of parameters: �� = 5 Number of dimensions: �� = 3 Number of (independent) dimensionless groups: �� − �� = 2 ��1 = ��(��2) �� ⁄�� (�� ����)
  • 3. = ��( �� ���� ) �� = ⁄�� (�� ����)��( �� ���� ) One-dimensional pure diffusion in an infinite pipe: �� ��, �� = �� �� ���� ��( �� ���� ) �� = �� ���� New variable: ���� ����
  • 4. = 1 ���� ���� ���� = �� − 1 2 ���� − 1 2−1�� = − �� 2�� �� ��, �� = �� �� ���� ��(��) ���� ���� = �� ���� �� �� ���� ��(��) = ��
  • 5. ���� �� �� ���� �� �� + �� �� ���� ���� ���� ���� ���� = − �� 2���� ���� �� + �� ���� ���� ��2�� ����2 = �� ������ ���� ��2�� ����2 (diffusion equation in 1D)�� ��2�� ����2 = ����
  • 6. ���� One-dimensional pure diffusion in an infinite pipe: �� �� ������ ���� ��2�� ����2 = − �� 2���� ���� �� + �� ���� ���� ��2�� ����2 + 1 2 �� + �� ���� ���� = 0 ��2�� ����2 + 1 2 �� + ��
  • 7. ���� ���� = �� ���� ���� ���� + 1 2 ���� = 0 ��� ���� ���� + 1 2 ���� = ����� ���� ���� + 1 2 ���� = ��0 = 0 1 �� ���� = − 1 2 ������ �� = ��1������ − ��2 4
  • 8. (it can be shown that ��0 = 0 satisfies initial and boundary conditions) One-dimensional pure diffusion in an infinite pipe: �� ��, �� = �� �� ���� ��(��) �� = � �� �� ��, �� ���� = � −∞ ∞ � 0 �� �� �� ���� ��(��) 2������������ ���� = ���� ���� (balance of mass) 1 = � −∞
  • 9. ∞ �� �� ���� ��(��) = ��1������ − ��2 4 � −∞ ∞ ��1������ − ��2 4 ���� = 1 ��1 = 1 ∫−∞ ∞ ������ −�� 2 4 ���� = 1 2 �� (solution from the integral tables)
  • 10. One-dimensional pure diffusion in an infinite pipe: ��1 = 1 2 �� ��(��) = ��1������ − ��2 4 �� ��, �� = �� �� ���� �� �� �� = �� ���� �� ��, �� = �� �� 4������ ������ − ��2 4���� Gaussian distribution, with a standard deviation, ��2 = 2����
  • 11. Solution for Diffusion+Advection in 1D: (translating with u)