Modeling diffusive processes via a constant effective diffusivity value taken to represent realistic uncertainty or heterogeneity is entrenched in scientific and engineering applications. This brings forth the question, to what extent does the flow pattern changes when symmetry is broken by anisotropy. This study supplies the answer by deriving a class of tridimensional solutions to the steady nonlinear diffusion equation in a spherical domain divided into an arbitrary number of meridian sectors with distinct diffusivities and generation rates. The new family of solutions permits flexible modeling, where traditionally only isotropic radial transport was considered. The flow patterns support an extensive variety of topological terrain via tesseral and sectoral harmonics. The anisotropy gives rise to an unconventional type of a fixed point combining both node and saddle attributes. The contours are nonsmooth on the contiguity planes between sectors and might or might not be localized in the polar angle \(\varphi\) and/or azimuthal angle \(\theta\), implying a particle might remain confined to a relatively small neighborhood or meander over the sphere. The impact on motion trajectories and thus transport efficiency implies that the energy required to sustain a steady flow is starkly underestimated when symmetry is assumed for simplicity despite the presence of anisotropy.


  1. nonlinear diffusion equation
  2. exact solutions
  3. anisotropy
  4. spherical coordinates
  5. sectoral and tesseral harmonics
  6. transport efficiency

MSC codes

  1. 35Q35
  2. 76E30
  3. 76N10
  4. 76S05

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The authors thank Prof. Richard Taylor of Thompson Rivers University, British Columbia, Canada, for insightful discussions.

Supplementary Materials

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Effects of Anisotropy in Tridimensional Diffusion: Flow Patterns and Transport Efficiency
Authors: Piyush Awasthi, Mahendhar Kumar, and Yana Nec
File: supplementary.pdf
Type: PDF
Contents: The file contains a brief manual to the application that implements the reported family of solutions for custom anisotropy settings to be chosen by the user. The application will be published in the Thompson Rivers University data depository TRUSpace. The file further contains explanations on how the singular linear systems were solved numerically.
File: supplementary2.zip
Type: ZIP
Contents: This archive provides 3D spherical projections for the respective planar projections given in the manuscript. The figures are interactive, allowing zoom, rotation, etc. Files named figure*_3Dproj.fig open as MATLAB figures. Files named figure*_3Dproj.ofig open in GNU Octave, a free source counterpart of MATLAB.


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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 460 - 483
ISSN (online): 1095-712X


Submitted: 8 February 2022
Accepted: 31 August 2022
Published online: 17 April 2023


  1. nonlinear diffusion equation
  2. exact solutions
  3. anisotropy
  4. spherical coordinates
  5. sectoral and tesseral harmonics
  6. transport efficiency

MSC codes

  1. 35Q35
  2. 76E30
  3. 76N10
  4. 76S05



Piyush Awasthi
School of Biomedical Engineering, University of British Columbia, Vancouver, BC, Canada.
Mahendhar Kumar
School of Mechanical Engineering, Vellore Institute of Technology, Tamilnadu, India.
Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, British Columbia, Canada.

Funding Information

Funding: This work was supported by the MITACS Globalink Research Internship program (Canada).

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