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Encyclopedia

Knowledge card

Similarity Transformation

Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory

Symbol

The change of variables that collapses (x, y, t) into η and reduces the PDEs to ODEs.

Essence

S1It maps the PDEs onto ODEs in the single variable η, valid because the profiles keep their shape in time. [1.1]

Dimension I

Definition

What it is — and what it is not

What it is
  • S2A self-similar change of variables that removes the explicit x, y, and t dependence.

Dimension II

In practice

How to deal with it

How to deal with it
  • S3Introduce η, the stream function ψ (hence f), and θ, then substitute to obtain the ODEs.

Sub-topics 12 / 37

Prerequisite

Related

References

  1. ↗

    Problem 1 draft synthesis; Aziz et al. (2021) is partial context only, DOI 10.1007/s10973-020-10210-2

    • 1.1

      Model-scope comparison — The seeded η, ψ, B(t), L(t), h_f(t), and ΔT₀ thermal scaling combine several sources. The actual Aziz article does not establish this complete unsteady map.

      AmbiguityTenuous

      The cited Aziz article supports only a partial, different configuration; the complete equation is an audited synthesis.

  2. Sharidan, Mahmood & Pop (2006), IJAME 11, 647–654.

    • 2.1

      IJAME 11, 647–654 — The (1−γt) unsteady stretching similarity transformation and its benchmark f″(0) values.

      Established
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