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        "description": "For n variables with dimensional rank k, there are n-k independent products; k is the rank of the declared variables, not automatically three or every conceivable base unit. The variable list, geometric/material ratios and dimensionful constants must be complete, and similarity concerns a common physical law, not unrelated mechanisms with matching numbers. The theorem does not determine the unknown function, numerical coefficients or a globally unique solved form. Some heat-transfer examples are hypothetical; turbulence/viscosity simplifications are conditional approximations. Propeller scaling excludes cavitation and may neglect skin friction. In the same fluid and gravity, matching both Reynolds and Froude ratios at different size is obstructed unless an effect is legitimately omitted or resources change. The final gravitational-unit discussion uses a declared physical constraint, not a universal modern two-unit rule. Full formula/proof and experimental replay were not certified.",
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      "statement": "# Buckingham: dimensionless products and physically similar systems\n\nBuckingham organizes a complete dimensionally homogeneous physical relation into independent dimensionless products and derives conditional scaling rules for geometrically and physically similar systems.\n\nA small model can help predict a full-sized machine only when the relevant physical ratios are preserved. The paper explains how a complete list of variables and independent dimensions yields a smaller set of dimensionless products. Experiments or theory must still determine the relation between those products. Its heat-transfer and propeller examples show both the usefulness of scaling and the practical limit: in the same fluid and gravity, a differently sized propeller cannot generally preserve viscous and gravity similarity at once.\n\n## Scientific scope\n\nFor n variables with dimensional rank k, there are n-k independent products; k is the rank of the declared variables, not automatically three or every conceivable base unit. The variable list, geometric/material ratios and dimensionful constants must be complete, and similarity concerns a common physical law, not unrelated mechanisms with matching numbers. The theorem does not determine the unknown function, numerical coefficients or a globally unique solved form. Some heat-transfer examples are hypothetical; turbulence/viscosity simplifications are conditional approximations. Propeller scaling excludes cavitation and may neglect skin friction. In the same fluid and gravity, matching both Reynolds and Froude ratios at different size is obstructed unless an effect is legitimately omitted or resources change. The final gravitational-unit discussion uses a declared physical constraint, not a universal modern two-unit rule. Full formula/proof and experimental replay were not certified.\n\n## People and systems\n\nE. Buckingham\n\n\n\n## Original sources\n\n- [Buckingham: dimensionless products and physically similar systems — original article scan](https://pdodds.w3.uvm.edu/research/papers/others/1914/buckingham1914a.pdf)\n",
      "summary": "Buckingham organizes a complete dimensionally homogeneous physical relation into independent dimensionless products and derives conditional scaling rules for geometrically and physically similar systems.\n\nA small model can help predict a full-sized machine only when the relevant physical ratios are preserved. The paper explains how a complete list of variables and independent dimensions yields a smaller set of dimensionless products. Experiments or theory must still determine the relation between those products. Its heat-transfer and propeller examples show both the usefulness of scaling and the practical limit: in the same fluid and gravity, a differently sized propeller cannot generally preserve viscous and gravity similarity at once.",
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    "id": "buckingham-dimensional-similarity-1914",
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