Math 2.0

Fluid mechanics Definition

Fluid mechanics

Builds on Vector fieldPartial differential equation

Definition

A fluid is a material that deforms without limit under any shear stress, however small. In the continuum model, a fluid fills a region Ω⊆ℝ3\Omega \subseteq \R^3. At each time tt it is described by a density field ρ(x,t)>0\rho(x, t) > 0, a velocity vector field 𝐮(x,t)\mathbf{u}(x, t) and a pressure field p(x,t)p(x, t). Fluid mechanics is the branch of mechanics that studies these fields through the conservation laws of mass, momentum and energy, written as partial differential equations. It consists of fluid statics, where 𝐮=0\mathbf{u} = 0, and fluid dynamics, where the fluid moves.

Remarks

Example: in a fluid at rest of constant density ρ\rho under gravity gg, the balance of forces gives ∂p/∂z=−ρg\partial p / \partial z = -\rho g, so p=p0−ρgzp = p_0 - \rho g z with zz the height. This is a problem of fluid statics. Non-example: a steel beam under a small shear stress deforms by a fixed small amount and then stops, so it is a solid and not a fluid. The continuum model fails when the mean free path of the molecules is comparable to the size of the flow, as in very thin gases.

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Differential equation, concept, no evidence levelDifferential equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptOpen set, concept, no evidence levelOpen setconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptPartial derivative, concept, no evidence levelPartial derivativeconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptVector field, concept, no evidence levelVector fieldconceptPartial derivative, definition, no evidence levelPartial derivativedefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionMulti-index notation, concept, no evidence levelMulti-index notationconceptPartial differential equation, definition, no evidence levelPartial differential …definitionFluid mechanics, definition, no evidence levelFluid mechanicsdefinition
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  1. Differential equation concept
  2. Euclidean space object
  3. Function concept
  4. Open set concept
  5. Function of several variables concept
  6. Vector-valued function concept
  7. Partial derivative concept
  8. Limit of a function of several variables concept
  9. Vector field concept
  10. Partial derivative definition
  11. Higher partial derivatives definition
  12. Multi-index notation concept
  13. Partial differential equation definition
  14. Fluid mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Fluid_mechanics
About this page
TagAX5RQ0MX88
URIm2://AX5RQ0MX88
Kinddefinition
WritingW0 stub
MSC76A02, 76-01
DefinesFluid mechanics
UsesVector field, Partial differential equation
Changed3 November 2026

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