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Rayleigh–Plesset equation

Rayleigh–Plesset equation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Rayleigh–Plesset equation rather than just read about it. In short: In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is a nonlinear ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. Its general form is usually written aswhere ρ L {\displaystyle \rho _{L}} is the density of the surrounding liquid, assumed to be constant R ( t ) {\displaystyle R(t)} is the radius of the bubb…

Rayleigh–Plesset equation — main illustration
Rayleigh–Plesset equation — illustration

Key takeaways

  • Rayleigh–Plesset equation belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Rayleigh–Plesset equation to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Rayleigh–Plesset equation from memory before moving on to harder problems.

Reference excerpt

In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is a nonlinear ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. Its general form is usually written aswhere

ρ L {\displaystyle \rho _{L}} is the density of the surrounding liquid, assumed to be constant

R ( t ) {\displaystyle R(t)} is the radius of the bubble

ν L {\displaystyle \nu _{L}} is the kinematic viscosity of the surrounding liquid, assumed to be constant

σ {\displaystyle \sigma } is the surface tension of the bubble-liquid interface

Δ P ( t ) = P ∞ ( t ) − P B ( t ) {\displaystyle \Delta P(t)=P_{\infty }(t)-P_{B}(t)} , in which, P B ( t ) {\displaystyle P_{B}(t)} is the pressure within the bubble, assumed to be uniform and P ∞ ( t ) {\displaystyle P_{\infty }(t)} is the external pressure infinitely far from the bubble Provided that P B ( t ) {\displaystyle P_{B}(t)} is known and P ∞ ( t ) {\displaystyle P_{\infty }(t)} is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius R ( t ) {\displaystyle R(t)} . The Rayleigh–Plesset equation can be derived from the Navier–Stokes equations under the assumption of spherical symmetry. It can also be derived using an energy balance.

History Neglecting surface tension and viscosity, the equation was first derived by W. H. Besant in his 1859 book with the problem statement stated as An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant (in fact, Besant attributes the problem to Cambridge Senate-House problems of 1847). Besant predicted the time required to fill an empty cavity of initial radius R 0 {\displaystyle R_{0}} to be

… excerpt ends here. Continue reading the full article.

Illustrations

Rayleigh–Plesset equation: The Rayleigh–Plesset equation is often applied to the study of cavitation bubbles, shown here forming behind a propeller.
The Rayleigh–Plesset equation is often applied to the study of cavitation bubbles, shown here forming behind a propeller.
Rayleigh–Plesset equation: Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to oscillatory pressure at its natural frequency undergoes expansion and then collapses.
Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to oscillatory pressure at its natural frequency undergoes expansion and then collapses.
Rayleigh–Plesset equation: Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to pressure-drop undergoes expansion and then collapses.
Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to pressure-drop undergoes expansion and then collapses.

Worked examples

Example 1 — a first encounter with Rayleigh–Plesset equation

Start with the simplest possible case. Write down what Rayleigh–Plesset equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rayleigh–Plesset equation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rayleigh–Plesset equation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rayleigh–Plesset equation

In research
Rayleigh–Plesset equation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Rayleigh–Plesset equation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Rayleigh–Plesset equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Equations of fluid dynamics, Ordinary differential equations, so understanding it makes those chapters shorter.
In everyday life
Look for Rayleigh–Plesset equation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Rayleigh–Plesset equation in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Rayleigh–Plesset equation means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Rayleigh–Plesset equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rayleigh–Plesset equation in simple terms?

In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is a nonlinear ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. Its general form is usually written aswhere ρ L {\displaystyle \rho _{L}}…

Why does Rayleigh–Plesset equation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rayleigh–Plesset equation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rayleigh–Plesset equation.

Tags

  • Equations of fluid dynamics
  • Ordinary differential equations

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