Problem Statement
Explain the Marangoni effect and give an example.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The Marangoni effect is fluid flow driven by gradients in surface tension. Since $\sigma = \sigma(T, c)$ depends on temperature and concentration, any gradient in $T$ or $c$ along a surface creates a tangential stress that drives flow from low-$\sigma$ to high-$\sigma$ regions.
Step 2 — Apply the relevant physical law or equation: Thermal Marangoni: A hot spot on a water surface has lower $\sigma$ → fluid flows away from hot spots. This drives flows in welding pools and semiconductor crystal growth.
Step 3 — Solve algebraically for the unknown: Solutal Marangoni (“tears of wine”): Ethanol has lower $\sigma$ than water ($\sigma_{EtOH}\approx22\ \text{mN/m}$, $\sigma_{H_2O}\approx73\ \text{mN/m}$). Wine climbs the glass wall, ethanol evaporates (raising $\sigma$), and the surface tension gradient pulls liquid up. Droplets form and run back down — the “tears.”
Step 4 — Substitute numerical values with units: Quantitatively: Marangoni flow velocity $\sim |\nabla\sigma|/\eta$.
Worked Calculation
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\sim |\nabla\sigma|/\eta}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
The Marangoni effect is fluid flow driven by gradients in surface tension. Since $\sigma = \sigma(T, c)$ depends on temperature and concentration, any gradient in $T$ or $c$ along a surface creates a tangential stress that drives flow from low-$\sigma$ to high-$\sigma$ regions.
Thermal Marangoni: A hot spot on a water surface has lower $\sigma$ → fluid flows away from hot spots. This drives flows in welding pools and semiconductor crystal growth.
Solutal Marangoni (“tears of wine”): Ethanol has lower $\sigma$ than water ($\sigma_{EtOH}\approx22\ \text{mN/m}$, $\sigma_{H_2O}\approx73\ \text{mN/m}$). Wine climbs the glass wall, ethanol evaporates (raising $\sigma$), and the surface tension gradient pulls liquid up. Droplets form and run back down — the “tears.”
Quantitatively: Marangoni flow velocity $\sim |\nabla\sigma|/\eta$.
Answer
$$\boxed{\sim |\nabla\sigma|/\eta}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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