Consider two fluids, one with a large coefficient of volume expansion and the other with a small one. In what fluid will a hot surface initiate stronger natural convection currents? Why? Assume the viscosity of the fluids to be the same.

Natural convection currents are the circulation of fluid caused by the differences in fluid density due to temperature changes. When a fluid is heated, it expands, and its density reduces. As a result, the heated fluid becomes lighter and rises, while the cooler, denser fluid sinks. This creates a continuous circulation of the fluid, forming convection currents.

The coefficient of volume expansion (denoted as \(\beta\)) is a measure of how much a fluid expands when its temperature increases. It is defined as the fractional change in volume per unit change in temperature at constant pressure: $$\beta = \frac{1}{V} \frac{dV}{dT}$$ A larger coefficient of volume expansion means that, for a given temperature change, the fluid will experience a larger volumetric expansion. This results in a larger change in fluid density (\(\Delta \rho\)) due to the temperature change. Since natural convection is driven by differences in fluid densities, this will directly affect the strength of convection currents.

Now that we have established the relationship between the coefficient of volume expansion, density change, and natural convection currents, we can compare the two fluids. Fluid 1: Large coefficient of volume expansion (\(\beta_1\)) Fluid 2: Small coefficient of volume expansion (\(\beta_2\)) Assuming the same temperature change: As the coefficient of volume expansion of Fluid 1 is larger than Fluid 2, the change in density (\(\Delta \rho_1\)) for Fluid 1 will be larger than that of Fluid 2 (\(\Delta \rho_2\)). The larger change in density will result in stronger buoyancy forces acting on the fluid, causing it to rise more quickly. This will, in turn, create stronger natural convection currents in Fluid 1 as compared to Fluid 2.

Based on the analysis above, we can conclude that a hot surface will initiate stronger natural convection currents in the fluid with a large coefficient of volume expansion compared to the fluid with a small coefficient of volume expansion, assuming the viscosity of the fluids to be the same. This is because the larger coefficient of volume expansion results in a more significant change in fluid density, leading to stronger buoyancy forces and, consequently, stronger natural convection currents.