A beaker contains a fluid of density ρ kg/m³, specific heat S J/kg°C and viscosity η. The beaker is filled up to height h. To estimate the rate of heat transfer per unit area (∂Q/A) by convection when the beaker is put on a hot place, a student proposed that it should depend on η, (SΔθh) and (1/ρg) when Δθ (in °C) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for (∂Q/A) is

The dimensions of ∂Q/A is J/s m² = [ML²T⁻³]L²T⁻¹ = [MT⁻³]
Checking units of options
A) ηSΔθh = [ML⁻¹T⁻¹][ML²T⁻²][K][L] = [MT⁻³] option A is correct
B) The dimensions of option B can be derived from option A by multiplying dimension of 1/ρg
[MT⁻³] × 1/[ML⁻³][LT⁻²] = [L²T⁻¹]
C) The dimensions of option C can be derived from option B by multiplying dimension of η²
[L²T⁻¹] × [ML⁻¹T⁻¹]² = [M⁻¹L⁴T⁻³]
D) The dimensions of option D can be derived from option C by multiplying dimension of ρg
[M⁻¹L⁴T⁻³] × [ML⁻³][LT⁻²] = [M⁻¹L²T⁻⁵]
Hence correct answer is option A.