A beaker contains a fluid of density $\rho \ kg / m^3$, specific heat $S J / \ kg^oC$ and viscosity $\eta $. The beaker is filled upto height $h$. To estimate the rate of heat transfer per unit area $(Q / A)$ by convection when beaker is put on a hot plate, a student proposes that it should depend on $\eta ,\left( {\frac{{S\Delta \theta }}{h}} \right)$ and $\left( {\frac{1}{{\rho g}}} \right)$ when $\Delta \theta ($ in $^\circ C)$ is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $(Q / A)$ is
JEE MAIN 2015, Diffcult
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Let $\frac{Q}{A}=\eta^a\left(\frac{S \Delta \theta}{h}\right)^b\left(\frac{1}{\rho g}\right)^c$
Using dimensional method
$M T^{-3}=\left[M L^{-1} T^{-1}\right]^a\left[L T^{-2}\right]^b\left[M^{-1} L^2 T^2\right]^c$
or, $M T^{-3}=\left[M^{a-c} L^{-a+b+2 c} T^{-a-2 b+2 c}\right]$
Equating powers and solving we get, $a=1, b=1, c=0$
$\therefore \frac{Q}{A}=\eta \frac{S \Delta \theta}{h}$
Using dimensional method
$M T^{-3}=\left[M L^{-1} T^{-1}\right]^a\left[L T^{-2}\right]^b\left[M^{-1} L^2 T^2\right]^c$
or, $M T^{-3}=\left[M^{a-c} L^{-a+b+2 c} T^{-a-2 b+2 c}\right]$
Equating powers and solving we get, $a=1, b=1, c=0$
$\therefore \frac{Q}{A}=\eta \frac{S \Delta \theta}{h}$
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