Question 5 (15 Points): Using "Dimensional Analysis" find the dimensionless equation, if prod =f(rho infty ,V_(infty ),c,mu _(infty )) and prod =rho _(infty )V_(infty )^hc^i doo' are given.

To find the dimensionless equation using dimensional analysis, we need to equate the dimensions of the two given expressions for $\prod$.<br /><br />The dimensions of the given expressions are:<br /><br />$\prod = f(\rho_{\infty}, V_{\infty}, c, \mu_{\infty})$<br /><br />The dimensions of the variables are:<br /><br />$\rho_{\infty}$: density (M L^-3)<br />$V_{\infty}$: velocity (L T^-1)<br />$c$: specific heat capacity (M L^-2 T^-1)<br />$\mu_{\infty}$: dynamic viscosity (M L^-1 T^-1)<br /><br />So, the dimensions of $\prod$ are:<br /><br />$\text{dim}(\prod) = \text{dim}(f(\rho_{\infty}, V_{\infty}, c, \mu_{\infty}))$<br /><br />$\text{dim}(\prod) = \text{dim}(\rho_{\infty}) + \text{dim}(V_{\infty}) + \text{dim}(c) + \text{dim}(\mu_{\infty})$<br /><br />$\text{dim}(\prod) = (M L^{-3}) + (L T^{-1}) + (M L^{-2} T^{-1}) + (M L^{-1} T^{-1})$<br /><br />$\text{dim}(\prod) = M^1 L^{-3} T^{-1}$<br /><br />Now, let's consider the second expression for $\prod$:<br /><br />$\prod = \rho_{\infty} V_{\infty}^{h} c^{i}$<br /><br />The dimensions of this expression are:<br /><br />$\text{dim}(\prod) = \text{dim}(\rho_{\infty}) + \text{dim}(V_{\infty})^{h} + \text{dim}(c)^{i}$<br /><br />$\text{dim}(\prod) = (M L^{-3}) + (L T^{-1})^{h} + (M L^{-2} T^{-1})^{i}$<br /><br />To make the dimensions equal, we need to find the values of $h$ and $i$ such that:<br /><br />$\text{dim}(\rho_{\infty}) + \text{dim}(V_{\infty})^{h} + \text{dim}(c)^{i} = \text{dim}(\rho_{\infty}) + \text{dim}(V_{\infty})^{h} + \text{dim}(c)^{i}$<br /><br />Equating the powers of $L$ and $T$, we get:<br /><br />$-3 = -3$<br /><br />$-1h = -1$<br /><br />$-1i = -1$<br /><br />Solving for $h$ and $i$, we get:<br /><br />$h = 1$<br /><br />$i = 1$<br /><br />Therefore, the dimensionless equation is:<br /><br />$\text{dim}(\prod) = \text{dim}(\rho_{\infty}) + \text{dim}(V_{\infty}) + \text{dim}(c)$<br /><br />$\text{dim}(\prod) = M^1 L^{-3} T^{-1}$<br /><br />Hence, the dimensionless equation is:<br /><br />$\frac{\prod}{\rho_{\infty} V_{\infty} c} = \text{dimensionless}$