Using "Dimensional Analysis" find the dimensionless equation, if prod =f(rho _(infty ),V_(infty ),C,mu _(infty )) (oo) and prod =rho _(infty )V_(infty )^hc^imu _(infty )^j are given.

To find the dimensionless equation using dimensional analysis, we need to ensure that the dimensions of both sides of the equation are the same. Let's start by identifying the dimensions of each variable in the given equation:<br /><br />1. $\rho_{\infty}$: Density (M L$^{-3}$)<br />2. $V_{\infty}$: Velocity (L T$^{-1}$)<br />3. $C$: Concentration (M L$^{-3}$)<br />4. $\mu_{\infty}$: Dynamic viscosity (M L$^{-1}$ T$^{-1}$)<br /><br />Now, let's rewrite the given equation with dimensions:<br /><br />$\Pi = \rho_{\infty} V_{\infty}^h C^i \mu_{\infty}^j$<br /><br />The dimensions of the left side of the equation are [M$^0$ L$^0$ T$^0$], which is dimensionless.<br /><br />To make the right side dimensionless as well, we need to find the values of $h$, $i$, and $j$ such that the dimensions of the right side match the dimensions of the left side. <br /><br />The dimensions of the right side are:<br /><br />[M$^{1+h+i+j}$ L$^{h-3i-j}$ T$^{-h-j}$]<br /><br />To make the dimensions of the right side match the dimensions of the left side, we need to set $h = 3i + j$ and $-h - j = 0$. Solving these equations, we get $h = 3i + j$ and $h = j$.<br /><br />Now, we can rewrite the equation as:<br /><br />$\Pi = \rho_{\infty} V_{\infty}^h C^i \mu_{\infty}^j$<br /><br />Since the dimensions of both sides of the equation are now the same, we can say that the equation is dimensionless.<br /><br />Therefore, the dimensionless equation is:<br /><br />$\Pi = \rho_{\infty} V_{\infty}^h C^i \mu_{\infty}^j$