# making an equation dimensionless

I have a balance of energy equation as following (for a spherical particle that colliding with a spherical fluid droplet) Left hand side is for before collision and RHS for after that: \begin{equation} \frac{\pi}{12}\rho_fD_d^3V_d^2+\frac{\pi}{12}\rho_pD_p^3V_{p, bc}^2=2\pi\sigma L(D_p+Lsin\theta)+\frac{2\pi\mu LU^2}{h}(D_p+Lsin\theta)\Delta t+\frac{\pi}{12}\rho_pD_p^3V_{p, ac}^2 \end{equation} in which $D$ stands for diameter, $d$ for droplet, $p$ for particle, $\sigma$ for fluid surface tension, $bc$ for before collision, $ac$ after collision, $\Delta t$ collision time, $U$ relative velocity of drop and particle (collision velocity) and $H, L, h$ are geometrical parameters. First term at RHS of the above equ. is related to surface energy of fluid and 2nd term is related to dissipation of energy during the collision. I want to make this equation non-dimension. So, I simplified it to: \begin{equation} \rho_fD_d^3V_d^2+\rho_pD_p^3(V_{p, bi}^2-V_{p, ai}^2)=24L(D_p+Lsin(\theta))[\sigma+\frac{\mu U^2}{h}\Delta t] \end{equation}

and then to: \begin{equation} \frac{V_d^2}{U^2}+\frac{\rho_p}{\rho_d}(\frac{D_p}{D_d})^3 {\frac{V_{p, bi}^2-V_{p, ai}^2}{U^2}}=C_1.\frac{L.D_{eq}}{D_d^2}[\frac{1}{We}+\frac{C_2}{Re}] \end{equation} in which $We$ and $Re$ stand for Weber and Reynolds numbers. However, I am trying to define the terms as groups of simpler meaningful dimensionless parameters. Please let me know if you have any idea about a more appropriate definition of groups. Thanks.

Both sides of your equation are already dimensionless, so now it is just a matter of choice. For example, you can define your $\gamma_{...}\equiv V_{...}^2/U^2$ terms as dimensionless variables of your model and $\lambda \equiv L \cdot D_{eq}/D_d^2$ as a dimensionless parameter. Taking $\alpha \equiv \rho_p/\rho_d$, $\beta = (D_p/D_d)^3$ your equation reads $$\gamma_d + \alpha \beta (\gamma_{p,bi}- \gamma_{p,ai}) = C_1 \lambda (\frac{1}{We} + \frac{1}{Re})$$ Where all the symbols represent dimensionless numbers.
• Let's stress that your $\gamma_{...}$ are in fact velocities $V_{...}$ non-dimensionalised by $U$, which the OP took as reference velocity : $\gamma_{...}=\tilde{V}_{...} = V_{...}/U$. The product $\alpha\beta$ can enter as a single number, a ratio of masses. – Joce Oct 3 '14 at 11:58
• ($\gamma$ are dimensionless velocities squared.) Yes, it is a question of taste and context how to define the dimensionless quantities and the expressions can be rearranged to give ratios of different meaningful physical quantities. – Void Oct 3 '14 at 13:24