that is: $$\label{irrot}
\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}=0$$ which is satisfied by the velocity potential $\phi$ where: $$v=\frac{\partial \phi}{\partial y}; w=\frac{\partial \phi}{\partial z}$$

The next step is to relate $\rho$ to the velocity, so that Equation \[mass\_cons\] can be expressed purely in velocity derivatives, that is to say purely in derivatives of $\phi$. For a perfect gas under adiabatic contraction/expansion that has a magnetic field perpendicular to the plane of the flow, the MHD-Bernoulli’s equation can be written as: $$\frac{1}{2}\left(v^2+w^2\right)+\frac{1}{\gamma-1}\frac{p\gamma}{\rho}+\frac{B_x^2}{4\pi\rho}=C(S)$$ where $C(S)$ is used to show that the value is constant along streamlines. Therefore, the values of the physical variables at each point in the flow can related to the free stream values (the values of the flow at an infinite distance from the circular cylinder). $$\label{ENERGY}
\frac{\gamma}{\gamma-1}\frac{p p_{\infty}}{\rho \rho_{\infty}}+\frac{1}{2}\left( v^2 +w^2 \right)U^2_{\infty}+\frac{B_x^2}{\rho}\frac{B_{x \infty}^2}{4\pi \rho_{\infty}}=\frac{\gamma}{\gamma-1}\frac{p_{\infty}}{\rho_{\infty}}+\frac{1}{2}U^2_{\infty}+\frac{B_{x \infty}^2}{4\pi \rho_{\infty}}$$ where the subscript $\infty$ implies the free stream values that are used for normalisation.

If we define the free stream speed of sound as $C_s=\sqrt{\gamma p_{\infty}/\rho_{\infty}}$, the free stream Alfven velocity as $V_a=B_{x\infty}/\sqrt{4\pi \rho_{\infty}}$ and denote the free stream Mach number as $M_{\infty}=U_{\infty}/C_s$ then equation \[ENERGY\] becomes: $$\frac{1}{\gamma-1}\frac{p}{\rho}+\frac{1}{2}M_{\infty}^2(v^2+w^2)+\frac{B_x^2}{\rho}\frac{V_A^2}{C_s^2}=\frac{1}{\gamma-1}+\frac{1}{2}M_{\infty}^2+\frac{V_A^2}{C_s^2}.$$ We can then define a relation between the Alfven velocity and the sound speed as 