Question

Prove that for a two dimensional motion to be possible the stream function and the velocity Potential satisfy the cauchys-Riemanns Equation. Moreover, deduce that both satisfies laplaces equatiin

Answer 1

Answer to the question 83980, Math / Differential Equations

Denote by $\overline{v} = (v_{1}, v_{2})$ the stream function. Since the flow is incompressible we have $div(\overline{v}) = 0$. Since it is potential we have that there exists the velocity potential $u$ such that $v_{1} = \frac{\partial u}{\partial x_{1}}$, $v_{2} = \frac{\partial u}{\partial x_{2}}$. Thus from $div(\overline{v}) = 0$ we obtain that $\Delta u = 0$. As $v_{1} = \frac{\partial u}{\partial x_{1}}$ we also have $\Delta v_{1} = 0$. As $v_{2} = \frac{\partial u}{\partial x_{2}}$ we also have $\Delta v_{2} = 0$.

From $div(\overline{v}) = 0$ we have $\frac{\partial v_1}{\partial x_1} = -\frac{\partial v_2}{\partial x_2}$. From $v_1 = \frac{\partial u}{\partial x_1}$, $v_2 = \frac{\partial u}{\partial x_2}$ we have $\frac{\partial v_1}{\partial x_2} = \frac{\partial v_2}{\partial x_1}$. Thus we obtain also Cauchy-Riemann equations.

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