How to find the determinant of system of PDE given below. Also explain the character of the system.

Question

Consider a two dimensional fow with Cartesian velocity components u and v. Assume that flow is steady, inviscid, incompressible and irrotational. The governing equations are then mass conservation, equation (1), and the irrotational condition, equation (2), $$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \hspace{0.5cm} \cdots (1)$$ $$\frac{\partial u}{\partial y} - \frac{\partial v}{\partial x} = 0 \hspace{0.5cm} \cdots (2)$$

Answer 1

HINT : $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 v}{\partial x\partial y} = 0 \hspace{0.5cm} $$ $$\frac{\partial^2 u}{\partial y^2} - \frac{\partial^2 v}{\partial x\partial y} = 0 \hspace{0.5cm} $$ $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ $$u(x,y)=f(x+iy)+g(x-iy)$$ $f$ and $g$ are arbitrary functions to be determined according to bondary conditions.

Then integrate $\frac{\partial v}{\partial x}=\frac{\partial u}{\partial y}$ with respect to $x$ in order to find $v(x,y)$.