Fluid Dynamics | Part IB, 2003

Define the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction ψ(x,y)\psi(x, y) as

u=(u,v)=(ψy,ψx)\mathbf{u}=(u, v)=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)

in Cartesian coordinates (x,y)(x, y). Show that the line integral

x1x2undl=ψ(x2)ψ(x1)\int_{\mathbf{x}_{1}}^{\mathbf{x}_{\mathbf{2}}} \mathbf{u} \cdot \mathbf{n} d l=\psi\left(\mathbf{x}_{\mathbf{2}}\right)-\psi\left(\mathbf{x}_{\mathbf{1}}\right)

along any path joining the points x1\mathbf{x}_{\mathbf{1}} and x2\mathbf{x}_{\mathbf{2}}, where n\mathbf{n} is the unit normal to the path. Describe how this result is related to the concept of mass conservation.

Inviscid, incompressible fluid is contained in the semi-infinite channel x>0x>0, 0<y<10<y<1, which has rigid walls at x=0x=0 and at y=0,1y=0,1, apart from a small opening at the origin through which the fluid is withdrawn with volume flux mm per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation

2ψx2+2ψy2=0\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0

and boundary conditions

if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain

ψ=my+2mπn=11nsinnπyenπx\psi=-m y+\frac{2 m}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin n \pi y e^{-n \pi x}

ψ=0 on y=0,x>0,ψ=m on x=0,0<y<1,ψ=m on y=1,x>0,ψmy as x,\begin{aligned} & \psi=0 \quad \text { on } y=0, x>0, \\ & \psi=-m \quad \text { on } x=0,0<y<1, \\ & \psi=-m \quad \text { on } y=1, x>0, \\ & \psi \rightarrow-m y \quad \text { as } x \rightarrow \infty, \end{aligned}

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