Fluid Dynamics | Part IB, 2001

A fluid motion has velocity potential ϕ(x,y,t)\phi(x, y, t) given by

ϕ=ϵycos(xt)\phi=\epsilon y \cos (x-t)

where ϵ\epsilon is a constant. Find the corresponding velocity field u(x,y,t)\mathbf{u}(x, y, t). Determine u\nabla \cdot \mathbf{u}.

The time-average of a quantity ψ(x,y,t)\psi(x, y, t) is defined as 12π02πψ(x,y,t)dt\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t.

Show that the time-average of this velocity field at every point (x,y)(x, y) is zero.

Write down an expression for the fluid acceleration and find the time-average acceleration at (x,y)(x, y).

Suppose now that ϵ1|\epsilon| \ll 1. The material particle at (0,0)(0,0) at time t=0t=0 is marked with dye. Write down equations for its subsequent motion and verify that its position (x,y)(x, y) at time t>0t>0 is given (correct to terms of order ϵ2\epsilon^{2} ) as

x=ϵ2(12t14sin2t)y=ϵsint\begin{aligned} &x=\epsilon^{2}\left(\frac{1}{2} t-\frac{1}{4} \sin 2 t\right) \\ &y=\epsilon \sin t \end{aligned}

Deduce the time-average velocity of the dyed particle correct to this order.

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