Consider constant-density pure-strain extensional (stretching) flow along the z-axis, uR = − (γ/2)R and uz = γz, where γ is the strain rate, in a steadily rotating frame of reference with Ω = Ωez. If ω = ωzez = 0 at t = 0, what is ωz(t)?

Question

Consider constant-density pure-strain extensional (stretching) flow along the z-axis, [latex]u_{R}=-(\gamma/2)R[/latex] and [latex]u_{z}=\gamma {z}[/latex], where γ is the strain rate, in a steadily rotating frame of reference with [latex]\Omega=\Omega\mathbf{e}_{z}[/latex]. If [latex]\omega_{z}\mathbf{e}_{z}=0[/latex] at t = 0, what is [latex]\omega _{z}(t)[/latex]?

Accepted Answer

To determine [latex]\omega _{z}(t)[/latex], (5.26) must be solved for the given flow field and initial condition. Here, the solution [latex]\omega _{z}(t)[/latex] is presumed to be independent of the spatial coordinates. So, for the given velocity field with ρ = constant, the z-direction component of (5.26) is:

[latex]\begin{array}{llll}\frac{D\omega_{n}}{Dt}&=&(\omega_{j}+2\Omega_{j})\frac{\partial u_{n}}{\partial x_{j}}+\frac{\varepsilon_{nqi}}{ ho^{2}}\frac{\partial ho}{\partial x_{q}}\frac{\partial p}{\partial x_{i}}+ u\frac{\partial^{2}\omega_{n}}{\partial x^{2}_{j}},& ext{or}\\frac{D\omega}{Dt}&=&(\omega+2\Omega)\cdot abla\mathbf{u}+\frac{1}{ ho^{2}} abla ho imes abla p+ u abla^{2}\omega,\end{array}[/latex] (5.26)

[latex]\frac{\partial\omega_{z}}{\partial t}=(\omega_{z}+2\Omega)\frac{\partial u_{z}}{\partial z}=(\omega_{z}+2\Omega)\gamma[/latex].

The two ends of this equality form a simple linear differential equation. Integrating after separating variables yields:

[latex]\ln(2\Omega+\omega_{z})=\gamma t+\mathrm{const}[/latex].

Here, the initial condition requires const. = ln(2Ω). A little algebra leads to the final answer:

[latex]\omega_{z}(t)=2\Omega(e^{\gamma t}-1)[/latex].

Therefore, when both Ω and γ are both positive, vorticity can arise from an apparently irrotational background. This example illustrates how vertical stretching can concentrate planetary vorticity.

Question 5.5: Consider constant-density pure-strain extensional (stretchin...

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