A fluid motion for which the Reynolds number is small (so that nonlinear terms in velocity are negligible) is known as a creeping flow or Stokes’s flow. For a steady creeping flow of an incompressible viscous fluid under zero body force, show that p is a harmonic function. Deduce that ψ defined by (10.14.19) is a biharmonic function in this case.

$\textbf{v}=\psi_{,2}\textbf{e}_{1}-\psi_{,1}\textbf{e}_{2}$ (10.14.19)

Question

A fluid motion for which the Reynolds number is small (so that nonlinear terms in velocity are negligible) is known as a creeping flow or Stokes’s flow. For a steady creeping flow of an incompressible viscous fluid under zero body force, show that p is a harmonic function. Deduce that ψ defined by (10.14.19) is a biharmonic function in this case.

$\textbf{v}=\psi_{,2}\textbf{e}_{1}-\psi_{,1}\textbf{e}_{2}$ (10.14.19)

[latex] extbf{v}=\psi_{,2} extbf{e}_{1}-\psi_{,1} extbf{e}_{2}[/latex] (10.14.19)

Accepted Answer

For creeping flow,

[latex]\frac{Dv}{Dt}=\frac{\partial extbf{v}}{\partial t}+( extbf{v}. riangledown) extbf{v}\approx \frac{\partial extbf{v}}{\partial t}[/latex] (10.14.27)

Then the Navier-Stokes equation (10.14.6) becomes

[latex]v riangledown^2 extbf{v}-\frac{1}{ ho} riangledown p+ extbf{b}=\frac{D extbf{v}}{Dt}[/latex] (10.14.6)

[latex]v riangledown^2 extbf{v}-\frac{1}{ ho} riangledown p+ extbf{b}=\frac{\partial extbf{v}}{\partial t}[/latex] (10.14.28)

For steady flow with zero body force, this equation reduces to

[latex]v riangledown^2 extbf{v}=\frac{1}{ ho} riangledown p[/latex] (10.14.29)

Taking the divergence of this equation and using the equation of continuity div v = 0, we get

[latex] riangledown^2 p=0[/latex] (10.14.30)

Thus, p is a harmonic function.
For ψ defined through the relation (10.14.19), we have

[latex]v_{1}=ψ_{,2},\ \ v_{2}=-ψ_{,1},\ \ v_{3}=0[/latex]

Using these in equation (10.14.29), we get

[latex]p_{,1}= ho v riangledown^2(\psi_{,2});\ \ p_{,2}=- ho v riangledown^2(\psi_{,1})[/latex] (10.14.31)

From these, we obtain, respectively,

[latex]p_{,12}= ho v riangledown^2(\psi_{,22});\ \ p_{,21}=- ho v riangledown^2(\psi_{,11})[/latex] (10.14.32)

so that

[latex]v ho riangledown^2(\psi_{,11}+\psi_{,22})=-(p_{,21}-p_{,12})=0\ or\ riangledown^2\psi=0[/latex]

Thus, ψ is a biharmonic function.

Question 10.14.3: A fluid motion for which the Reynolds number is small (so th...

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