0
$\begingroup$

I've seen plenty of derivations for stagnation point flow, but they all use strain rate [1/s] and do not explain how one calculates it. Is there an equation or procedure that is used to find the strain rate of the flow?

Conceivably I could do an experiment to find this, but I believe it should be possible to find a function of the fluid viscosity, $\nu$, incoming velocity, $u_\infty$ and distance from the stagnation point, $z_o$ which yields the strain rate $a$.

When solving the Blasius or Falkner Skan equations I get that the velocity far from the wall goes to $\infty$ which makes me wonder if I can only apply it near the wall.

If there is I'm surprised because I haven't seen anything addressing this as it is more common to be in a position where you know the incoming velocity rather than the strain rate.

$\endgroup$
  • $\begingroup$ Please give a reference for a derivation that you have come across. $\endgroup$ – Deep May 5 '18 at 4:21
  • $\begingroup$ @Deep I have seen it in lectures, but here is a derivation (in German):onlinelibrary.wiley.com/doi/epdf/10.1002/zamm.19360160304 $\endgroup$ – James Urban May 5 '18 at 17:21
  • $\begingroup$ Sorry, can't read German. $\endgroup$ – Deep May 6 '18 at 3:36
1
$\begingroup$

You can read about the strain rate tensor here. The strain rate tensor encodes the information about the local rate of deformation of the medium in all directions. This tensor can be defined as the symmetric part of the Jacobian matrix $\boldsymbol{J} = \nabla \boldsymbol{v}$, that is \begin{equation} \frac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^\mathrm{T}) \end{equation} For incompressible flows ($\nabla \cdot \boldsymbol{v} = 0$) this tensor is traceless, since its trace is precisely equal to the divergence of $\boldsymbol{v}$.

The particular problem that you mention (stagnation point flow) is usually studied under the hypotheses that the flow is irrotational (potential) and incompressible. The first hypothesis means that the flow velocity can be represented as the gradient of a scalar field function (irrotational flow) \begin{equation} \boldsymbol{v} = \nabla\phi \end{equation} Furthermore, using incompressibility one can furthermore prove that the flow satisfies the Laplace equation: \begin{equation} \Delta\phi = 0 \end{equation} which implies that the potential function must be linear on the coordinates. That is: \begin{equation} \phi = S_{ij}x_ix_j + A_{i}x_i + C,\quad S_{11} = - S_{22} \end{equation} Note that the axes are usually aligned with the two axes of symmetry of the flow (a straight stream impinging on an infinite plate) meaning that $A_{i} = 0$, since the velocity component $i$ must change when crossing the $j$-axis. $S_{12} = 0$ for the same reason. Note that in this case the constants $S_{ij}$ must coincide with the components of the strain rate tensor according to its definition. So in this particular flow problem, one knows that these numbers will represent the strain rate tensor from the start, taking the usual assumptions into account.

Now, the $v_2$-component of the velocity on the $x_2$-axis is given by $v_2 = \frac{\partial \phi}{\partial x_2} = 2S_{22}x_2$, which gives you a formula to calculate $S_{22} $ (let us call it $a$) as \begin{equation} |a| = \left|\frac{v_2}{2x_2}\right| \end{equation} In an experiment, you could measure the centerline velocity at $z_0 = x_2$ and use this formula to determine the strain rate.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.