Part XII: Near Field & Creeping Flow

Using the mathematical tools of asymptotics and perturbation theory, we’ve managed to find what the flow past rigid stationary objects—or the flow caused by rigid moving objects—looks like far away from those objects, in the region called the far field. Now we’d like to try and see if we can pull the same tricks to get the shape of flow in the region immediately next to an object; a region perhaps unsurprisingly referred to as the near field.

A rigid unmoving object is immersed in a uniform flow of constant pressure. We are currently interested in obtaining expressions for the flow in the region near the object, here surrounded by a transparent shell, called the near field. A spherical coordinate system has been illustrated for convenience.

One of the key reasons fluid mechanicists are interested in solving flows in the near field is to calculate forces on immersed objects. Just as we found in Part II, we’ll find that the force acting on an immersed object results from integrating the tractions \vec{t}_s on the surface of the object. The difference is that now we have additional contributions to the surface traction stemming from the flow-induced molecular forces in the fluid:

\displaystyle \vec{F} = \int_S \vec{t}_s\ dA = \int_S \left(-p \hat{n} + \boldsymbol \tau \cdot \hat{n}\right)\ dA

Because these tractions are being evaluated at the surface of the object, only the characteristics of the flow in the region immediately next to the object’s surface are going to influence the force on the object. Consequently, forces on objects are entirely determined by near field flow, and so we have good reason to try and characterize it.

Since we’re dealing with the same physical scenario as before, that of a steady uniform flow over a still rigid object (and its moving object counterpart, by extension), we can try to use the same perturbation flow approach we used before and expand the velocities and pressures into background and perturbation parts. This time, however, the background velocity we’re perturbing off of isn’t the velocity at infinity—we don’t care about what’s happening at infinity—it’s the velocity on the surface of the object.

Luckily, we have a very robust requirement for the velocity field near the object already. Our definition of fluid velocity we came up with in the very first Part, as the weighted average of the molecules in a tiny “molecular “net”, demands that the velocity transition continuously from the velocity of the object as we move away from it. This is called the no-slip condition, and for a still object, it means that \vec{v} = 0 on the surface of the immersed object. As a result, we can take our background velocity to be no velocity at all; \vec{v}_b = 0.

For the background pressure p_b, we don’t really have an obvious choice. The pressure assuredly changes over the surface of the object, so there isn’t a single value of the pressure at the surface that we can perturb off of. Consequently, we can simply take the background pressure to be the pressure of the flow at infinity, applied everywhere, since it’s the only pressure the physical set-up we’re considering is “giving” us. This has a considerable advantage in the sense that the net force across the surface of the object caused by the background pressure is identically zero, since the background pressure is the same everywhere on the surface of the object and integrating a surface normal over a closed surface is identically zero:

\displaystyle \vec{F}_{p_b} =  \int_S -p_b \hat{n}\ dA =  -p_b \int_S \hat{n}\ dA = 0

This also has the nice perk that \nabla p_b = 0, so it falls out of the perturbed Navier-Stokes equations.

Unfortunately, using the exact same physical set-up we used before for far field flow and these background quantities, we just wind up getting the original Navier-Stokes + conservation of mass system of equations for the perturbation flow:

\displaystyle \rho \nabla \vec{v}_\epsilon \cdot \left(\vec{v_b} + \vec{v_\epsilon}\right) = - \nabla p_\epsilon\ +\    \mu \nabla^2 \vec{v_\epsilon}

\displaystyle \nabla \cdot \vec{v}_\epsilon = 0

That certainly didn’t simplify things, but hopefully the following steps will.

We’d like to introduce a kinematic constraint just as we did for the far field case, but this time our constraint should be based on the behavior of the flow immediately next to the object, rather than the behavior far away from it. Because of the no-slip condition, the entire velocity field necessarily drops to zero as we move closer and closer to the object, which in this case is just the perturbation velocity field. This means that we can always define a region in which \rho\nabla\vec{v}\cdot\vec{v} is much smaller than \mu\nabla^2\vec{v}, no matter what the values of \nabla\vec{v} ,  \nabla^2\vec{v}, \rho and \mu are. This is equivalent to saying that the Reynolds tensor is effectively zero in this region, and it is precisely this region that we’ll consider as the near field.

With this in mind, it seems pretty obvious to take as our kinematic constraint that \nabla\vec{v}\cdot\vec{v} = 0. If we do that, we’ll obtain the following:

\displaystyle 0 = - \nabla p_\epsilon\ +\    \mu \nabla^2 \vec{v_\epsilon}

\displaystyle \nabla \cdot \vec{v}_\epsilon = 0

\nabla \vec{v}_\epsilon \cdot \vec{v_\epsilon} = 0

This form of the Navier-Stokes equation might look familiar—it’s exactly the form we had when we were solving for Hagen-Poiseuille flow. The difference is that now we don’t have the unidirectionality in the flow that we had when solving for pipe flow, so we can’t take this partial differential equation and turn it into an ordinary differential equation as we did before. Flows that we construct using this system of equations are usually called creeping flows or Stokes flows.

Doing a cute mathematical trick makes it really easy to solve for the pressure in a creeping flow. If you take the divergence of both sides of the Navier-Stokes equation for creeping flow, you get:

\displaystyle \nabla \cdot 0 = \nabla \cdot - \nabla p_\epsilon\ +\    \nabla \cdot \mu \nabla^2 \vec{v_\epsilon}

\displaystyle \nabla \cdot 0 =  \nabla \cdot - \nabla p_\epsilon\ +\    \mu \nabla^2 \left(\nabla \cdot \vec{v_\epsilon}\right)

\displaystyle  \nabla^2 p_\epsilon = 0

That last equation for the pressure is exactly the same equation that we had for the velocity potential in far field flow, Laplace’s equation! As a result, we know that the perturbation pressure has the same general expression as the velocity potential from far field flow:

\displaystyle p_\epsilon =  \sum \limits_{\ell=0}^{\infty}   \sum\limits_{m=-\ell}^{\ell}  \left(A_{\ell m}r^\ell + \frac{B_{\ell  m}}{r^{\ell+1}}\right)  Y^m_{\ell}(\theta,\varphi)

Using the general solution to Laplace’s equation in the form of spherical harmonics, you could then plug this solution for the perturbative pressure into the (simplified) Navier-Stokes equation, and you’d get your answer for the velocity field. Easier said than done, but I digress.

However, there’s a catch. The problem is that unlike irrotational flow, some Stokes flows don’t necessarily cause a perturbation in the pressure! To prove it, take a look at what happens if we a priori assume that the perturbative flow has a constant perturbative pressure (including the possibility that it’s zero), and so \nabla p_\epsilon = 0. With this assumption, we wind up finding:

\displaystyle 0 = - \nabla p_\epsilon\ +\    \mu \nabla^2 \vec{v_\epsilon}

\displaystyle 0 = \nabla^2 \vec{v_\epsilon}

\displaystyle \nabla^2 v_{\epsilon_r} = \nabla^2 v_{\epsilon_\theta} = \nabla^2 v_{\epsilon_\varphi} = 0

That means that, in the case of Stokes flows with uniform perturbative pressure, each component of the perturbation velocity satisfies Laplace’s equation, whose solutions have the exact same general forms as the solutions for the pressure—an infinite sum of solid harmonics. Most of the time, fluid dynamicists refer to each of these velocity fields as the homogeneous \vec{v}_h part for the uniform-pressure contribution and inhomogeneous \vec{v}_{\not h} part for the varying-pressure contribution of the near field flow:

\displaystyle \vec{v}_\epsilon = \vec{v}_h + \vec{v}_{\not h}

\displaystyle \nabla^2 \vec{v}_{h} = 0,\quad  \nabla^2 \vec{v}_{\not h} = \nabla p_\epsilon

p_\epsilon (r,\theta,\varphi) = p_h + p_{\not h}(r,\theta,\varphi)

Solving both equations for each of the components of the perturbation velocity nets you the following general form for creeping flow velocity fields, which is often called Lamb’s general solution. To skip you the trouble, I’ve written it below, where \vec{r} represents the position vector.* Be warned; this is really ugly.

\displaystyle \vec{v}_{\epsilon} = \sum_{n=0}^{n=\infty} \sum\limits_{m=-n}^{n} \left[ \frac{(n+3)r^2\nabla\left[A_{n m}r^n\, Y^m_{n}(\theta,\varphi)\right]}{2\mu(n+1)(2n+3)} -\frac{n\vec{r} A_{n m}r^n\, Y^m_{n}(\theta,\varphi)}{\mu(n+1)(2n+3)}\right] + \sum_{n=1}^{n=\infty} \sum\limits_{m=-n}^{n} \left[(n-2)r^2\nabla\frac{B_{n m}\, Y^m_{n}(\theta,\varphi)}{2r^{n+1}\mu n (1-2n)} +\frac{\left(n+1\right)\vec{r} B_{n m}\, Y^m_{n}(\theta,\varphi)}{r^{n+1}\mu n(2n-1)}\right]+ \frac{B_{0m}\vec{r}}{2r} + \sum_{n=-\infty}^{n=\infty} \sum\limits_{m=-n}^{n} \nabla \left[\left(C_{n m}r^n + \frac{D_{n m}}{r^{n+1}}\right) Y^m_{n}(\theta,\varphi)\right]+ \sum_{n=-\infty}^{n=\infty} \sum\limits_{m=-n}^{n} \nabla \times \left[\vec{r} \left(E_{n m}r^n + \frac{F_{n m}}{r^{n+1}}\right)\, Y^m_{n}(\theta,\varphi) \right]

Not only does this look like mathematical verborrhea, it also contains six distinct infinite sequences of coefficients. However, take a look at that \nabla \left(C_{n m}r^n + \frac{D_{n  m}}{r^{n+1}}\right) Y^m_{n}(\theta,\varphi) term inside of the second sum. This term, which is part of the homogeneous velocity field, has the same exact form as the general solution of far field flow in the previous Part; it’s the gradient of a velocity potential! As a result, the general solution for the flow velocity in the near field includes the general solution for the velocity in the far field. That doesn’t mean that we can accurately describe far field flows using the creeping flow equations, though; flows of this form always induce pressure variations in the far field thanks to Bernoulli’s equation, while the near field flows of this type never induce anything but uniform pressure changes everywhere by virtue of being part of the homogeneous velocity field.

*Technically, the B_{0m} term always has to be 0 since the corresponding flow fails to satisfy conservation of mass due to a mathematical quirk in vector calculus.

In any case, we could try to take the general solution we got above and use the boundary conditions in the problem to restrict it into some relatively specific form we can derive conclusions about, like we did for far field flow. Unfortunately, we’ll find that’s not possible in general for two reasons.

The first is that the boundary condition associated with the object in the near field, that the velocity drop to zero on the surface of the object, doesn’t yield a massive restriction on the coefficients in the general solution of creeping flows. This is because our solution for the flow can still go to infinity at the origin if the origin is within the object, since the flow isn’t defined there anyways. That means that we can have terms that both grow and decay as we move closer to the origin, unlike what we found for far field flow (which just decays when moving closer to infinity). We can still use the condition that \vec{v} = 0 on the object’s surface to restrict the form of the flow, but it can only be done on a case-by-case basis as prescribed by the geometry of the object.

The second reason is associated with what near field flow should look like as we move away from the near field—intuition might tell you that the near field flow needs to eventually look like the imposed uniform flow as we move away from the object. But this isn’t necessarily true! Remember that our expressions & assumptions for near field flow are only valid in the region immediately next to the object, so expecting the near field flow to turn into the flow we prescribe at infinity is foolish in general. The near field flow should transition into something in a region somewhere between the near field and the far field, which will then turn into a dipole flow in the far field, which will then turn into a uniform flow at infinity.

That being said, sometimes constraining the near field flow by what it does at infinity isn’t a terrible approximation—fluid mechanicists do it all the time, and often come up with neat mathematical tricks to make the solutions constrained by the flow at infinity more accurate. However, doing so often leads to apparent paradoxes and nonsense results, so one needs to tread lightly in case one of these paradoxes pops up in your studies.

To give you an example, take a look at two different steady flows over a rigid unmoving cylinder, when one of these flows has a larger uniform velocity at infinity:

Sketches of two experimentally observed flows past a sphere; the scenario on the right has a larger uniform velocity at infinity. The flow on the left matches the Stokes flow prediction everywhere in the near field when using the boundary condition that the perturbative flow decay at infinity. The flow on the right does not, even in the near field region where the Stokes equations are always valid.

The solution one finds from the general form of creeping flows, combined with applying boundary conditions based on the uniform flow at infinity, matches the flow with no vortices almost identically everywhere in the flow. However, it doesn’t really seem to match the flow with the vortices; which would be fine, if the flows were dissimilar only after some distance away from the object. However, the flows don’t match even in the region we’d think they should, right next to the object where we said that flows are always creeping flows. How could this be happening?

The reason is that the flow in that region that isn’t quite the near or the far field is different, which means that the boundary conditions for the creeping flow in the near field change! So the flow immediately next to either sphere is still a creeping flow, they’re just different creeping flows because the boundary conditions for the creeping flow induced by the flow in the not-quite-near/not-quite-far region are different. I haven’t seen this mentioned in other fluid mechanics books, but I like to call it the matching paradox.

All of this should be motivation enough to dig into what flows look like in the not-quite-near, not-quite-far region. Formally, this region is often called the boundary layer, and it contains perhaps the most notable feature of flows past rigid immersed objects; wakes.

Things To Think About

1. How would you estimate some characteristic length representing how far the near field extends from the object?

2. Given the form of \boldsymbol \tau we found before, what does \boldsymbol \tau \cdot \hat{n} look like as a function of the velocity? What does it convey physically about the flow-induced forces on the object?

3. What other choices for the background pressure could you make, and how would they be better or worse than the choice we made?

4. How are the assumptions that we made to get the creeping flow equations here different from the assumptions we made to get the same equations in pipe flow?

5. What parts of the perturbation pressure induce a net force on an immersed objects?

6. When solving for creeping flow over a sphere, one can find a unique solution that both matches the boundary conditions both at infinity and at the surface of the object. Why is this solution only valid in the near field, given what we know about the flow behavior in the far field?

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