Methods to solve the Navier–Stokes and continuity equations

Section 7.7 Methods to solve the Navier–Stokes and continuity equations

Subsection 7.7.1 Methods of solution

For real problems, it is notoriously difficult to solve the Navier–Stokes (7.5.1) and continuity (3.4.2) equations. Many researchers spend their entire careers analysing and solving these very equations! In practical terms, a variety of analytical and numerical techniques must be employed, and usually we only get an approximate solution. In this course we are only looking at analytical approaches, but you should be aware that there are many different approaches possible, and we describe a few in the following:

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A few problems in fluid mechanics have analytical solutions, meaning they can be solved exactly. For example the solution for Poiseuille flow in a pipe is exact, and has proved to be an excellent description of steady flow in a pipe, even in realistic, imperfect conditions.
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Other problems can be solved in the form of an infinite series. An example of this is boundary layer theory or lubrication theory, in which we choose a small parameter \(\epsilon\ll1\) (which is a dimensionless quantity with a physical meaning). For example, it could be that two lengthscales in the problem are very different from one another, and we choose \(\epsilon\) equal to the smaller lengthscale divided by the large one. We write the solution as an infinite series of terms, for example

\begin{align*} \bu=&\bu_0+\epsilon\bu_1+\epsilon^2\bu_2+\epsilon^3\bu_3+\dots,\\ p=&p_0+\epsilon p_1+\epsilon^2p_2+\epsilon^3p_3+\dots, \end{align*}

where the dots indicate terms multiplied by higher powers of \(\epsilon\) (that is, smaller terms). Examples include:
- flow in a curved pipe for small Dean numbers (proportional to the Reynolds number times the square root of the curvature of the pipe),
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- flow of a thin film of fluid. We take \(\epsilon\) equal to a typical film thickness divided by a typical lengthscale tangential to the film. Often, it is sufficient to compute only the leading nontrivial terms in the series, and the equations governing these terms represent a hugh simplification over the Navier–Stokes and continuity equations. Whilst the resulting equations generally need to be solved numerically, the spatial dimension of the problem is reduced by one for such flows, and this is a huge saving of computational time.
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Other problems can be solved approximately by neglecting terms that are known to be small. An example of this is high-Reynolds-number flow when we can solve the Euler equations and low-Reynolds-number flow when we can solve the Stokes equations instead of the full Navier–Stokes equations. Flow in a porous medium (such as flow of water through soil, oil through porous rocks or tissue fluid between cells in the body) is a different type of approximation in which we only consider average flow variables. This approximation is a simplification of the solution in the form of an infinite series, and effectively, we are just finding the first term of the infinite series, that is we are finding \(\bu_0\) and \(p_0\) from the series solutions given above.
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In many cases, we cannot solve the equations analytically, but we can find a solution by simulating them on a computer.
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In other cases we need to simplify the equations somewhat to solve them on a computer. An example is if there are sharp boundary layers, which need fine resolution for the solution to be found there.
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Finally, one can use an experimental approach, either by building a scale model, which should have dimensionless parameters such as the Reynolds number the same as the original, or by taking measurements on the original system. The big advantage is that, as long as the experiment is correctly performed, the flows should be realistic of the system being studied, but the disadvantages are that it is often hard to perform measurements of flow and pressure with sufficient accuracy or resolution.
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Subsection 7.7.2 Methods to solve the equations analytically

We start by looking at the special cases in which we can solve the equations analytically. In these cases, we typically first make some simplifying assumptions. Note that for these types of problems, we are only trying to find a solution that works (proving the uniqueness of the solutions we find is a much harder thing to do – i.e. it is hard to prove that the solution we found is the only one).

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Stability: Whether or not a given solution that we find is stable is a crucial question, because if it is not stable then it would not be seen in an experiment. To determine stability mathematically we would need to use techniques such as those taught in the course MA40045 Dynamical Systems. However, it is typically very difficult to do this in practice in the study of fluid dynamics because of the (infinite) dimensionality of the phase space. The methods we use in this course to find exact solutions – that is, making assumptions and finding a solution to the equations – get us a solution, but don’t know whether or not it is stable. A practical way of determining stability is to perform an experiment or numerical simulation (which is sometimes called a numerical experiment) – an unstable solution would not be able to be found this way, but these methods can be difficult if it is only very weakly unstable. Plus part of the reason for finding analytical solutions is to avoid having to do experiments or numerics, which are often much more time-consuming and difficult!

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Simplifying a problem by making assumptions: When considering a problem that involves solving the Navier–Stokes and continuity equations, we often make one or more (in fact as many as possible!) of the following simplifying assumptions:

Two-dimensional flow (in \((x,y)\)-plane): \(\partial u/\partial z=\partial v/\partial z =\partial w/\partial z=\partial p/\partial z=0\) (indeed \(\partial/\partial z=0\) for any variable) and \(w=0\text{.}\) Of course, this can be generalised to two-dimensional flow in the \((x,z)\)- or \((y,z)\)-planes and also two-dimensional flow in the \((r,\theta)\)-directions in cylindrical polar coordinates. For example we may do a lab experiment between closely separated parallel plates to ensure that the flow is approximately two-dimensional (Hele–Shaw cell).
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One-dimensional flow (in \(x\)-direction): \(\partial/\partial y=0\) and \(\partial/\partial z=0\) and \(v=w=0\text{.}\) This can be generalised to any direction. For example, blood flow in arteries has been successfully analysed using a one-dimensional approach in which the changes in flow and pressure that happen during a cardiac cycle can be seen as waves that travel around the body with a characteristic wave speed (that is different from the blood velocity).
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Independence of a coordinate: For example \(\partial/\partial x=0\) if nothing in the problem depends on \(x\) and there is no reason to assume that anything should depend on \(x\text{.}\)
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We used this for the Stokes boundary layer problem Example 7.8.12.
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Axisymmetric flow: \(\partial u_r/\partial\theta=\partial u_\theta/\partial\theta =\partial u_z/\theta=\partial p/\theta=0\) (indeed \(\partial/\partial\theta=0\) for all variables) and \(u_\theta=0\text{.}\) In addition, if there is a symmetry in a plane that includes the axis the This can alternatively be written in spherical coordinates (in which case \(\partial/\partial\phi=0\) and \(u_\phi=0\)).
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Rotationally symmetric flow: \(\partial u_r/\partial\theta=\partial u_\theta/\partial\theta =\partial u_z/\theta=\partial p/\theta=0\) (indeed \(\partial/\partial\theta=0\) for all variables) but \(u_\theta\) is not necesarily zero. For example a line vortex is rotationally symmetric but not axisymmetric. We can similarly write Problems of this sort can also be written in spherical coordinates (in which case \(\partial {}{\phi}=0\)).
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Spherically symmetric flow: \(\partial/\partial\theta=0\) and \(\partial/\partial\phi=0\) and \(u_\theta=u_\phi=0\text{.}\)
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Steady flow: \(\partial u/\partial t=\partial v/\partial t =\partial w/\partial t=\partial p/\partial t=0\) (indeed \(\partial/\partial t=0\) for any variable).
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Many examples that we cover make this assumption.
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Fully developed flow: \(\partial u/\partial z=\partial v/\partial z=\partial w/\partial z=0\text{,}\) but \(\partial p/\partial z\) is not, in general, zero.
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The pipe flow problem, Poiseuille flow Example 7.8.4 uses this assumption.
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Periodic and sinusoidal flow (with period \(T\)):

\begin{align*} u(x,y,z,t)=&u_0(x,y,z)e^{\im\omega t}+\overline{u_0(x,y,z)}e^{-\im\omega t},\\ v(x,y,z,t)=&v_0(x,y,z)e^{\im\omega t}+\overline{v_0(x,y,z)}e^{-\im\omega t},\\ w(x,y,z,t)=&w_0(x,y,z)e^{\im\omega t}+\overline{w_0(x,y,z)}e^{-\im\omega t},\\ p(x,y,z,t)=&p_0(x,y,z)e^{\im\omega t}+\overline{p_0(x,y,z)}e^{-\im\omega t}, \end{align*}

where the angular frequency \(\omega=2\pi/T\text{.}\)

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This assumption will be made in the Stokes boundary layer example Example 7.8.12.
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Most of these are assumptions on the geometry and spatial symmetries of the problem, but the assumptions of steady flow, fully developed flow and periodic and sinusoidal flow are different. The trick is to look carefully at the problem, and think which (if any) of the above assumptions might be reasonable. We can try making them, and then check that the problem is still consistent. If it is consistent that is fine, but if not it means that the assumption was wrong.

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