4.4.2.1 Example

An Equation We Will Require Later. We will look at an example of a Fourier-Bessel series expansion in order to get a better understanding of the concept. We assume that during the derivation of the solution to a Bessel differential equation, we are left with the following equation

${\displaystyle \sum _{i=1}^{\infty }{c}_i{J}_0\left({\lambda }_i\frac{r}{R}\right)=-{\upsilon }_{\max }\left(1-{\left(\frac{r}{R}\right)}^2\right)}$

This is actually not a random function. We will encounter it when deriving a solution to the time-dependent velocity profile in Hagen-Poiseuille flow. The equation is given by Eq. 18.33. On the way we encountered a Bessel differential equation that could be solved using a Bessel function (see section 18.3.2.2). This function is shown on the left-hand side of Eq. 4.52. Upon applying the boundary conditions, we perform an eigenvalue expansion (see section 18.3.2.2). This is the reason why we find a sum on the left-hand side of Eq. 4.52 and also the roots λ_i of J₀.

In the next step, we had to apply initial conditions that we obtained from the time-independent solution to the Navier-Stokes equation. This gives us the right-hand side of Eq. 4.52. So we know that a solution to our original differential equation will be a sum of Bessel functions that sum up to result in a profile, as given on the right-hand side of Eq. 4.52. Obviously, we are missing “only” the coefficients c_i. This is a classical example where we can apply a Fourier-Bessel series. Our problem is bound on 0 ≤ r ≤ R with r being the radius and the independent variable of our problem. The maximum radius is R. The function we are seeking to approximate is the velocity profile v (r), which is zero at the upper boundary, i.e., v (R) = 0 due to no-slip condition.

Setting Up. We now apply Eq. 4.51 to Eq. 4.52 and find

$c_i=\frac{2}{R^2{J}_{n+1}^2\left({\lambda }_i\frac{R}{R}\right)}{\int }_0^{R}r{J}_n\left({\lambda }_i\frac{r}{R}\right)f\left(r\right)dr$

From Eq. 4.52 we find n = 0; therefore Eq. 4.53 becomes

$c_i=\frac{2}{R^2{J}_1^2\left({\lambda }_i\frac{R}{R}\right)}{\int }_0^{R}r{J}_0\left({\lambda }_i\frac{r}{R}\right)f\left(r\right)dr$

The function we want to approximate is the right-hand side of Eq. 4.52. Therefore

$f\left(r\right)=-{\upsilon }_{\max }\left(1-{\left(\frac{r}{R}\right)}^2\right)$

in which case Eq. 4.54 becomes

$c_i=-\frac{2{\upsilon }_{\max }}{R^2{J}_1^5\left({\lambda }_i\frac{R}{R}\right)}{\int }_0^{R}r{J}_0\left({\lambda }_i\frac{r}{R}\right)\left(1-{\left(\frac{r}{R}\right)}^2\right)dr$

Integration. The only really problematic aspect about Fourier-Bessel series are the integrals that have to be evaluated. As it happens integrations of more complex Bessel functions always tend to be a bit tricky. However, we have noted several important properties of Bessel integrals in section 3.2.4.1 which will help us here. We find

$\begin{array}{l}{\int }_0^{R}r{J}_0\left({\lambda }_i\frac{r}{R}\right)\left(1-{\left(\frac{r}{R}\right)}^2\right)dr\hfill \\ ={\int }_0^{R}r{J}_0\left({\lambda }_i\frac{r}{R}\right)dr-\frac{1}{R^2}{\int }_0^R{r}^3{J}_0\left({\lambda }_i\frac{r}{R}\right)dr\hfill \end{array}$

with the first integral amounting to

$\begin{array}{ll}{\int }_0^{R}r{J}_0\left({\lambda }_i\frac{r}{R}\right)dr\hfill & ={\left[\frac{1}{\frac{{\lambda }_i}{R}}r{J}_1\left({\lambda }_i\frac{r}{R}\right)\right]}_0^R\hfill \\ \hfill & =\frac{R}{\frac{{\lambda }_i}{R}}{J}_i\left({\lambda }_i\frac{r}{R}\right)\hfill \end{array}$

where we have used Eq. 3.57 for the integral. The second integral is evaluated to

$\begin{array}{ll}-\frac{1}{{\lambda }_i\frac{R^2}{R}}{\int }_0^R{\lambda }_i\frac{r^3}{R}{J}_0\left({\lambda }_i\frac{r}{R}\right)dr\hfill & =\frac{1}{\frac{{\lambda }_i}{R}{R}^2}{\left[r^3{J}_1\left({\lambda }_i\frac{r}{R}\right)-\frac{2}{\frac{{\lambda }_i}{R}}{r}^2{J}_2\left({\lambda }_i\frac{r}{R}\right)\right]}_0^R\hfill \\ \hfill & =-\frac{R}{\frac{{\lambda }_i}{R}}{J}_i\left({\lambda }_i\frac{R}{R}\right)+\frac{2}{{\left(\frac{{\lambda }_i}{R}\right)}^2}{J}_2\left({\lambda }_i\frac{R}{R}\right)\hfill \end{array}$

where we have used Eq. 3.53 to simplify the integral. Summing up Eq. 4.57 and Eq. 4.58 we find

$\begin{array}{l}\frac{R}{\frac{{\lambda }_i}{R}}{J}_1\left({\lambda }_i\frac{R}{R}\right)-\frac{1}{{\left(\frac{{\lambda }_i}{R}\right)}^2}\left(2J_1\left({\lambda }_i\frac{R}{R}\right)-3{\lambda }_i\frac{R}{R}{J}_i\left({\lambda }_i\frac{R}{R}\right)\right)\hfill \\ =\frac{2}{{\left(\frac{{\lambda }_i}{R}\right)}^2}{J}_2\left({\lambda }_i\frac{R}{R}\right)\hfill \end{array}$

Here we can apply the recurrence relation for expressing the Bessel function of order 2 as Bessel functions of lower order according to Eq. 3.60, from which we find for ν = 1

$J_2\left({\lambda }_i\frac{R}{R}\right)=\frac{2}{{\lambda }_i\frac{R}{R}}\left(J_1\left({\lambda }_i\frac{R}{R}\right)-J_0\left({\lambda }_i\frac{R}{R}\right)\right)$

in which case Eq. 4.59 becomes

$\begin{array}{ll}\frac{2}{{\left(\frac{{\lambda }_i}{R}\right)}^2}{J}_2\left({\lambda }_i\frac{R}{R}\right)\hfill & =\frac{4R^2}{{\lambda }_i^3}\left(J_1({\lambda }_i)-J_0\left({\lambda }_i\right)\right)\hfill \\ \hfill & =\frac{4R^2}{{\lambda }_i^3}{J}_1({\lambda }_i)\hfill \end{array}$

where we use that fact the λ_i is the roots of J₀ and therefore J₀ (λ_i) = 0. We now find the coefficients according to Eq. 4.55 as

$\begin{array}{l}{c}_i=-\frac{2{\upsilon }_{\max }}{R^2{J}_1^2\left({\lambda }_i\frac{R}{R}\right)}\frac{4R^2}{{\lambda }_i^3}{J}_i\left({\lambda }_i\right)\hfill \\ =-\frac{8{\upsilon }_{\max }}{{\lambda }_i^3{J}_1\left({\lambda }_i\right)}\hfill \end{array}$

Solution. Returning to Eq. 4.52, we now find that

$\begin{array}{ll}{\displaystyle \sum _{i=1}^{\infty }{c}_i{J}_0\left({\lambda }_i\frac{r}{R}\right)}\hfill & =-{\upsilon }_{\max }\left(1-{\left(\frac{r}{R}\right)}^2\right)\hfill \\ {\displaystyle \sum _{i=1}^{\infty }\frac{8{\upsilon }_{\max }}{{\lambda }_i^3{J}_1\left({\lambda }_i\right)}{J}_0\left({\lambda }_i\frac{r}{R}\right)}\hfill & =-{\upsilon }_{\max }\left(1-{\left(\frac{r}{R}\right)}^2\right)\hfill \end{array}$

With Eq. 4.61 we have found a Bessel series that approximates very closely the parabolic target function, i.e., the right-hand side of Eq. 4.61. It shows very neatly the analogy to the Fourier series. In a Fourier series expansion, we find series of sine and cosine functions that approximate the target function. Here we found a series of Bessel functions to do this.

Visualization. In the next step, we will visualize how well the series we found approximates our target function. In Fig. 4.17 you can see the parabolic target function and the Bessel series of Eq. 4.61 with the expansion orders i_max = 1, i_max = 2, and i_max = 5. As you can see, even for very low expansion orders, the function is approximated correctly.

Remarks. This concludes this example. As stated, the target function was not chosen randomly. As we will see later, this is a function we require when discussing time-dependent phenomena in Hagen-Poiseuille flow (see section 18.3).

4.4.2.2 Expansion on Different Intervals and Using Different Eigenvalues

Several other scenarios lead to different equations for the Fourier-Bessel expansia. For example, the eigenvalues of the Bessel function may not be known. However, it may be possible to obtain them for the derivative of the function. In this case we would have Neumann boundary conditions instead of the (more common case) of Dirichlet boundary conditions. In this case, the series expansion looks different. We will not detail this (and other) cases here because they are less common.