# Numerical solution of the integral for the profile of a vortex

I am trying to solve numerically the integral extracted from the article by Vanyó et al. (https://doi.org/10.1103/PhysRevE.90.013002) $$h(r') = \int_{0}^{r^{\prime}} \frac{\partial p}{\partial r} / \frac{\partial p}{\partial z} dr$$ My goal is to get the graph:

where $$z(r) = H_{w} - h (r)$$. Parameters: $$H_{w} = 30 cm$$, $$f = 10 Hz$$, $$n = 7$$ and $$u_{z,max} = 10.8$$ cm/s. Using Mathematica, I managed to calculate the partial derivatives that are replaced in the integral:

A0 = 1;
freq = 10;
cc = 1;
uzmax = 10.8;
II = 0;
\[Omega] = 1;
R0 = 1;
n = 7;
H = 30;

III[t] = 1 + II  Sen (\[Omega] t);

(* auxiliary functions *)
fff[r, t] = III[t] [ 1 - (r/R0)^2 ];
g[z] = 1 - Abs[(2 z)/H - 1]^n;
F[r] = III[t] ( r/2 - r^3/(4 R0^2));
G[z] = \!$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]$$g[z]$$\);

(* vortex force *)
\[CapitalGamma] = A0 2 \[Pi] freq;

(* 1) tangential component *)
ut[r] = \[CapitalGamma]/r (1 - E^(-(r^2/cc^2)));
(* 2) Componente vertical *)
uz[r, z] = - uzmax fff[r, t] g[z];

ur[r, z] = uzmax F[r] G[z];

(* partial derivatives *)

\[Rho] = 1;
\[Nu] = 0.01;
grav = 981;

Pr = \[Rho] [ - ur[r, z] \!$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]$$ur[r, z]$$\) - uz[r, z] \!$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]$$ur[r, z]$$\) + ut[r]^2/
r + \[Nu] ( 1/r \!$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]$$ur[r, z]$$\) + \!$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]ur[r, z]\)\) + \!$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]ur[r, z]\)\) - ur[r, z]/r^2)];
Pz = \[Rho] [ - ur[r, z] \!$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]$$uz[r, z]$$\) - uz[r, z] \!$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]$$uz[r, z]$$\) + \[Nu] ( 1/r \!$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]$$uz[r, z]$$\) + \!$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]$$\*SubscriptBox[\(\[PartialD]$$, $$r$$]uz[r, z]\)\) + \!$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]$$\*SubscriptBox[\(\[PartialD]$$, $$z$$]uz[r, z]\)\)) + grav];


I'm trying to solve the integral using NIntegrate but I can't find any results. Any suggestions?

• Hello and welcome. It would be more helpful for future users if you could use a more appropriate title. In this context, by more appropriate I mean a title that describes/specifies the problem in more suitable terms, rather than using physics terms that many users here cannot understand.
– bmf
Feb 17 at 12:53
• @Soliton-104 Where can one find the definition of h[r]? Feb 17 at 15:39
• Prior to copy and paste into this forum, convert your code to InputForm so that it is not cluttered with the box structures. Feb 17 at 17:47
• None of your function definitions have patterns on the LHS. For example, g[z] = 1 - Abs[(2 z)/H - 1]^n; should read g[z_] = 1 - Abs[(2 z)/H - 1]^n; In the definition of fff you appear to be using function brackets in the place of parentheses. See four kinds of bracketing used in the Wolfram Language. Feb 17 at 17:56