Analytical and experimental study of the substance transport in the vortex flow

Abstract

The characteristics of the vortex flow with a free surface formed in a vertical cylindrical container filled with water, where the source of motion is the disk at the bottom endwall, are studied experimentally and analytically. Different types of oils and petroleum are used in experiments as admixtures. The problem of an “oil body” form in a complex vortex flow is considered on the basis of the analysis of equations for the mechanics of immiscible liquids of different densities with physically based boundary conditions. The calculations of the free surface forms are carried out in cases of one- and two-component fluid. The relations obtained are in satisfactory agreement with the experimentally registered forms of the interfaces. The flow structure near a rotating disk is studied, the shapes of the trajectories of moving liquid particles in the water column near the disk are determined. The coincidence of the types of spiral movement of liquid particles on the surface and near the disk indicates that the characteristic features of the vortex flow are determined in the boundary layer on the rotating disk and then transferred to the entire spatial region occupied by the liquid. Experimental and theoretical study of the vortex flow showed that the trajectories of moving liquid particles near the water surface are spatial spirals along which these particles move from the periphery to the center. The expressions obtained for the form of trajectories of liquid particles accelerated by the rotating disk coincide with the type of registered tracers’ trajectories in the depth of liquid and on the free surface of the flow.

Appendices

Appendix A

The substitution of velocity field representation

$$\begin{aligned} {u}=r{A^{\prime }}_z , {v}={\omega }rB(z), {w}=-2A(z) \end{aligned}$$

into the system (4.2) reforms it to the form

$$\begin{aligned} \begin{array}{l} \nu {B^{\prime \prime }}_{zz} +2(A{B^{\prime }}_{z} -B{A^{\prime }}_{z} )=0 \\ -\frac{1}{\rho }{p^{\prime }}_{r} =r\left( {\nu {A^{\prime \prime \prime }}_{zzz} +2A{A^{\prime \prime }}_{zz} -{A^{\prime }}_z^2 +\omega ^{2} B^{2}} \right) \\ -\frac{1}{\rho }{p^{\prime }}_z =2\left( {\nu {A^{\prime \prime }}_{zz} +2A{A^{\prime }}_{z} } \right) +g \\ \end{array} \end{aligned}$$

(A.1)

The integrals of the second and third equations of system (A.1) produce the ratios

$$\begin{aligned} p= & {} -\frac{{\rho } r^{2}}{2}({\upnu } {A^{\prime \prime \prime }}_{zzz} +2AA^{\prime \prime }_{zz} -{A^{\prime }}_z^2 +{\omega }^{2}B^{2})+f(z), \end{aligned}$$

(A.2)

$$\begin{aligned} p= & {} -2{\rho }({\upnu } {A^{\prime }}_z +A^{2})-{\rho }gz+h(r), \end{aligned}$$

(A.3)

where f and h are arbitrary functions of their arguments.

The consistency condition for (A.2) and (A.3) leads to the requirement

$$\begin{aligned} \begin{array}{l} f(z)=-\,2{\rho }({\upnu } {A^{\prime }}_z +A^{2})-{\rho }gz+c_1 , {\upnu } {A^{\prime \prime \prime }}_{zzz} +2A{A^{\prime \prime }}_{zz} -{A^{\prime }}_z^2 +{\omega }^{2}B^{2}=c_2 \\ h(r)=-\,c_2 \frac{{\rho } r^{2}}{2}+c_1 , \\ \end{array} \end{aligned}$$

(A.4)

where $c_{1, 2}$ are some constant values.

In the absence of disk rotation (${\omega }=0)$, the flow does not occur, and therefore should be $A=0$, $p=-\rho gz+c_1$, $c_2 =0$. As a result, the system (A.1) can be reduced into two equations

$$\begin{aligned} {\upnu } {B^{\prime \prime }}_{zz} +2(A{B^{\prime }}_z -B{A^{\prime }}_z )=0, \quad {\upnu } {A^{\prime \prime \prime }}_{zzz} +2A{A^{\prime \prime }}_{zz} -{A^{\prime }}_z^2 +{\omega }^{2}B^{2}=0 \end{aligned}$$

(A.5)

herewith $p=-2\rho (\nu A^{\prime }_z +A^{2})-\rho gz+c_1$.

To satisfy the non-slip conditions

$$\begin{aligned} \left. {A(z)} \right| _{z=0} =\left. {{A^{\prime }}_z (z)} \right| _{z=0} =0, \left. {B(z)} \right| _{z=0} =1 \end{aligned}$$

and to produce the solution of the (A.5), the functions A and B near the disk surface are set as

$$\begin{aligned} \begin{array}{l} A(z)=a_2 z^{2}+a_3 z^{3}+a_4 z^{4}+a_5 z^{5}+\cdots \\ B(z)=1+b_1 z+b_2 z^{2}+b_3 z^{3}+b_4 z^{4}+b_5 z^{5}+\cdots \\ \end{array} \end{aligned}$$

(A.6)

Substitution of (A.6) into (A.5) determines the coefficients $a_i$ and $b_i$ so that

$$\begin{aligned} \begin{array}{l} A(z)=\frac{3}{2}{\upnu }b_3 z^{2}-\frac{{\omega }^{2}z^{3}}{6{\upnu }}-\frac{b_1 {\omega }^{2}z^{4}}{12{\upnu }}-\frac{b_1^2 {\omega }^{2}z^{5}}{60{\upnu }}-\frac{b_3 {\omega }^{2}z^{6}}{120{\upnu }}+\cdots \\ B(z)=1+b_1 z+b_3 z^{3}+\left( {b_1 b_3 -\frac{{\omega }^{2}}{3{\upnu }^{2}}} \right) \frac{z^{4}}{4}-\frac{b_1 {\omega }^{2}z^{5}}{15{\upnu }^{2}}+\cdots \\ \end{array} \end{aligned}$$

(A.7)

Values $b_1$ and $b_3$ imply free parameters of the solution. Along with A(z) and B(z), it is convenient to determine the value $C(z)=A^{\prime }_z$

$$\begin{aligned} C(z)=3{\upnu }b_3 z-\frac{{\omega }^{2}z^{2}}{{2\upnu }}-\frac{b_1 {\omega }^{2}z^{3}}{{3\upnu }}-\frac{b_1^2 {\omega }^{2}z^{4}}{{12\upnu }}-\frac{b_3 {\omega }^{2}z^{5}}{20{\upnu }}+\cdots \end{aligned}$$

(A.8)

The values $b_1$ and $b_3$ cannot be put equal to zero, as it violates the physical meaning of the obtained solutions. If $b_3 =0$, then the main term of the expansion A(z) equals $- {\omega }^{2}z^{3}/6{\upnu }$. Since $w=-2A(z)$, then it follows that vertical fluid flow is directed away from the disk, while the experiments show the opposite. If the vertical flow is directed to the disk, the first line of system (A.7) leads to the condition $b_3 >0$. This result overlaps with the expression (A.8) for the value C(z) when liquid is tossed along the disk from the axis of rotation, in full accordance with the experiment.

If $b_1 =0$, then from the expression for B(z) follows that the azimuthal velocity component increases with vertical distance from the disk rotation plane. In fact, it can only decrease, as the presence of any velocity is determined by the viscous friction, so that the maximum value of the azimuthal velocity is located directly on disk surface. Thus, from the second relation of (4.10) system necessarily follows $b_1 <0$.

For further analysis presented in the article, the values ${\alpha }={b_1 }/{{\updelta }_{{\upnu }} } < 0$ and ${\beta }={b_3 }/{{\updelta }_{{\upnu }}^3 } > 0$ are introduced where $\delta _\nu =\sqrt{{\upnu }/{\omega }}$.

Appendix B

The substitution of principal terms from the expansions (4.4) into Eq. (4.5) produces the system

$$\begin{aligned} {\dot{r}}=3{\beta \omega }r(t)x(t), {\dot{\varphi }}={\omega }(1+{\alpha } x(t)), {\dot{x}}=-3{\omega \beta } x^{2}(t) \end{aligned}$$

(B.1)

From the third equation of (B.1), it follows

$$\begin{aligned} x(t)=\frac{x_0 }{1+3{\omega \beta } x_0 t}, \end{aligned}$$

(B.2)

where $x_0$ is the initial vertical coordinate of a fluid element.

Substitution of (B.2) into the first and second equations of the system (B.1) represents the radial and azimuthal coordinates of fluid element by the expressions

$$\begin{aligned} r(t)= & {} r_0 (1+3{\omega \beta } x_0 t), \end{aligned}$$

(B.3)

$$\begin{aligned} {\varphi }-{\omega }t= & {} \frac{{\alpha }}{3{\beta }}\ln (1+3{\omega \beta } x_0 t)+{\varphi }_0 , \end{aligned}$$

(B.4)

where $r_0$ and ${\varphi }_0$ are the initial radial and azimuthal coordinates of a fluid element.

The use of (B.3) allows us to rewrite (B.4) as $3{\beta } ({\varphi }-{\omega } t-{\varphi }_0 )/{\alpha }=\ln r/r_0$ so that the final result has the form

$$\begin{aligned} r=r_0 \exp \left( {\frac{3{\beta }}{{\alpha }}({\varphi }-{\omega }t-{\varphi }_0 )} \right) , \end{aligned}$$

(B.5)

which is presented in the text of the article.