1. Introduction

The capacity of cloaking to conceal arbitrary objects from detection promises potential applications in the fields of energy and military, and thus has attracted increasing interests from academic communities [1–7]. Various cloaking solutions have been proposed and demonstrated, and in most of cases, the theory of transformation optics (TO) has been applied to design the distribution of local permeability and permittivity to realize the detouring of light beam from the objects which require stealth [8–12]. To achieve functions of tunable cloaking, exterior electric field can be exerted on the metasurface to adjust the local parameters, with which cloaking of various object sizes have been realized [13–15]. However, in most cases, the involved metasurfaces with complex distribution of permittivity and permeability are challenging to fabricate [16–19]. To ease the complicated fabrication process to realize spatially varying material parameters, one is lead to adopt a metamaterial-free solution to adopt naturally ready materials to achieve the routing/cloaking purpose.

To date, optofluidic techniques have rapidly developed for its superiority in favorable adjustability of optical properties via hydrodynamically modulate fluidic boundaries or distributions of flow field [20–25]. Microflow, due to its facile manipulation, is regarded as a promising alternative for metamaterials to designate the route of light propagation using transformation optofluidics (TOF) method [26–28]. It was originally proposed and demonstrated by employing convection-diffusion process of flowing streams to realize beam focusing [27,29], splitting [26,30], and cloaking [31]. However, designing the convection-diffusion in microflow with inhomogeneous refractive index still required complicated computational fluid dynamic (CFD) processes. As an alternative, recently proposed streamline tracing-based transformation optofluidics (STTOF) method merely required analytically designating the light-carrying streamlines to manipulate the light path, wherein the light ray and the streamline follow the same trajectory [32,33].

In this paper, we extend the STTOF method for cloaking an object via analytically designating and detouring of light propagation. The proof-of-concept demonstration is achieved on an optofluidic chip, based on our theory STTOF [24,25]. Therein a cylinder on an optofluidic chip surrounded by micro-flow mimics the object to be hidden. A beam of light is directed and shed on the cylinder, and the micro-flow developed with liquid core liquid cladding (L²) configuration re-directs the light beam and steers it to bypass the cylinder with an analytically tractable route. As the light is transporting along with the flow streamlines, the theoretical prediction of light routing can be achieved by a simple analysis of flow field around the cylinder, which is elaborated in the following theory section.

2. Theory

In our previous work, the proposed STTOF method provides feasibility to manipulate the light routing in a given microfluidic domain, in which an L² configuration was utilized to constrain the light beam transmitting along a specific streamline from fluidic/optical source to the sink, and the light path could be analytically controlled by adjusting the flow rate. Previously reported cloaking devices function via bending the light beam to bypass the object to be hidden [2,4], herein we demonstrate on an optofluidic chip that the STTOF method can also be leveraged to properly design the light path to realize the cloaking function.

Firstly we developed a model, in which a cylinder is assumed to be the object under cloaking. The cylinder is surrounded by a microflow that is bounded in a circular domain (Fig. 1(a)). The microflow consists of a liquid core and liquid cladding structure (Fig. 1(b)) in which the flow rate of liquid cladding is much higher than that of the liquid core. In such a way, the squeezed liquid core can be treated approximately as a streamline in the flow field as shown in Fig. 1(c). Fluids and light can be coupled into this circular domain via a pair of inlet and outlet (source and sink) symmetrically located at the boundary of the circular domain. As the L² structure promises the light can transmit by following a specific streamline in the fluidic domain, analytically solving the governing equation on the flow field could give a full landscape of the light steering and cloaking. Firstly, the complex potential of a flow field without boundary and the cylinder (Fig. 1(a)), where a pair of source and sink with strength of Q and –Q is respectively located at (-R,0) and (R,0) can be expressed as [34]:

 

Fig. 1. (a) The conceptual schematic of cloaking where the blue circular area indicates the area under cloaking, and the light pattern remains consistent with the input after bypassing the cloaking area (CUG denotes the abbreviation of China University of Geosciences). The yellow line denotes one of the light trajectory and the iridescent dash lines denote more options for light paths; (b) schematic of the liquid core/ liquid cladding (L²) configuration; (c) geometry of the configuration, where point A, B, C, E, and D represent the intersections between the y-axis and the boundaries and one light path guided by a streamline. Three different stages of the streamline transformation process by conformal mapping are illustrated in: (d) a pair of fluidic/optical source-sink with a distance of 2R in an unbounded domain; (e) setting a circular boundary with radius of R to the source-sink pair; (f) adding a cylinder with radius of a into the flow field;

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After adding a circular boundary with radius of R on this flow field, the complex potential can be calculated by superposition of a pair of imaginary source and sink generating circular boundary constraints [35]:

If a cylinder with radius of a is introduced into this flow field, the complex potential W₂ can be calculated by the Milne–Thomson circle theorem [36]:

In order to identify the relationship between the light path and the flow rate ratio, the velocity field of the flow inside the domain needs to be analyzed. By extracting the real part of Eq. (4) and developing the partial derivative in terms of x, y, the velocity components along x and y axis can be represented respectively as:

To simplify the calculation, here we only consider the positive half plane (x>0). The flow rates of two claddings can be calculated by integrating the velocity x-component along y axis (x=0) shown as Fig. 1(c), described as:

We consider a more practical situation that the cylindrical object has an adjustable diameter a (0<a < R) and the center of the cylinder is placed at an arbitrary point z₁ (z₁=r·cosθ+ i·r·sinθ), as illustrated in Fig. 2. By coordinates transforming of complex potential W₂ to move the whole flow field in complex space and superimposing the boundary condition of barrier cylinder, transformed complex potential W₄ can be given as follows:

By extracting the imaginary part of Eq. (8), the streamline function can be obtained (Eq. (10) in Appendix). As the position of the cylinder with a diameter of a is shifted from Z₁(-0.4R, 0) to Z₅(0.4R, 0), the streamlines in the flow field can be analytically calculated using Eq. (10), illustrated in the insets of Fig. 2 respectively. Extracting the real part of Eq. (8), the velocity potential function can be used to develop its partial derivatives in terms of x, y to obtain the velocity field (Eqs. (11) and (12) in Appendix). Subsequently the flow rate ratio between two the cladding flows (r_fr=Q_A/Q_B) can be calculated by integrating the velocity x-component along y axis (x=0), illustrated as Eq. (13) in Appendix. The corresponding relationship between streamlines and flow rate ratio can be simply given as:

 

Fig. 2. Schematic of streamlines when the position of internal boundary of the cylinder with a radius of a is varied from Z₁(-0.4R, 0) to Z₅(0.4R, 0).

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3. Experimental setup

To verify our theory, optofluidic devices were designed and fabricated. The device consists of a circular cell with a diameter of 2 mm, and two ports, served as optical and fluid source/sinks, connected to the circular cell (Fig. 3(a)). The width of the ports are designed to have a much smaller dimension (130 µm) compared to the diameter of circular cell, so that the geometry of the device can approximately satisfies the assumptions of the proposed theory. At the source port, core inlets and cladding inlets are designed that core fluid can be sandwiched by two cladding fluids to build up the L² configuration. The connecting microchannels were set at 100 µm(width)×125 µm(height). A reserved microchannel was employed for placing optical fiber with coupling the laser into the L² structure. In the first group of test, the position of cylinder under cloaking was kept at the center of cloaking cell, whereas the diameter of cylinder is set to 300 µm, 400 µm and 600 µm respectively. In the second group of test, the cylinder under cloaking with constant diameter of 400 µm respectively were located at (-400 µm, 0), (-200 µm, 0), (0, 0) respectively.

 

Fig. 3. (a) A schematic for experimental setup; (b) a snapshot of the fabricated optofluidic chip.

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As shown in Fig. 3(b), the optofluidic chips were fabricated using PDMS (polydimethylsiloxane). Firstly, the mixture of two PDMA component with a mix ratio of 1:10 was put into a vacuum pot for 5min vacuum treatment. Secondly, the PDMS mixture was poured into a culture dish with a customized SU-8 silicon wafer inside, and the whole dish was put into an incubator treating at 70 °C for one hour. Then, the solidified PDMS blocks were peeled off from the wafer and punched with 0.75 mm holes on each port. Finally, the patterned PDMS is bonded to a PDMS substrate after treating with oxygen plasma for 2min and heating at 80 °C for an hour.

In the experiment, four syringe pumps (LEAD FLUID TYD02-02) were connected to the optofluidic chips via plastic tubes for fluid pumping and quantitative flow rate control, two pumps for core fluid injection and the other two for cladding fluid. Cinnamaldehyde is chosen as the core fluid with a refractive index of n_cin=1.622 and the cladding fluid is mixture of ethylene glycol and glycerin with mixing ratio of 2:1 with a refractive index of n_mix=1.446. To visualize the light path, Rhodamine B (Sigma-Aldrich R6626, absorption peak at 553 nm, emission peak at 619 nm) was added into the core fluid. And an optical fiber (THORLABS, AFS105/125Y) was employed to couple the laser (Laserwave, LWGL 532-100mW-F) into the optofluidic chip. The observation of light path was carried out using an inverted fluorescence microscope (SHUOGUANG CFM-500E) with an attached CCD (SONY U3CCD06000KPA).

4. Result and discussion

Firstly, we demonstrate the concept of cloaking using our proposed STTOF method on the optofluidic chip under conditions of various object sizes and positions. In the experiment, laser light was initially introduced into chip via an optical fiber, and then coupled into the fluidic cell via the L² structure. The emission from the fiber was initially and directly shed on the cylinder standing inside the fluidic cell. After coupling into the L² structure, light can be detoured and bypass the cylinder by properly setting the flow rates of core and cladding flows, and in this case, the flow rates of cladding A and cladding B were fixed at 20 µL/min and 40 µL/min respectively, and the core flow rate was kept at 5 µL/min. As the core fluid was doped with fluorescent molecules, various cloaking cases can be achieved and visualized, when the flow rate setting was kept the same, as shown in the fluorescent images of Fig. 4. Figures 4(a) and 4(b) show two cloaking cases with different object sizes, that the radii of cylinders are 400 µm and 600 µm respectively. The fluorescent intensity profiles along line a and b at the inlet and outlet are shown with their Gaussian fittings in Figs. 4(d) and 4(e), respectively. The comparison shows that the cylinder with a larger size results in a higher transmission loss at the same flow rate condition. Figure 4(c) shows the detoured light path bypassing the cylinder that is located 400 µm away from the center of the fluidic cell. Figure 4(f) shows the corresponding intensity profiles at the inlet and outlet for this case, which indicates a higher transmission loss also compared to the case when the cylinder is located at the center. We believe that either the increased size or the shift of cylinder will lead to an increased streamline curvature if flow rate condition remains, and thus introduce more bending loss.

 

Fig. 4. Fluorescent images of three different cloaking cases: (a) when the circular cloaking area is set at the center of the fluidic cell (a) with a radius of 400 µm, (b) with a radius of 600 µm; (c) when the cloaking area with a radius of 400 µm is shifted 400 µm away from center of the fluidic cell; (d), (e) and (f) illustrate the fluorescent intensity profiles along line a and b denoted in (a), (b), and (c) respectively.

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To further validate our proposed theory to analytically designate of light path for the cloaking, we carried out tests in two groups on the designed optofluidic chips. The first group of test is aimed at investigating the relationship between the light path and the flow rate ratio under conditions of various cylinder sizes. In the test, the cylinder was located at the center of the fluidic cell with three different diameters of 300 µm, 400 µm and 600 µm. The flow rate of the core fluid (Q_core) was kept at 5 µL/min, the flow rate of cladding B (Q_B) was fixed at 20 µL/min, and the flow rate of cladding A (Q_A) was increased from 30 µL/min to 100 µL/min, which resulted in a varied flow rate ratio between cladding A and B (r_fr) ranging from 1.5 to 5 to make the light route in the negative half plane. For routing in the positive half plane, the flow rate of cladding A (Q_A) was settled at 20µL/min and the flow rate of cladding B (Q_B) was raised from 30 µL/min to 100 µL/min (r_fr range from 0.2 to 0.67). With the variation of the flow rate ratio, the incident light from the fiber can be detoured to bypass the cylinder following different streamlines. We label the light paths using the coordinate of the intersection between the light path and the y axis as y_B (Fig. 1(c)). In Fig. 5, the measured y_B was plotted against the flow rate ratios (r_fr) under conditions of three different cylinder sizes, and it is found that the measurement agrees well with our theoretical predictions (Eq. (9)). The insets of Fig. 5 also show a good agreement between experimental ray tracing and calculated streamlines under the flow condition of Q_core=5 µL/min, Q_A=20 µL/min and Q_B=40 µL/min. It should be noted that the streamline of core flow can approach the cylinder with an infinitely small gap in between according to our proposed model as show in Fig. 1(a), but we observed that fluctuation of flow occurs when drawing the core stream close to the cylinder in practice. This might be a result of induced surface tension to disturb the pressure-driven flow locally, as the core fluid is hydrophilic to the PDMS material. Also considering that the width of the core flow cannot be squeezed to infinitely smallness due to the mismatch between the dynamic viscosities of core and cladding fluids, we tested the flow rate ratio ranging from 1.5 to 5 in the negative half plane and from 0.2 to 0.67 in the positive half plane in practice for more effective cloaking of the cylinder.

 

Fig. 5. The relationship between flow rate ratio and y_B (the point coordinate of the intersection between the light path and the y axis) for both theoretical prediction and experimental measurements. The red, green, blue colors represent the conditions when the cylinder has a diameter of 300µm 400µm and 600µm respectively. Comparisons between experimental ray tracing and corresponding calculated streamlines (dashed lines) are shown in the insets.

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We also verify the manipulation of light path along the mathematically predicted streamlines under conditions of the cylinder at various positions. In this group of test, three different positions of the cylinder (diameter of 400 µm) were taken into account, in which the center of cylinder is positioned at three coordinates (-400 µm, 0), (-200 µm, 0), (0, 0). In the experiment, the flow rate of the core flow (Q_core) was kept at 5 µL/min and the flow rate ratio (r_fr) between two cladding flows was increased from 0.2 to 5 as the implementation in the first group of test. As illustrated in Fig. 6, the measured y_B that is dependent on the flow rate ratio (r_fr) agrees well with the theoretical curve plotted using Eq. (9). When the flow rates of Q_core, Q_A and Q_B were set at 5 µL/min, 20 µL/min and 40 µL/min respectively, the fluorescent images of the three cloaking cases are shown in the insets in Fig. 6, and all of the visualized light paths can be managed to bypass the cylinder via predicted routings using Eq. (10).

 

Fig. 6. The relationship between flow rate ratio of two cladding flows and y_B (the point coordinate of the intersection between the light path and the y axis) for both theoretical prediction and experimental measurements. The red, green, blue colors represent the conditions when the center of the cylinder is positioned at (-400 µm, 0), (-200 µm, 0), and (0, 0), respectively. Comparisons between experimental ray tracing and corresponding calculated streamlines (dashed lines) are shown in the insets.

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The experimental validation shows that the proposed STTOF method can be used for cloaking object surrounded by microflows, and the manipulation of the light path can be analytically designed. Therein, in the preliminary experiment (shown in Fig. 4), we demonstrated the proof-of-concept of cloaking using our proposed STTOF method on the optofluidic chip under conditions of various object sizes and positions. Subsequently, the additional two experiments (shown in Fig. 5 and Fig. 6) were carried out to validate that the light routing agree well with the theoretical prediction using Eq. (9) when the sizes and locations of circular cloaking region were varied respectively. Compared with the metamaterial cloaks enabled by spatially designing the corresponding material parameters, the proposed STTOF method can be used to greatly simplify the processes of device design and fabrication for cloaking purpose [3–7]. Compared with other cloaking devices enabled by optofluidic technologies, our proposed theory can be used to analytically calculate the streamlines and velocity field to easily guide the design process for different cloaking scenarios instead of using CFD simulation for analysis on a case-by-case basis [27,28,31].

5. Conclusion and outlook

In conclusion, we propose an optofluidic model employing the STTOF method to hydrodynamically reconfigure light propagation in a given flow field for object cloaking purpose. The concept is demonstrated on an optofluidic chip, and experiments are carried out to validate our proposed theory. Experimental results show that the proposed STTOF method, leveraging L2 configuration, can be used to successfully detour the light path from the object under cloaking. Based on our derived complex potentials of the flow field, streamlines as well as velocity field can be analytically retrieved to offer a guidance for mathematically planning of the light path. The scenarios when the cloaked object has variable sizes and positions are taken into account in our model, and experimentally validated on the optofluidic chip. Our demonstration of cloaking is realized on an optofluidic chip at micro-scale in this case, but the fundamental mathematics of potential flow theory used for the mapping of the streamlines and calculation of velocity field of flow can be generally applicable to situations at multi-scale. Therefore, we prudentially and optimistically envisage that the proposed method can be further extended for the cloaking of underwater infrastructures at macro-scale via manipulating and configuring their surrounding flow field.

Appendix

This appendix provides the derivation process of Eq. (9). As shown in Fig. 7(a), (a) cylindrical object has an adjustable diameter a (0<a < R) and location z₁ (z₁=r·cosθ+ i·r·sinθ) in a circular bounded dipole flow field. By extracting the imaginary part of Eq. (8), streamline equation can be written as:

(10)$$\begin{array}{r} \textrm{arctan}\frac{{(y + r \cdot \sin \theta )}}{{(x + r \cdot \cos \theta ) + R}} + \textrm{arctan}\frac{{(y + r \cdot \sin \theta )}}{{R - (x + r \cdot \cos \theta )}} - \arctan \frac{{{a^2} \cdot y + r \cdot \sin \theta ({x^2} + {y^2})}}{{{a^2} \cdot x + (r \cdot \cos \theta + R)({x^2} + {y^2})}}\\ - \arctan \frac{{{a^2} \cdot y + r \cdot \sin \theta ({x^2} + {y^2})}}{{(R\textrm{ - }r \cdot \cos \theta )({x^2} + {y^2}) - {a^2} \cdot x}} = {C_i}(i = 1,2,3\ldots ) \end{array}$$

Subsequently we extract the real part of (Eq. (8)) and develop the partial derivative in terms of x, y directions, the velocity components along x and y axis can be represented respectively as:

(11)$$\left\{ {\begin{array}{{c}} \begin{array}{l} u = \frac{{\textrm{2(}x + R + r \cdot \cos \theta )}}{{{{\textrm{(}x + r \cdot \cos \theta + R)}^2} + {{(y + r \cdot \sin \theta )}^2}}} - \frac{{\textrm{2(}x - R + r \cdot \cos \theta )}}{{{{(x + r \cdot \cos \theta - R)}^2} + {{(y + r \cdot \sin \theta )}^2}}}\\ + \frac{{{a^4}x + 2{{(r \cdot \cos \theta + R)}^2}({x^3} + x \cdot {y^2}) + (r \cdot \cos \theta + R)(3{a^2}{x^2} + {a^2}{y^2}) + 2{{(r \cdot \sin \theta )}^2}({x^3} + x \cdot {y^2}) + 2{a^2}x \cdot y(r \cdot \sin \theta )}}{{{{[{a^2}x + (r \cdot \cos \theta + R)({x^2} + {y^2})]}^2} + {{[{a^2}y + (r \cdot \sin \theta )({x^2} + {y^2})]}^2}}}\\ - \frac{{{a^4}x + 2{{(r \cdot \cos \theta - R)}^2}({x^3} + x \cdot {y^2}) + (r \cdot \cos \theta - R)(3{a^2}{x^2} + {a^2}{y^2}) + 2{{(r \cdot \sin \theta )}^2}({x^3} + x \cdot {y^2}) + 2{a^2}x \cdot y(r \cdot \sin \theta )}}{{{{[{a^2}x + (r \cdot \cos \theta - R)({x^2} + {y^2})]}^2} + {{[{a^2}y + (r \cdot \sin \theta )({x^2} + {y^2})]}^2}}} \end{array}\\ \begin{array}{l} v = \frac{{\textrm{2(}y + r \cdot \sin \theta )}}{{{{\textrm{(}x + r \cdot \cos \theta + R)}^2} + {{(y + r \cdot \sin \theta )}^2}}} - \frac{{2(y + r \cdot \sin \theta )}}{{{{\textrm{(}x + r \cdot \cos \theta - R)}^2} + {{(y + r \cdot \sin \theta )}^2}}}\\ + \frac{{{a^4}y + 2{{(r \cdot \cos \theta + R)}^2}({y^3} + {x^2} \cdot y) + 2{a^2}x \cdot y(r \cdot \cos \theta + R) + 2{{(r \cdot \sin \theta )}^2}({y^3} + {x^2} \cdot y) + 2{a^2}(r \cdot \sin \theta )({x^2} + 3{y^2})}}{{{{[{a^2}x + (r \cdot \cos \theta + R)({x^2} + {y^2})]}^2} + {{[{a^2}y + (r \cdot \sin \theta )({x^2} + {y^2})]}^2}}}\\ - \frac{{{a^4}y + 2{{(r \cdot \cos \theta - R)}^2}({y^3} + {x^2} \cdot y) + 2{a^2}x \cdot y(r \cdot \cos \theta - R) + 2{{(r \cdot \sin \theta )}^2}({y^3} + {x^2} \cdot y) + 2{a^2}(r \cdot \sin \theta )({x^2} + 3{y^2})}}{{{{[{a^2}x + (r \cdot \cos \theta - R)({x^2} + {y^2})]}^2} + {{[{a^2}y + (r \cdot \sin \theta )({x^2} + {y^2})]}^2}}} \end{array} \end{array}} \right.$$

To calculate the flow rates of two cladding flows by integrating the velocity x-component along y axis, when x=0 the velocity component can be written as:

(12)$$\left\{ {\begin{array}{{c}} \begin{array}{l} u = \frac{{\textrm{2(}R + r \cdot \cos \theta )}}{{{{\textrm{(}r \cdot \cos \theta + R)}^2} + {{(y + r \cdot \sin \theta )}^2}}} - \frac{{\textrm{2(}r \cdot \cos \theta - R)}}{{{{(r \cdot \cos \theta - R)}^2} + {{(y + r \cdot \sin \theta )}^2}}}\\ + \frac{{{a^2}{y^2}(r \cdot \cos \theta + R)}}{{{y^\textrm{4}}{{(r \cdot \cos \theta + R)}^2} + {{[{a^2}y + {y^2}(r \cdot \sin \theta )]}^2}}}\\ - \frac{{{a^2}{y^2}(r \cdot \cos \theta - R)}}{{{y^\textrm{4}}{{(r \cdot \cos \theta - R)}^2} + {{[{a^2}y + {y^2}(r \cdot \sin \theta )]}^2}}} \end{array}\\ \begin{array}{l} v = \frac{{\textrm{2(}y + r \cdot \sin \theta )}}{{{{\textrm{(}r \cdot \cos \theta + R)}^2} + {{(y + r \cdot \sin \theta )}^2}}} - \frac{{2(y + r \cdot \sin \theta )}}{{{{\textrm{(}r \cdot \cos \theta - R)}^2} + {{(y + r \cdot \sin \theta )}^2}}}\\ + \frac{{{a^4}y + 2{y^3}{{(r \cdot \cos \theta + R)}^2} + 2{y^3}{{(r \cdot \sin \theta )}^2} + 3{a^2}{y^2}(r \cdot \sin \theta )}}{{{y^\textrm{4}}{{(r \cdot \cos \theta + R)}^2} + {{[{a^2}y + {y^2}(r \cdot \sin \theta )]}^2}}}\\ - \frac{{{a^4}y + 2{y^3}{{(r \cdot \cos \theta - R)}^2} + 2{y^3}{{(r \cdot \sin \theta )}^2} + 3{a^2}{y^2}(r \cdot \sin \theta )}}{{{y^\textrm{4}}{{(r \cdot \cos \theta - R)}^2} + {{[{a^2}y + {y^2}(r \cdot \sin \theta )]}^2}}} \end{array} \end{array}} \right.$$

 

Fig. 7. (a) A specific streamline in the flow field where cylindrical object has an adjustable radius a (0<a < R) and adjustable location z₁ (z₁=r·cosθ+ i·r·sinθ). (b) A case study: the mapping of streamlines according to Eq. (10) when the cylindrical object with a radius of 0.2R is set at a position of Z (0.2R, 0.2R).

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To simplify the calculation, here we only consider the positive half plane (x>0). The flow rate of two cladding flows Q_A and Q_B are respectively described as:

(13)$$\left\{ \begin{array}{l} Q\textrm{A = }\int_B^A {u \cdot dy = \textrm{2}(\arctan \frac{{\sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} }}{{R + r \cdot \cos \theta }} - \arctan \frac{{\sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} }}{{r \cdot \cos \theta - R}} - \arctan \frac{{yB + r \cdot \sin \theta }}{{R + r \cdot \cos \theta }} + \arctan \frac{{yB + r \cdot \sin \theta }}{{r \cdot \cos \theta - R}})} \\ + (\frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2}] \cdot (\sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} - r \cdot \sin \theta ) + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta ]} }}\\ - \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2}] \cdot (\sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} - r \cdot \sin \theta ) + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } ]}}\\ - \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2}] \cdot yB + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta ]} }}\\ + \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2}] \cdot yB + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } ]}}\\ Q\textrm{B = }\int_C^B u \cdot dy\textrm{ + }\int_E^D {u \cdot dy} = \textrm{2}(\arctan \frac{{yB + r \cdot \sin \theta }}{{R + r \cdot \cos \theta }} + \arctan \frac{{yB + r \cdot \sin \theta }}{{r \cdot \cos \theta - R}} - \arctan \frac{{a + r \cdot \sin \theta }}{{R + r \cdot \cos \theta }} + \arctan \frac{{a + r \cdot \sin \theta }}{{r \cdot \cos \theta - R}}\\ + \arctan \frac{{r \cdot \sin \theta - a}}{{R + r \cdot \cos \theta }} - \arctan \frac{{r \cdot \sin \theta - a}}{{r \cdot \cos \theta - R}} + \arctan \frac{{\sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} }}{{R + r \cdot \cos \theta }} - \arctan \frac{{\sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} }}{{r \cdot \cos \theta - R}})\\ + (\frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2}] \cdot yB + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta ]} }}\\ - \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2}] \cdot yB + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } ]}}\\ - \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2}] + a \cdot r \cdot \sin \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } }}\\ + \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2}] + a \cdot r \cdot \sin \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } }}\\ \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{a \cdot r \cdot \sin \theta - [{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2}]}}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } }}\\ - \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{a \cdot r \cdot \sin \theta - [{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2}]}}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } }}\\ - \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2}] \cdot ( - \sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} - r \cdot \sin \theta ) + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(R + r \cdot \cos \theta )}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta ]} }}\\ + \frac{{R + r \cdot \cos \theta }}{{\sqrt {{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - r \cdot \sin \theta } }}\arctan \frac{{[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2}] \cdot ( - \sqrt {{R^2} - {{(r \cdot \cos \theta )}^2}} - r \cdot \sin \theta ) + {a^2}r \cdot \sin \theta }}{{\sqrt {{a^2}[{{(r \cdot \cos \theta - R)}^2} + {{(r \cdot \sin \theta )}^2} - {a^2}r \cdot \sin \theta } ]}} \end{array} \right.$$

The relationship between flow rate ratio (r_fr=Q_A/Q_B) and streamlines can be solved by combining (Eq. (10)) and Eq. (13).

As a case study, Fig. 7(b) illustrates the streamlines of the flow field when the cylindrical object with a radius of a=0.2R is set at a position of Z (0.2R, 0.2R). Therein the streamlines can be analytically calculated by Eq. (10), and the homologous flow rate configuration can be solved by Eq. (13).

Funding

National Natural Science Foundation of China (61804138, 61905224, 11804087); Wuhan Municipal Science and Technology Bureau (2018010401011297).

Disclosures

The authors declare no conflicts of interest.

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