On the Theory of Magneto-Induced Circulations in Trombosed Channels

Abstract

The mathematical model and the method of its approximate analysis dealing with flows induced by a traveling magnetic field in a channel occupied by non-magnetic fluid with an embedded ferrofluid drop are proposed. One of the ends of the channel is assumed to be closed (thrombotic). The aim of the study is to develop the scientific basis for magnetically induced intensification of drug transport through thrombosed blood vessels.

APPENDIX

To calculate the components of the amplitudes ${{H}_{{01}}}$, ${{H}_{{02}}}$, ${{H}_{{03}}}$, and ${{H}_{{04}}}$ of the magnetic fields generated by the solenoids in Fig. 1, the Biot–Savart–Laplace law is used in formula (1.1):

$${{H}_{{01x}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{-\left( {h + a} \right)}^{-a} dz'\left( {\mathop \int \limits_0^{2\pi } \frac{{\left( {z - z{\kern 1pt} '} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {z - z{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {x-{{x}_{0}} - \frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{01z}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{-\left( {h + a} \right)}^{-a} dz'\left( {\mathop \int \limits_0^{2\pi } \frac{{\frac{D}{2}-\left( {x-{{x}_{0}}} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {z - z{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {x-{{x}_{0}} - \frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{02x}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{a + l}^{a + l + h} dz'\left( {\mathop \int \limits_0^{2\pi } \frac{{\left( {z - z{\kern 1pt} '} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {z - z{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {x - {{x}_{0}}-\frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{02z}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{a + l}^{a + l + h} dz'\left( {\mathop \int \limits_0^{2\pi } \frac{{\frac{D}{2}-\left( {x-{{x}_{0}}} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {z-z{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {x-{{x}_{0}}-\frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{03x}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{-\left( {b + {{x}_{0}}} \right)}^{-\left( {h + b + {{x}_{0}}} \right)} dx'\left( {\mathop \int \limits_0^{2\pi } \frac{{\frac{D}{2} - \left( {z + l{\text{/}}2} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {x-x{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {\left( {z + l{\text{/}}2} \right)-\frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{03z}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{-\left( {b + {{x}_{0}}} \right)}^{-\left( {h + b + {{x}_{0}}} \right)} dx'\left( {\mathop \int \limits_0^{2\pi } \frac{{\left( {x-x{\kern 1pt} '} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {x-x{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {\left( {z + l{\text{/}}2} \right)-\frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{04x}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{b-{{x}_{0}}}^{h + b-{{x}_{0}}} dx'\left( {\mathop \int \limits_0^{2\pi } \frac{{\frac{D}{2} - \left( {z + l{\text{/}}2} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {x-x{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {\left( {z + l{\text{/}}2} \right)-\frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right),$$

$${{H}_{{04z}}}\left( {x,z} \right) = \frac{{IDN}}{{8\pi h}}\mathop \int \limits_{b-{{x}_{0}}}^{h + b-{{x}_{0}}} dx{\kern 1pt} '\left( {\mathop \int \limits_0^{2\pi } \frac{{\left( {x-x{\kern 1pt} '} \right)\cos {{\varphi }}}}{{{{{\left[ {{{{\left( {x - x{\kern 1pt} '} \right)}}^{2}} + {{{\left( {\frac{D}{2}\sin {{\varphi }}} \right)}}^{2}} + {{{\left( {\left( {z + \frac{l}{2}} \right)-\frac{D}{2}\cos {{\varphi }}} \right)}}^{2}}} \right]}}^{{\frac{3}{2}}}}}}d{{\varphi }}} \right).$$

where I is the electric current in the solenoid, h is the height of the solenoid, N is the number of turns in it, а is the distance from the vertical solenoids to the vessel, b is the distance from the horizontal solenoids to the center of the ferrofluid cloud whose coordinate along the channel axis is equal to  –x₀.