A Microscale Mathematical Blood Flow Model for Understanding Cardiovascular Diseases

Abstract

The blood plasma flow through a swarm of red blood cells in capillaries is modeled as an axisymmetric Stokes flow within inverted prolate spheroidal solid-fluid unitary cells. The solid internal spheroid represents a particle of the swarm, while the external spheroid surrounds the spheroidal particle and contains the analogous amount of fluid that corresponds to the fluid volume fraction of the swarm. Analytical expansions for the components of the flow velocity are obtained by introducing a stream function ψ which satisfies the fourth-order partial differential equation E⁴ψ = 0. We assume nonslip conditions on the internal inverted spheroidal boundary which is also impermeable, while on the external spheroidal surface, we assume continuity of the tangential velocity component and nil vorticity. In order to solve the problem at hand, we employ the method of Kelvin inversion, under which, the initial problem, formulated in the inverted prolate spheroidal coordinates, is transformed to an equivalent one in the prolate spheroidal coordinates, where the solution space of the equation E⁴ψ = 0 is already known from our previously published work. The solution for the original problem is obtained by using the inverse Kelvin transformation and the effect of this transform to the Stokes operator (Dassios, IMA J Appl Math 74:427-438, 2009). Finally, the analytical solution for the stream function ψ is given through a series expansion of specific combinations of Gegenbauer functions of mixed order, multiplied by the Euclidean distance on the first and on the third power, in a so-called R-separable form.

Appendix A Functions g2n

The functions g_2n are

$$ {g}_2\left(\tau \right)={A}_2{G}_2\left(\tau \right)+{B}_2{H}_2\left(\tau \right)+{C}_2{G}_0\left(\tau \right)+{D}_2{G}_1\left(\tau \right)+{E}_2{G}_4\left(\tau \right)+{F}_2{H}_4\left(\tau \right) $$

(28)

and for n ≥ 2

$$ {\displaystyle \begin{array}{l}{g}_{2n}\left(\tau \right)={A}_{2n}{G}_{2n}\left(\tau \right)+{B}_{2n}{H}_{2n}\left(\tau \right)+{C}_{2n}{G}_{2n-2}\left(\tau \right)+{D}_{2n}{H}_{2n-2}\left(\tau \right)\\ {}\operatorname{}+{E}_{2n}{G}_{2n+2}\left(\tau \right)+{F}_2{H}_{2n+2}\left(\tau \right).\end{array}} $$

(29)

The coefficients A_n, B_n, C_n, D_n, E_n, and F_n are derived by applying the boundary conditions (16), (17), (18), and (19) and by geometrical reduction of the obtained stream function in the analogous form of the spherical system of coordinates. For instance, we have that

$$ {g}_2\left(\tau \right)=\frac{c^2}{D}\left[{\Lambda}_2{G}_2\left(\tau \right)+{\Lambda}_3\left(\frac{5{G}_4\left({\tau}_a\right)}{G_1\left({\tau}_a\right)}{G}_1\left(\tau \right)+{G}_4\left(\tau \right)\right)+{\Lambda}_4{H}_2\left(\tau \right)\right], $$

(30)

$$ {g}_4\left(\tau \right)=-\frac{c^2}{2D}{G}_2\left(\tau \right)+\frac{\Lambda_2^{(4)}{G}_6\left(\tau \right)+{M}_2^{(4)}{H}_6\left(\tau \right)+{\Lambda}_0^{(4)}{G}_4\left(\tau \right)+{M}_0^{(4)}{H}_4\left(\tau \right)}{D^{(4)}}. $$

(31)

The coefficients that we use to define the stream functions ψ_q(τʹ, ζʹ ) are presented in (32), (33), (34), (35), (36), (37), (38), (39), and (40) where the primes denote the first derivative of the functions.

$$ {\Lambda}_2={G}_4\left({\tau}_{\beta}\right){H}_2^{\prime}\left({\tau}_{\beta}\right)-{G}_4^{\prime}\left({\tau}_{\beta}\right){H}_2\left({\tau}_{\beta}\right)-\frac{5}{\tau_a}{G}_4\left({\tau}_a\right)\left[{H}_2\left({\tau}_{\beta}\right)-{\tau}_{\beta }{H}_2^{\prime}\left({\tau}_{\beta}\right)\right], $$

(32)

$$ {\Lambda}_3={G}_2^{\prime}\left({\tau}_{\beta}\right){H}_2\left({\tau}_{\beta}\right)-{G}_2\left({\tau}_{\beta}\right){H}_2^{\prime}\left({\tau}_{\beta}\right), $$

(33)

$$ {\Lambda}_4={G}_2\left({\tau}_{\beta}\right){G}_4^{\prime}\left({\tau}_{\beta}\right)-{G}_2^{\prime}\left({\tau}_{\beta}\right){G}_4\left({\tau}_{\beta}\right)+\frac{5}{\tau_a}{G}_4\left({\tau}_a\right)\left[{G}_2\left({\tau}_{\beta}\right)-{\tau}_{\beta }{G}_2^{\prime}\left({\tau}_{\beta}\right)\right], $$

(34)

$$ D=\frac{1}{2{G}_2\left({\tau}_a\right)}\left[{\Lambda}_2{G}_2\left({\tau}_a\right)+6{\Lambda}_3{G}_4\left({\tau}_a\right)+{\Lambda}_4{H}_2\left({\tau}_a\right)\right], $$

(35)

$$ {D}^{(4)}=\left|\begin{array}{cccc}{G}_6\left({\tau}_{\beta}\right)& {H}_6\left({\tau}_{\beta}\right)& {G}_4\left({\tau}_{\beta}\right)& {H}_4\left({\tau}_{\beta}\right)\\ {}{G}_6^{\prime}\left({\tau}_{\beta}\right)& {H}_6^{\prime}\left({\tau}_{\beta}\right)& {G}_4^{\prime}\left({\tau}_{\beta}\right)& {H}_4^{\prime}\left({\tau}_{\beta}\right)\\ {}{G}_6\left({\tau}_{\alpha}\right)& {H}_6\left({\tau}_{\alpha}\right)& {G}_4\left({\tau}_{\alpha}\right)& {H}_4\left({\tau}_{\alpha}\right)\\ {}30{G}_6\left({\tau}_{\alpha}\right)& 30{H}_6\left({\tau}_{\alpha}\right)& 12{G}_4\left({\tau}_{\alpha}\right)& 12{H}_4\left({\tau}_{\alpha}\right)\end{array}\right|, $$

(36)

$$ {\Lambda}_2^{(4)}=-\frac{c^2}{2D}\left|\begin{array}{cccc}{G}_2\left({\tau}_{\beta}\right)& {H}_6\left({\tau}_{\beta}\right)& {G}_4\left({\tau}_{\beta}\right)& {H}_4\left({\tau}_{\beta}\right)\\ {}{G}_6^{\prime}\left({\tau}_{\beta}\right)& {H}_6^{\prime}\left({\tau}_{\beta}\right)& {G}_4^{\prime}\left({\tau}_{\beta}\right)& {H}_4^{\prime}\left({\tau}_{\beta}\right)\\ {}{G}_2\left({\tau}_{\alpha}\right)& {H}_6\left({\tau}_{\alpha}\right)& {G}_4\left({\tau}_{\alpha}\right)& {H}_4\left({\tau}_{\alpha}\right)\\ {}2{G}_2\left({\tau}_{\alpha}\right)& 30{H}_6\left({\tau}_{\alpha}\right)& 12{G}_4\left({\tau}_{\alpha}\right)& 12{H}_4\left({\tau}_{\alpha}\right)\end{array}\right|, $$

(37)

$$ {M}_2^{(4)}=-\frac{c^2}{2D}\left|\begin{array}{cccc}{G}_6\left({\tau}_{\beta}\right)& {G}_2\left({\tau}_{\beta}\right)& {G}_4\left({\tau}_{\beta}\right)& {H}_4\left({\tau}_{\beta}\right)\\ {}{G}_6^{\prime}\left({\tau}_{\beta}\right)& {G}_2^{\prime}\left({\tau}_{\beta}\right)& {G}_4^{\prime}\left({\tau}_{\beta}\right)& {H}_4^{\prime}\left({\tau}_{\beta}\right)\\ {}{G}_6\left({\tau}_{\alpha}\right)& {G}_2\left({\tau}_{\alpha}\right)& {G}_4\left({\tau}_{\alpha}\right)& {H}_4\left({\tau}_{\alpha}\right)\\ {}30{G}_6\left({\tau}_{\alpha}\right)& 2{G}_2\left({\tau}_{\alpha}\right)& 12{G}_4\left({\tau}_{\alpha}\right)& 12{H}_4\left({\tau}_{\alpha}\right)\end{array}\right|, $$

(38)

$$ {\Lambda}_0^{(4)}=-\frac{c^2}{2D}\left|\begin{array}{cccc}{G}_6\left({\tau}_{\beta}\right)& {H}_6\left({\tau}_{\beta}\right)& {G}_2\left({\tau}_{\beta}\right)& {H}_4\left({\tau}_{\beta}\right)\\ {}{G}_6^{\prime}\left({\tau}_{\beta}\right)& {H}_6^{\prime}\left({\tau}_{\beta}\right)& {G}_2^{\prime}\left({\tau}_{\beta}\right)& {H}_4^{\prime}\left({\tau}_{\beta}\right)\\ {}{G}_6\left({\tau}_{\alpha}\right)& {H}_6\left({\tau}_{\alpha}\right)& {G}_2\left({\tau}_{\alpha}\right)& {H}_4\left({\tau}_{\alpha}\right)\\ {}30{G}_6\left({\tau}_{\alpha}\right)& 30{H}_6\left({\tau}_{\alpha}\right)& 2{G}_2\left({\tau}_{\alpha}\right)& 12{H}_4\left({\tau}_{\alpha}\right)\end{array}\right|, $$

(39)

$$ {M}_0^{(4)}=-\frac{c^2}{2D}\left|\begin{array}{cccc}{G}_6\left({\tau}_{\beta}\right)& {H}_6\left({\tau}_{\beta}\right)& {G}_4\left({\tau}_{\beta}\right)& {G}_2\left({\tau}_{\beta}\right)\\ {}{G}_6^{\prime}\left({\tau}_{\beta}\right)& {H}_6^{\prime}\left({\tau}_{\beta}\right)& {G}_4^{\prime}\left({\tau}_{\beta}\right)& {G}_2^{\prime}\left({\tau}_{\beta}\right)\\ {}{G}_6\left({\tau}_{\alpha}\right)& {H}_6\left({\tau}_{\alpha}\right)& {G}_4\left({\tau}_{\alpha}\right)& {G}_2\left({\tau}_{\alpha}\right)\\ {}30{G}_6\left({\tau}_{\alpha}\right)& 30{H}_6\left({\tau}_{\alpha}\right)& 12{G}_4\left({\tau}_{\alpha}\right)& 2{G}_2\left({\tau}_{\alpha}\right)\end{array}\right|. $$

(40)