Basic idea for the problem of a magnetohydrodynamic flow of blood in a porous channel

Magntohydrodynamic (MHD) is the part of mechanics concerned with the study of moving electrically conductive fluids in the presence of magnetic and electric fields. The importance of the MHD lies on providing several possibilities of application such as in metallurgy, electricity generation, and electromagnetic boosters. In the same way, MHD is used to study the flow of arterial blood under the action of an applied magnetic field (^{Misra et al., 2011}). In the medical field, these investigations are of great value in the treatment of cardiovascular disorders, considering the case of accelerated circulation. Given that blood erythrocytes have small negative charges associated to them, it is expected that the application of a magnetic field can influence the movement of the erythrocytes and, as a consequence, that the flow of blood can possibly be affected too.

In effect, the behaviour of the blood flow subjected to different situations has been reported by several authors. For example, ^{Misra & Shit (2007)} proposed a model in order to study the blood flow through a stenosed arterial segment considering the slip velocity at the wall of the artery. ^{Fung & Sobin (1969)} reported their investigation about the flow of blood between two endothelial layers, and the mass transference problem in an alveolar sheet was studied by ^{Fung & Tang (1975)}. In the same way, ^{Misra et al. (2008)} and ^{Misra & Shit (2009)} mathematically modelled the blood flow in a channel with stretching walls, considering the blood as a non-Newtonian fluid. On the other hand, ^{Fukada & Kaibara (1980)} noted that under some conditions blood could exhibit viscoelastic behaviour due to the viscoelastic properties of the erythrocytes.

This paper is dedicated to get an analytical approximate solution for the MHD boundary layer flow of blood in the aortic arch (^{Misra et al., 2011}). The flow is assumed to obey Walter’s liquid-B model, and this supposition results in a highly nonlinear partial differential equation (NPDE). Nevertheless, after introducing some non-dimensional quantities, it is possible to express the above mentioned NPDE in terms of a complicated and long nonlinear ordinary differential equation (NODE). Despite the difficulty of the nonlinear ODE to be solved, this work will provide a handy analytical approximate solution that is ideal for practical applications.

Figure 1 shows the physical model of a steady blood flow in a channel provided with porous boundaries bounded by two thin parallel plates which is under the effect of a transverse magnetic field. The erythrocytes (red blood cells) are the particles influenced by the magnetic field because they have a small negative charge. The x-axis is chosen along the centre line of the channel, and the y-axis is in the transverse direction. In the same way, the flow is assumed to be symmetric about x-axis, and the porous walls are represented by the values y=R/2 and y=-R/2, where R is the channel width. The fluid injection (extraction) is given through the porous with velocity V/2(V>0 suction and V<O injection).

Source: Author’s own elaboration.

Figure 1 Figure shows the physical model of a porous channel through which blood flows under the action of a magnetic transverse field. 

Let u, v be the x component and y component of the velocity; Bo is the strength of the applied magnetic field; and ρ, μ, σ, Ko are respectively the density, kinematic viscosity, electrical conductivity, and coefficient of viscoelasticity of the fluid. Therefore, the MHD boundary layer flows for an incompressible fluid (Walter’s Liquid-B fluid) in terms of the non-dimensional variables:

is given by the nonlinear equation:

where Re=VRν is the Reynolds number, M=Ra2=σB02R2ν denotes the square of the Hartmann number, and the viscoelastic parameter is given by K1=VK0Rν.

In terms of the non-dimensional distance η (from equation 1), the boundary conditions for equation 2 are expressed as:

Therefore, the problem consists in approximating equation 2 subjected to boundary conditions (equation 3).

Basic idea of the novel boundary value problems Picard method (BVPP)

The basic idea of Picard iteration method (PIM) consists in expressing the problem:

where, in accordance with the local existence and uniqueness Picard’s theorem, y(t) is a function whose graph passes through (t0,A) and satisfies y'=ft,y in some neighbourhood of t0, in terms of the following integral equation derived from equation 4 (^{Elsgolts, 1977}; ^{Filobello-Nino et al., 2016a}; ^{Simmons, 2016}; ^{Zill, 2012}):

As a matter of fact, PIM expresses the solution of equation 5 as the limit of a sequence of functions yn(t) (n→∞) defined through the recurrence formula:

A relevant point about equation 6 occurs when f(t,y) is continuous in all its arguments and satisfies a Lipschitz condition in the y variable on a band defined by a≤x≤b and -∞<y<∞. If the above conditions are satisfied, then let (x0,y0) be any point of the above mentioned band, then the initial value problem (equation 4) has one and only one solution in a≤x≤b. In a sequence, it is known that independently of the selected initial function y0(t), the sequence yn(t) caused by the iterative procedure (equation 6) converges to the solution of equation 4 in the above mentioned interval. This convergence highlights the effectiveness of the Picard method for solving differential equations, with a more in-depth discussion available in works like those of ^{Filobello-Nino et al. (2016a)} and ^{Simmons (2016)}, the latter being an essential resource for a comprehensive understanding of the method.

Next, we focus on the case of boundary value problems (BVP) with Dirichlet boundary conditions, assuming that the values of the sought solution are given at two points t0 and t1. Therefore, let us consider the following problem:

To begin, let us approach the problem, assuming that we know the value of y'(t0) (although it is unknown), and that the right hand side of equation 7 is a continuous function. On the other hand, the freedom of BVPP method is employed with the purpose of choosing an adequate trial function y0(t), aiming to include the two boundary values and to accelerate the convergence of the procedure (^{Filobello-Nino et al., 2016a)}. BVPP method proposes as trial function a polynomial function P(t), which contains one or more adjusting parameters D,E,F,... to be determined, that is:

In accordance to equation 7, BVPP employs the following integral equation:

where, as it was already mentioned, the value of y'(t0)=β is unknown for the time being, and it is determined by the BVPP alghorithm.

The solution for equation 9 can be expressed as the limit of a sequence of functions yn(t) (n→∞), in accordance to the following recurrence formula:

Assuming that f(t,y,y') is continous in all its arguments and also that it satisfies a Lipschitz condition in y and y' variables, in a neighbourhood of the point (t0,y0,y'0), then, irrespective of equation 8, the successive approximations yn(t) which follow from equation 10 converge to a solution of the following problem, which resembles equation 7 (^{Filobello-Nino et al., 2016a}; ^{Simmons, 2016}).

in a small neighbourhood of the point t0.

Nevertheless, sometimes it occurs that f(t,y,y') satisfies a Lipschitz condition in the band t0≤t≤t1, -∞≤y≤∞, and -∞≤y'≤∞. In this case, if (t0,y0,y'0) is any point into the above mentioned band, then the equation 11 has one and only one solution: y=y(t) in t0≤t≤t1 (^{Simmons, 2016}).

Next, in order to ensure that the n-th iteration of BVPP (equation 10) is also an approximate solution for equation 7, the values of β,D,E,F,… are chosen in order to guarantee that the approximate solution satisfies y(t1)=C and, for the same reason, equation 7. Although equation 7 and equation 11 are related in this way, in practice it is not necessary to explicitly consider the auxiliary equation 11. There is still the question of calculating the earlier mentioned parameters. Although ^{Filobello-Nino et al. (2016a)} indicate three manners to optimally calculate their values, in this work we will only employ the first of them. The basic rudiments of this procedure, denominated method 1, are as follows (^{Filobello-Nino et al., 2016a}).

In practical situations, it is assumed that the nth approximation (for some n) is sufficient, then we will symbolically write equation 10 as follows:

where H(t,β,D,E,F,..) represents the function obtained from the mentioned iterative process.

This method assumes as known, for instance, the numerical solution of ecuation 7 or somehow a known set of points of the sought solution curve (such as it will occur later in this work); hence, equation 12 is evaluated at as many points within the interval t0,t1 as parameters to be determined:

where t2,t3,t4∈ (t0,t1) and the values of yn(t0), yn(t1), yn(t2), yn(t3), yn(t4),… are assumed to be known.

Equation 13 is a system of algebraic equations, whose solution allows to know the value of the parameters β,D,E,F,… It is expected that the vectorial function F of the component equations of the system (equation 13) F:Rn→Rn(β,D,E,F)→(H(t0,β,D,E,F,…)-yn(t0),…,H(tn,β,D,E,F,…)-yn(tn)) is of class C1(Rn) and its Jacobian matrix is not zero at any point in order to guarantee the existence of a unique solution of equation 13. In addition, it is noted that equation 12 should provide a good approximation, after considering several inner points, because it is ensured that it will pass through n-points of the exact solution.

On the other hand, the above mentioned procedure for BVPP can be extended for the case of nonlinear differential equations of order greater than two. In this case, the boundary conditions of the problem will not only specify special values of the sought solution, as it occurs in equation 7, but in the same way they will contain derivative values evaluated in some points of the problem domain. In this case, we have to add other equations to the system of algebraic equations 13, obtained after differentiating equation 12, in order to guarantee that the approximate solution (equation 12) satisfies all the boundary conditions of the proposed problem.

Application of boundary value problems Picard method (BVPP) to obtain an explicit analytic approximate solution for the magnetohydrodynamic flow of blood in a porous channel

In this section, we will get an explicit analytical approximate solution for the problem afore explained, emphasizing the ease with which a handy and precise analytical approximated solution is obtained for the highly nonlinear problem (equations 2 and 3), despite the difficulty of the nonlinear differential equation 2.

To start, equation 2 is rewritten in terms of the following integral equation:

where A is the unknown value of f'(η=0).

In accordance to the proposed method, the following iterative process is introduced from equation 14:

n=1,2,3,4....

The first iteration for equation 15 corresponds to n=1:

Next, taking advantage of the flexibility of the BVPP method, the following polynomial is proposed as the initial approximation:

Therefore, after substituting equation 17 into equation 16 and performing elementary integrations, we get:

Two solutions are provided from equation 18 for the two set of parameters M, Re, and K1. As a matter of fact, in accordance with the proposed method, the numerical solution of equations 2 and 3 is employed in order to set up a nonlinear system to calculate the values of A, B, C, D, E and F.

The first case study assumes the following values of the parameters (^{Vazquez-Leal, 2020}):

From the numerical solution of this problem, for this set of parameters (equation 19) (^{Vazquez-Leal, 2020}), we obtain the following points:

After substituting the points from equation 20 into equation 18, and also into the necessary derivatives of equation 18, we get a system of nonlinear algebraic equations for the unknown values A, B, C, D, E and F, whose solution results in the following approximate solution.

Table 1 compares the exact solution (^{Vazquez-Leal, 2020}), the approximated solution proposed for the novel continuum-cancelation Leal method (CCLM) (^{Vazquez-Leal, 2020}) for the same problem with parameters (equation 19), and the BVPP solution (equation 21). It is worth noting that equation 21 provides not only a handy solution but an accurate one (see Discussion section).

The second case study proposes the following values of the parameters:

Again, from the numerical solution for this set of parameters (equation 22) we get:

Next, we substitute equation 23 into equation 18, and also into the necessary derivatives of equation 18, aiming to obtain a set of nonlinear algebraic equations for the unknown A, B, C, D, E and F whose solutions provide the following approximate solution:

Table 2 compares the exact solution of the proposed problem with parameters (equation 22) and approximated solution (equation 24). It is clear that equation 24 provides not only a handy solution but an accurate one.