To non-dimensionalize the obtained governing equations, we introduce the non-dimensional variables as follows:

$$\begin{gathered} \;\tilde{u} = \frac{{uu_{0} \delta }}{b},\;\;\tilde{r} = rd_{0\;} ,\;\;\tilde{z} = zb\;,\;\;\tilde{v} = wu_{0\;} ,\;\;\tilde{h} = hd_{0} ,\;\;\tilde{P} = \frac{{u_{0} b\mu_{0} p}}{{d_{0}^{2} }}\;, \hfill \\ {\text{Re}} = \;\frac{{\rho bu_{0} }}{{\mu_{0} }},\,\,\theta = \frac{{T - T_{0} }}{{T_{0} }},\,\,\;\Pr = \frac{{\mu c_{p} }}{k},\;\,\,{\text{Ec}} = \frac{{\mu_{0}^{2} }}{{c_{{pT_{0} }} }},\;\,\,Z = \frac{{k_{1} }}{{d_{0}^{2} }}\;, \hfill \\ M = \frac{{\sigma_{1} H_{0}^{2} d_{0}^{2} }}{{\mu_{0} }}\;,\;\,\,Q = A\frac{{d_{0}^{2} }}{K}\;,\;\,G_{r} = \frac{{g\alpha d_{0}^{3} T_{0} }}{{v^{2} }},\,\;N^{2} = \frac{{4d_{0}^{2} \alpha_{v}^{2} }}{k}\;,\,\,G_{{\text{N}}} = \frac{{\alpha_{2} }}{{\alpha_{1} }}T_{0} . \hfill \\ \end{gathered}$$

(11)

where \(\Pr ,\;Z\;,\;N\;,\;{\text{Re}} \;,\;\theta ,\;Z\;\;{\text{Gr}}{\kern 1pt} \;M\;,\;{\text{Ec}}\;{\text{and}}\;\;G_{{\text{N}}},\) respectively, represent the Prandtl number, porosity parameter, radiation absorption parameter, Reynolds number, temperature parameter, Grashof number, magnetic field parameter, Eckert number, and nonlinear thermal convection. In the case of aortic stenosis \(\frac{\delta }{{d_{0} }}{ \ll }1\) and the other additional conditions,

$${\text{Re}} \frac{{\delta^{*} n^{{\frac{1}{n - 1}}} }}{b} \ll 1,$$

(12)

assuming the following approximation:

$$\frac{{d_{0\;} n^{{\frac{1}{n - 1}}} }}{b}\; \sim \;O\left( 1 \right),$$

(13)

To non-dimensionalize the continuity equation, we substitute the non-dimensional quantities in (11) into (1) to obtain:

$$\frac{{u_{0} \delta }}{{bd_{0} }}\frac{\partial u}{{\partial r}} + \frac{{uu_{0} \delta }}{{rbd_{0} }} + \frac{{u_{0} }}{b}\frac{\partial w}{{\partial z}} = 0,$$

(14)

Since \(\frac{\delta }{{d_{0} }}{ \ll }1\),

$$\frac{{u_{0} }}{b}\frac{\partial w}{{\partial z}} = 0,$$

(15)

$$\frac{\partial w}{{\partial z}} = 0.$$

(16)

To non-dimensionalize the momentum equation (\(r\)-direction), substitute (11) into (2) to obtain:

Also, since \(\frac{\delta }{{d_{0} }}{ \ll }1\), \(\frac{\partial w}{{\partial z}} = 0\),

$$- \frac{{u_{0} b\mu_{0} }}{{d_{0}^{3} }}\frac{\partial p}{{\partial z}} = 0,$$

(17)

$$\frac{\partial p}{{\partial z}} = 0.$$

(18)

Also, substituting the non-dimensional variables in (11) and (7) into the momentum equation (*z*-direction):

$$\begin{aligned} \frac{\partial p}{{\partial z}} & = \left( {1 + \frac{1}{\beta }} \right)\left[ {H_{r} \left( {\frac{1}{r} - r^{m - 1} \left( {m + 1} \right)} \right)} \right]\frac{\partial w}{{\partial r}} + \left( {1 + \frac{1}{\beta }} \right)\left[ {1 + H_{r} \left( {1 - r^{m} } \right)} \right]\frac{{\partial^{2} w}}{{\partial r^{2} }}wM^{2} \\ & \quad + G_{r} \left[ {\theta + G_{{\text{N}}} \theta^{2} } \right]\cos \gamma \; - w\left( {1 + \frac{1}{\beta }} \right)\frac{{H_{r} }}{Z}\left( {1 - r^{m} } \right) - \frac{{b^{*} w^{2} d_{0}^{2} }}{{k_{1} }}, \\ \end{aligned}$$

(19)

where \(G_{N} = \frac{{\alpha_{2} T_{0} }}{{\alpha_{1} }}\) is the nonlinear thermal convection.

Also, using the non-dimensional variables in Eq. (11), the energy equation becomes

$$\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial \theta }{{\partial r}}} \right) + \;{\text{Ec}}\Pr \left( {1 + \frac{1}{\beta }} \right)\left[ {1 + H_{r} \left( {1 - r^{m} } \right)} \right]\left( {\frac{\partial w}{{\partial r}}} \right)^{2} - N^{2} \theta = 0,$$

(20)

where \(B_{r} = \;{\text{Ec}}\Pr\) Brinkman number (\(B_{r}\)) is a dimensionless number used to study viscous flow. The corresponding boundary conditions are

$$\;\frac{\partial \theta }{{\partial r}} = 0,\;\,\,\,\frac{\partial w}{{\partial r}} = 0,\;\;\;\;at\;\;r = 0,$$

(21)

and the no-slip boundary conditions (assuming that at a solid boundary, the fluid will have zero velocity relative to the boundary) at the artery wall

$$\;w = 0,\,\,\,\theta = 0,\quad {\text{at}}\;\;r = h\left( z \right),$$

(22)

where *h*(*z*) is defined by

$$\begin{gathered} h(z) = \left( {1 + \xi^{\prime } z} \right)\left[ {1 - \eta_{1} \left( {(z - a^{\prime } ) - (z - a^{\prime } )^{n} } \right)} \right], \hfill \\ \;\;{\text{where}}\;a^{\prime } \le z \le a^{\prime } + 1 \hfill \\ \end{gathered}.$$

(23)

With the use of the Legendre collocation method, we have to define some functions. Let \(P_{n} (x)\) be the Legendre polynomial function of degree \(n\). We recall that \(P(x)\) is the solution (eigenfunction) of the Sturm–Liouville problem as follows:

$$\begin{aligned} & \left[ {\left( {1 - x^{2} } \right)P_{n}^{^{\prime}} \left( x \right)} \right]^{^{\prime}} + n\left( {n + 1} \right)p_{n} \left( x \right) = 0, \\ & x \in \left[ { - 1,1} \right],\;n = 0,1,2,3, \ldots \\ \end{aligned}.$$

(24)

Equation (24) satisfies the recursive relations:

$$P_{0} \left( x \right) = 1\;,\;P_{1} \left( x \right) = x\;,\;P_{2} \left( x \right) = \frac{1}{2}\left( {3x^{2} - 1} \right),$$

(25)

$$P_{n} \left( x \right) = \;\frac{2n - 1}{n}\;x\;P_{n - 1} \left( x \right) - \;\frac{n - 1}{n}P_{n - 2} \left( x \right)\;\;;n \ge 1,$$

(26)

The set of Legendre polynomials from a [− 1,1] orthogonal set is

$$\int_{ - 1}^{1} P_{n} \left( x \right)P_{m} \left( x \right)\;w\left( x \right){\text{d}}x = \;\frac{2}{2n + 1}\delta_{m,n} ,$$

(27)

where \(\delta_{m,n}\) is the Kronecker delta function. To apply the Legendre polynomial to the problem with a semi-infinite domain, we introduce algebraic mapping

$$x = \frac{2\varsigma }{h} - 1,[ - 1,1] \to [0,h],$$

(28)

the boundary value problem is solved within the region [0, \(h\)] in place of [0,\(\infty\)), whereas the scaling parameter is taken to be sufficiently large enough to evaluate the thickness of the boundary layer. Therefore, the real solutions \(f\left( \varsigma \right)\) and \(\theta \left( \varsigma \right)\) are expressed as the basis of the Legendre polynomial function as

$$f\left( \varsigma \right) = \sum\limits_{j = 0}^{N} a_{j} P_{j} \left( \varsigma \right),\;\;\,\,\theta \left( \varsigma \right) = \sum\limits_{j = 0}^{N} b_{j} P_{j} \left( \varsigma \right),\,\,\,{\text{for}}\quad j = 0,1,2,3, \ldots ,$$

(29)

Hence,

$$\begin{gathered} f\left( \varsigma \right) = a_{0} P_{0} \left( \varsigma \right) + a_{1} P_{1} \left( \varsigma \right) + a_{2} P_{2} \left( \varsigma \right) + \cdots , \hfill \\ \theta \left( \varsigma \right) = b_{0} P_{0} \left( \varsigma \right) + b_{1} P_{1} \left( \varsigma \right) + b_{2} P_{2} \left( \varsigma \right) + \cdots , \hfill \\ \end{gathered}$$

(30)

where \(P_{0} \left( \varsigma \right)\), \(P_{1} \left( \varsigma \right)\), \(P_{2} \left( \varsigma \right)\),…,\(P_{n} \left( \varsigma \right)\) are generated from recursive relation in (26) and

$$P_{0} \left( \varsigma \right) = 1\;,\;P_{1} \left( \varsigma \right) = \varsigma \;,\;P_{2} \left( \varsigma \right) = \frac{1}{2}\left( {3\varsigma^{2} - 1} \right), \ldots$$

(31)

Hence, substituting Eq. (31) into (30)

$$f\left( \varsigma \right) = a_{0} + a_{1} \left( {\frac{2\varsigma }{h} - 1} \right) + \frac{{a_{2} }}{2}\left[ {3\left( {\frac{2\varsigma }{h} - 1\;} \right)^{2} - 1} \right] + \cdots ,$$

(32)

$$\theta \left( \varsigma \right) = b_{0} + b_{1} \left( {\frac{2\varsigma }{h} - 1} \right) + \frac{{b_{2} }}{2}\left[ {3\left( {\frac{2\varsigma }{h} - 1\;} \right)^{2} - 1} \right] + \cdots ,$$

(33)

for \(h = 6\) and \(N = 6\), Eqs. (32–33) become

$$f\left( \varsigma \right) = a_{0} + \frac{{a_{1} }}{6}\left( { - 6 + 2\varsigma } \right) + \frac{{a_{2} }}{6}\left[ {6 - 6\varsigma + \varsigma^{2} } \right] + \cdots$$

(34)

$$\theta \left( \varsigma \right) = b_{0} + \frac{{b_{1} }}{6}\left( { - 6 + 2\varsigma } \right) + \frac{{b_{2} }}{6}\left[ {6 - 6\varsigma + \varsigma^{2} } \right] + \cdots$$

(35)

We assumed that \(w(r)\) and \(\theta (r)\) are the Legendre base trial functions, defined by

$$w(z) = \sum\limits_{j = 0}^{N} a_{j} P_{j} \left( {\frac{2z}{h} - 1\;} \right),\;\theta (r) = \sum\limits_{j = 0}^{N} b_{j} P_{j} \left( {\frac{2r}{h} - 1\;} \right),$$

(36)

where \(a_{j}\) and \(b_{j}\) are constants to be determined and \(P_{j} \left( {\frac{2r}{h} - 1} \right)\) is the shifted Legendre function from \([ - 1,1]\) to \([0,h]\). Substituting (36) into the boundary conditions in (21) and (22), respectively, we have

$$\begin{array}{*{20}l} {\left[ {\frac{{\text{d}}}{{{\text{d}}r}}\sum\limits_{j = 0}^{N} a_{j} P_{j} \left( {\frac{2r}{h} - 1} \right)} \right]_{r = 0} = 0} \hfill \\ \end{array} ,\;\begin{array}{*{20}l} {\left[ {\frac{{\text{d}}}{{{\text{d}}r}}\sum\limits_{j = 0}^{N} b_{j} P_{j} \left( {\frac{2r}{h} - 1} \right)} \right]_{r = 0} = 0} \hfill \\ \end{array} ,$$

(37)

$$\begin{array}{*{20}l} {\left[ {\sum\limits_{j = 0}^{N} a_{j} P_{j} \left( {\frac{2r}{h} - 1} \right)} \right]_{r = h\left( z \right)} = 0} \hfill \\ \end{array} ,\;\begin{array}{*{20}l} {\left[ {\sum\limits_{j = 0}^{N} b_{j} P_{j} \left( {\frac{2r}{h} - 1} \right)} \right]_{r = h\left( z \right)} = 0} \hfill \\ \end{array} ,$$

(38)

Residues \(D_{w} \left( {r,a_{j} ,b_{j} } \right)\) and \(D_{\theta } \left( {r,a_{j} ,b_{j} } \right)\) are derived from the above (39) and (40) accordingly.

The residues are minimized close to zero using the collocation method as follows:

$$\begin{array}{*{20}l} {{\text{for}}\;\delta \left( {r - r_{j} } \right) = \left\{ {\begin{array}{*{20}c} {1,} & {t = t_{j} } \\ {0,} & {{\text{otherwise}},} \\ \end{array} } \right.} \hfill \\ \end{array}$$

$$\begin{array}{*{20}l} {\int_{0}^{1} D_{w} \delta \left( {r - r_{j} } \right){\text{d}}r = D_{f} \left( {r_{j} ,a_{k} ,b_{k} } \right) = 0,\;\;\;{\text{for}}\;j = 1,\;2, \ldots N - 1} \hfill \\ \end{array}$$

(39)

$$\begin{array}{*{20}l} {\int_{0}^{1} D_{\theta } \delta \left( {r - r_{j} } \right)dr = D_{\theta } \left( {r_{j} ,a_{k} ,b_{k} } \right) = 0,\;\;\;{\text{for}}\;j = 1,\;2,..N - 1} \hfill \\ \end{array}$$

(40)

The above procedure sought the unknown constant coefficients \(a_{j} ,\) and \(b_{j}\) which are then substituted in Eq. (36) as the required solution.