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Effect of Thermal Radiation, Temperature Jump and Inclined Magnetic Field on the Peristaltic Transport of Blood Flow in an Asymmetric Channel with Variable Viscosity and Heat Absorption/Generation

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Abstract

This article is aimed at describing the effect of thermal radiation, heat source/sink and thermal slip on the peristaltic flow of blood in an asymmetric channel. Mathematical analysis has been carried out in the presence of aligned magnetic field. The physical problem is first modeled, and then, the solutions of coupled nonlinear differential equations are obtained by means of regular perturbation method. Assumptions of long wavelength and low Reynolds number approximations were adopted. Physical characteristics of various embedded parameters such as Hartmann number, radiation parameter, aligned magnetic field parameter and heat source/sink parameter are illustrated and discussed graphically.

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Abbreviations

\(a_1, b_1\) :

Amplitudes of the wavy walls

a, b :

Amplitude ratios

\(d_1+d_2\) :

Width of channel

\(\lambda\) :

Wavelength

c :

Wave speed

\(t^{\prime }\) :

Time

u :

Axial velocity

d :

Mean half width of the channel

\(B_0\) :

Intensity of external magnetic field

\(\rho\) :

Constant density

g :

Acceleration due to gravity

\(\Theta\) :

Aligned magnetic field

K :

Thermal conductivity

\(\sigma\) :

Electrical conductivity of the fluid

\(c_p\) :

Specific heat

M :

Hartmann number

p :

Pressure

\(u^{\prime },v^{\prime }\) :

Velocity components in wave frame

\(\gamma\) :

Thermal slip parameter

\(k^{\prime }\) :

Mean absorption coefficient

\(\theta\) :

Temperature distribution

\(\alpha\) :

Reynolds model viscosity parameter

\(Q^{\prime }\) :

Heat generation coefficient

\(\mu _0\) :

Constant viscosity

\(q_r\) :

Radiative heat flux

R :

Thermal radiation parameter

\(\mu ^{\prime }(Y^{\prime })\) :

Viscosity function

\(\alpha ^{\prime }\) :

Coefficient of thermal expansion

Re :

Reynolds number

\(p^{\prime }\) :

Pressure in wave frame

\(\delta\) :

Wave number

Pr :

Prandtl number

Gr :

Grashof number

\(\beta\) :

Heat source/sink parameter

Q :

Volume flow rate

\(\sigma ^{\prime }\) :

Stefan–Boltzmann constant

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Author information

Authors and Affiliations

  1. Department of Mathematics, Annamalai University, Annamalainagar, 608 002, India

    Ajaz Ahmad Dar

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  1. Ajaz Ahmad Dar
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Correspondence to Ajaz Ahmad Dar.

Appendix

Appendix

$$\begin{aligned}&C_1=\frac{\beta \left( \gamma h_1-\gamma h_2-h_1 h_2\right) }{2 \left( 1+R\right)}.\\&C_2=-\frac{-\beta h_1-\beta h_2}{2 \left( 1+R\right) }.\\&C_3=\frac{e^{\sqrt{A_2} \left( h_1+h_2\right) } \left( A_4+A_3+\sqrt{A_2} Gr \left( h_1-h_2\right) {}^2 \left( A_6\right) \right) }{A_5}.\\&C_4=\frac{A_4-A_3+\sqrt{A_2} Gr \left( h_1-h_2\right) {}^2 \left( A_7\right) }{-A_5}.\\&C_5=\frac{A_8-A_9+A_{10}+A_{11}-\sqrt{A_2} Gr \left( A_{12}+A_{13}\right) }{6 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) A_2^{3/2} \left( A_{14}\right) }.\\&C_6=\frac{A_{17}-A_{16}+A_{15}}{6 A_2 A_{14}}.\\&C_7=\frac{-2 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) {}^2 A_2 \left( 4 C_3-e^{\sqrt{A_2} \left( h_1+h_2\right) } C_4\right) -A_{20}+A_{21}+A_{33}+A_{30}}{8\left( -e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) A_2^{3/2} A_{18}}.\\&C_8=\frac{e^{-\sqrt{A_2} \left( h_1+h_2\right) } A_{27}}{8 \left( -e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) A_2^{3/2} A_{18}}.\\&C_9=\frac{e^{-\sqrt{A_2} \left( h_1+h_2\right) } \left( 8 \left( e^{\sqrt{A_2} \left( 3 h_1+h_2\right) }-e^{\sqrt{A_2} \left( h_1+3 h_2\right) }\right) A_1 Gr \left( h_1-h_2\right) +A_{44}\right) }{4 \left( -e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) A_2^{5/2} A_{18}}.\\&C_{10}=\frac{e^{-\frac{3}{2} \sqrt{A_2} \left( h_1+h_2\right) } \left( A_{48}+4 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }+e^{\sqrt{A_2} \left( h_1+2h_2\right) }\right) A_2^{5/2} F_1+A_{50}+A_{46}+A_{47}\right) }{A_{45}}.\\&A_1=\frac{\beta }{1+R}, \,\,A_2=M^2\cos ^2\Theta .\\&A_3=3 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) Gr\left( h_1-h_2\right) \left( -2 C_2+A_1 \left( h_1+h_2\right) \right) .\\&A_4=-6 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) A_2^{3/2} \left( F_0+h_1-h_2\right) .\\&A_5=6 \left( -e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) \sqrt{A_2} \left( e^{\sqrt{A_2} h_1} \left( -2+\sqrt{A_2} \left( h_1-h_2\right) \right) +e^{\sqrt{A_2} h_2} \left( 2+\sqrt{A_2} \left( h_1-h_2\right) \right) \right) .\\&A_6=e^{\sqrt{A_2} h_2} \left( 3 C_2-A_1 \left( 2 h_1+h_2\right) \right) e^{\sqrt{A_2} h_1} \left( 3 C_2-A_1 \left( h_1+2 h_2\right) \right) .\\&A_7=e^{\sqrt{A_2} h_2} \left( -3 C_2+A_1 \left( h_1+2 h_2\right) \right)+e^{\sqrt{A_2} h_1} \left( -3 C_2+A_1 \left( 2 h_1+h_2\right) \right) .\\&A_8=-6 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) {}^2 A_2^{3/2} \left( h_1+h_2\right) .\\&A_9=3 \left( e^{2 \sqrt{A_2} h_1}-e^{2 \sqrt{A_2} h_2}\right) A_2^2 F_0 \left( h_1+h_2\right) .\\&A_{10}=\left( e^{2 \sqrt{A_2} h_1}-e^{2 \sqrt{A_2} h_2}\right) A_2 Gr h_1 h_2 \left( -h_1+h_2\right) \left( -3 C_2+A_1 \left( h_1+h_2\right) \right) .\\&A_{11}=3 \left( e^{2 \sqrt{A_2} h_1}-e^{2 \sqrt{A_2} h_2}\right) Gr \left( h_1-h_2\right)\left( -2 C_2+A_1 \left( h_1+h_2\right) \right) .\\ \end{aligned}$$
$$\begin{aligned} &A_{12}=\left( e^{2 \sqrt{A_2} h_1}+e^{2 \sqrt{A_2} h_2}\right) \left( h_1^2-4 h_1 h_2+h_2^2\right)\left( -3 C_2+A_1 \left( h_1+h_2\right) \right) .\\&A_{13}=2 e^{\sqrt{A_2} \left( h_1+h_2\right) } \left( -3 C_2 \left( h_1^2+h_2^2\right) +2 A_1 \left( h_1^3+h_2^3\right) \right) .\\&A_{14}=e^{\sqrt{A_2} h_1} \left( -2+\sqrt{A_2} \left( h_1-h_2\right) \right) +e^{\sqrt{A_2} h_2} \left( 2+\sqrt{A_2} \left( h_1-h_2\right) \right) .\\&A_{15}=\left( e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) \sqrt{A_2} Gr \left( h_1-h_2\right) \left( -3 C_2 \left( h_1+h_2\right) +A_1 \left( h_1^2+h_1h_2+h_2^2\right) \right) .\\&A_{16}=3 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) Gr \left( -2 C_2 \left( h_1+h_2\right) +A_1 \left( h_1^2+h_2^2\right) \right) .\\&A_{17}=12 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) A_2+6 \left( e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) A_2^{3/2} F_0.\\&A_{18}=e^{\sqrt{A_2} h_1} \left( -2+\sqrt{A_2} \left( h_1-h_2\right) \right) +e^{\sqrt{A_2} h_2} \left( 2+\sqrt{A_2} \left( h_1-h_2\right) \right) .\\&A_{19}=-2 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) {}^2 A_2 \left( C_3-4 e^{\sqrt{A_2} \left( h_1+h_2\right) } C_4\right) .\\&A_{20}=16 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }-e^{\sqrt{A_2} \left( h_1+2 h_2\right) }\right)A_1 Gr \left( h_1-h_2\right) .\\&A_{21}=8 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }+e^{\sqrt{A_2} \left( h_1+2 h_2\right) }\right)A_1 \sqrt{A_2} Gr \left( h_1-h_2\right) {}^2.\\&A_{22}=-5 \left( e^{\sqrt{A_2} \left( 3 h_1+h_2\right) }-e^{\sqrt{A_2} \left( h_1+3 h_2\right) }\right)C_4 \left( h_1-h_2\right) .\\&A_{23}=4 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) C_3 \left( -e^{\sqrt{A_2} h_2} h_1+e^{\sqrt{A_2} h_1} h_2\right) .\\&A_{24}=\left( h_1-h_2\right) \left( 4 e^{\sqrt{A_2} h_1}-4 e^{\sqrt{A_2} h_2}+e^{2 \sqrt{A_2} h_1} C_4 h_1^2-e^{2 \sqrt{A_2} h_2} C_4 h_2^2+C_3\left( h_1^2-h_2^2\right) \right) .\\&A_{25}=2 e^{\sqrt{A_2} \left( h_1+h_2\right) } A_2^{5/2} \left( 4 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) F_1+A_{24}\right) .\\&A_{26}=C_4 \left( \left( h_1+h_2\right) \left( e^{2 \sqrt{A_2} h_1} h_1+e^{2 \sqrt{A_2} h_2} h_2\right) -2 e^{\sqrt{A_2} \left( h_1+h_2\right) } \left( h_1^2+h_2^2\right) \right) .\\&A_{27}=A_{19}-A_{20}+A_{21}+A_2^{3/2} \left( A_{22}-A_{23}\right) +A_{25}-2 e^{\sqrt{A_2} \left( h_1+h_2\right) } A_2^2 \left( C_3 \left( h_1-h_2\right) {}^2+A_{26}\right) .\\&A_{28}=-4 e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }+4 e^{\sqrt{A_2} \left( h_1+2 h_2\right) }+e^{2 \sqrt{A_2} \left( h_1+h_2\right) } C_4 \left( h_1^2-h_2^2\right) \\&A_{29}=\left( h_1-h_2\right) \left( A_{28}+C_3 \left( e^{2 \sqrt{A_2} h_2} h_1^2-e^{2 \sqrt{A_2} h_1} h_2^2\right) \right) .\\&A_{30}=2 A_2^{5/2} \left( -4 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right)}-e^{\sqrt{A_2} \left( h_1+2 h_2\right) }\right) F_1+A_{29}\right) .\\&A_{31}=C_3 \left( \left( h_1+h_2\right) \left( e^{2 \sqrt{A_2} h_2} h_1+e^{2 \sqrt{A_2} h_1} h_2\right) -2 e^{\sqrt{A_2} \left( h_1+h_2\right) } \left( h_1^2+h_2^2\right) \right) .\\&A_{32}=5 \left( e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) C_3 \left( h_1-h_2\right)-4 C_4 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) } h_1-e^{\sqrt{A_2}\left( h_1+2 h_2\right) } h_2\right) .\\&A_{33}=\left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) A_2^{3/2} \left( A_{32}\right)+2 A_2^2 \left( e^{2 \sqrt{A_2} \left( h_1+h_2\right) } C_4 \left( h_1-h_2\right) {}^2+A_{31}\right) .\\&A_{34}=C_3 \left( e^{\sqrt{A_2} \left( h_1+2 h_2\right) } h_1-e^{\sqrt{A_2} \left( 2 h_1+h_2\right) } h_2\right)+C_4 \left( e^{\sqrt{A_2} \left( 3h_1+2 h_2\right) } h_1-e^{\sqrt{A_2} \left( 2 h_1+3 h_2\right) } h_2\right) .\\&A_{35}=2 A_2^{5/2} \left( h_1+h_2\right) \left( -2 e^{\sqrt{A_2} \left( h_1+h_2\right) } \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) {}^2+\left( h_1-h_2\right) \left( A_{34}\right) \right) .\\&A_{36}=e^{\sqrt{A_2} \left( 2 h_1+3 h_2\right) } h_1^2-\left( e^{\sqrt{A_2} \left( 4 h_1+h_2\right) }+e^{\sqrt{A_2} \left( h_1+4 h_2\right) }\right) h_1 h_2+e^{\sqrt{A_2} \left( 3 h_1+2 h_2\right) } h_2^2.\\&A_{37}=C_3 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) } h_1^2-\left( e^{3 \sqrt{A_2} h_1}+e^{3 \sqrt{A_2} h_2}\right) h_1 h_2+e^{\sqrt{A_2} \left( h_1+2h_2\right) } h_2^2\right) -C_4 \left( A_{36}\right) .\\&A_{38}=8 e^{2 \sqrt{A_2} \left( h_1+h_2\right) } A_1 \sqrt{A_2} Gr \left( h_1^2+h_2^2+{\text{Cosh}}\left[ \sqrt{A_2} \left( h_1-h_2\right) \right] \left( h_1^2-4 h_1 h_2+h_2^2\right) \right) .\\&A_{39}=e^{\sqrt{A_2} \left( h_1+h_2\right) } \left( \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) C_4+4 A_1 Gr h_1 \left( h_1-h_2\right) h_2\right) .\\ \end{aligned}$$
$$\begin{aligned}&A_{40}=\left( e^{2 \sqrt{A_2} h_1}-e^{2 \sqrt{A_2} h_2}\right) A_2 \left( \left( -e^{\sqrt{A_2} h_1}+e^{\sqrt{A_2} h_2}\right) C_3+A_{39}\right) .\\&A_{41}=\left( e^{3 \sqrt{A_2} h_1}-2 e^{3 \sqrt{A_2} h_2}+e^{\sqrt{A_2} \left( h_1+2 h_2\right) }\right) h_1 +\left( -2 e^{3 \sqrt{A_2} h_1}+e^{3\sqrt{A_2} h_2}+e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }\right) h_2.\\&A_{42}=2 e^{\sqrt{A_2} \left( 4 h_1+h_2\right) } h_1-e^{\sqrt{A_2} \left( 3 h_1+2 h_2\right) } h_1-e^{\sqrt{A_2} \left( h_1+4 h_2\right) } \left( h_1-2h_2\right) .\\&A_{43}=A_2^{3/2} \left( C_4 \left( A_{42}-e^{\sqrt{A_2} \left( 4 h_1+h_2\right) } h_2-e^{\sqrt{A_2} \left( 2 h_1+3 h_2\right) } h_2\right) -C_3 \left( A_{41}\right) \right) .\\&A_{44}=2 \left( e^{\sqrt{A_2} \left( 3 h_1+h_2\right) }-e^{\sqrt{A_2} \left( h_1+3 h_2\right) }\right) A_2^3 F_1 \left( h_1+h_2\right) +A_{43}-A_{40}-A_{38}+2A_2^2 \left( A_{37}\right) -A_{35}.\\&A_{45}=8 A_2^2 \left( -2 \text {Sinh}\left[ \frac{1}{2} \sqrt{A_2} \left( h_1-h_2\right) \right] +{\text {Cosh}}\left[ \frac{1}{2} \sqrt{A_2} \left( h_1-h_2\right) \right] \sqrt{A_2} \left( h_1-h_2\right) \right) .\\&A_{46}=2 A_2^{3/2} \left( C_3 \left( e^{2 \sqrt{A_2} h_2} h_1-e^{2 \sqrt{A_2} h_1} h_2\right)+C_4 \left( e^{\sqrt{A_2} \left( 3 h_1+h_2\right) }h_1-e^{\sqrt{A_2} \left( h_1+3 h_2\right) } h_2\right) \right) .\\&A_{47}=2 e^{\sqrt{A_2} \left( h_1+h_2\right) } A_2^2 \left( 4 \left( e^{\sqrt{A_2} h_1}-e^{\sqrt{A_2} h_2}\right) +\left( C_3-e^{\sqrt{A_2} \left( h_1+h_2\right) }C_4\right) \left( h_1^2-h_2^2\right) \right) .\\&A_{48}=-\left( e^{2 \sqrt{A_2} h_1}-e^{2 \sqrt{A_2} h_2}\right) A_2 \left( C_3+e^{\sqrt{A_2} \left( h_1+h_2\right) } C_4\right) .\\&A_{49}=8 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }-e^{\sqrt{A_2} \left( h_1+2 h_2\right) }\right) A_1 Gr \left( h_1+h_2\right) .\\&A_{50}=A_{49}-4 \left( e^{\sqrt{A_2} \left( 2 h_1+h_2\right) }+e^{\sqrt{A_2} \left( h_1+2 h_2\right) }\right)A_1 \sqrt{A_2} Gr \left( h_1^2-h_2^2\right) . \end{aligned}$$

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Dar, A.A. Effect of Thermal Radiation, Temperature Jump and Inclined Magnetic Field on the Peristaltic Transport of Blood Flow in an Asymmetric Channel with Variable Viscosity and Heat Absorption/Generation. Iran J Sci Technol Trans Mech Eng 45, 487–501 (2021). https://doi.org/10.1007/s40997-020-00349-6

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