Analytical model of solutions of (2+1)-D heat convection equations in a shape memory alloy device immersed in a blood vessel

We investigate a bio-system composed of a shape memory alloy (SMA) immersed and subjected to heat convection in a blood vessel, affected by heart beats that create a wave motion of long wavelength. The tackled model in (2+1)-D is based on the continuity and momentum equations for the fluid phase, besides; the state of the SMA are described via previous works in the form of statistical distributions of energy for both Martensite and Austenite phases. The solution based on the reductive perturbation technique gives a thermal diffusion-like equation as a key for expressing the temperature and velocity components of the blood. In terms of two cases concerning the difference between the wave numbers in the perpendicular directions, it is found that the system’s temperature increases nonlinearly from a minimum initial temperature 293 K (20 °C) up to a maximum value about 316.68 K (43.68 °C), then tends to decrease along the blood flow (anisotropy of K and L) direction. In both cases it is observed that the SMA acquires most of this temperature raising not the blood because of its conventional biological limits (37–40 °C). The range of the heart beats wave numbers characteristic for each person plays an important role in realizing phase changes in the anisotropic case leading to the formation of the hysteresis loops Martensite-Austenite-Martensite or vice versa, according to the energy variation. The entropy generation σ is investigated for the system (Blood + SMA), it predicts that along the flow direction the system gains energy convectively up to a maximum value, then reverses his tendency to gradually loosing energy passing by the equilibrium state, then the system looses energy to the surroundings by the same amount which was gained beforehand. The loss diminishes but stops before arriving to equilibrium again. For certain differences in wave numbers the system starts to store energy again after it passes by the state of equilibrium for the second time. In the curves of σ the common points of intersections can be looked for as the positions where the phase changes take place. It is observed that the effect of heat transfer is dominated over the viscous dissipation substantially; this is illustrated by the irreversibility distribution ratio ϕ and the Bejan number. On the other hand this is assured by the smallness of the ratio between the initial effect of shear viscosity to the initial thermal effect in the alloy (Γ ≈ 10−8). Furthermore, this allows the SMA