mathematical models for biomagnetic fluid flow and applications

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model is used to obtain numerical solution of the differential equations describing the fluid flow (blood) in a rectangular channel under the action of a magnetic field. The obtained numerical results, presented graphically, showed that the flow is appreciably influenced by the magnetic field. These results indicate that application of a magnetic field, in the flow of a biomagnetic fluid, could be useful for medical and engineering applications. 3. MATHEMATICAL FORMULATION The mathematical model for the Biomagnetic Fluid Dynamics is based on the modified Stokes principles and on the assumption that besides the three thermodynamic variables P, ρ and T the biomagnetic fluid behavior is also a function of magnetization M [6]. Under these assumptions, the governing equations for incompressible fluid flow are similar to those derived for FerroHydroDynamics (FHD) [7], and are given by : Continuity Equation ∇⋅ V= 0 (1) Linear Momentum DV 2 2 ρ =−∇ p+ρ F+η∇ V+ξ( ∇ V+ 2∇×ω ) +µ o ( M⋅∇ ) H+ ⎡∇× H⎤× Dt ⎣ ⎦ B (2) Angular Momentum Dω 2 ρ I =µ oM× H+η∇ ′ ω+ 2ξ( ∇× V−2ω ) (3) Dt Magnetization DM 1 =ω× M− ⎡M−M ⎤ o Dt τ ⎣ ⎦ (4) Maxwell Equations (Amperes law and Gauss second law) ∇× H= J =σ ( V× B) (5) ∇⋅ B=∇⋅ H+ M = 0 ( ) Magnetization Equations These equations describe the dependence of saturation or static magnetization M o on the applied magnetic field intensity H and temperature T and the appropriate expressions are given below. In the above equations V is the velocity field, ρ is the fluid density, P is the pressure, F is the body force per unit volume, η and ξ are the coefficients of dynamical and rotational viscosity, respectively, ω is the angular velocity, µ is the magnetic permeability, M is the magnetization, H is the magnetic field intensity, B is the magnetic induction, I is the moment of inertia, per unit mass, η ′ is the shear spin viscosity, τ is the magnetic relaxation time, M is the saturation magnetization and σ is the electrical conductivity of the fluid. o 3.1 Equilibrium Flow The above set of equations is a very complicated system and simplifications must be made in order to solve it for a specific problem. As a first approximation we can consider that biofluids are poor conductors and the induced current is negligibly small. Thus, unlike MHD, Lorentz forces (the last term in eq.(2)) are much smaller in comparison to the magnetization force (the last but one term in eq.(2)). Also, ο
for medium shear rates, like blood flow in artery, the diffusion of the spin term is much smaller than that of the magnetic torque or the exchange between internal and external momentum. The major simplification however, can take place if we consider that the magnetic fluid has either achieved instantaneous magnetization or time has elapsed beyond the relaxation time, after the flow has exposed to the magnetic field. In this situation the flow can be considered as equilibrium flow and once the particle reaches saturation magnetization it will not have addition magnetization even if the magnetic field is further increased. Under the equilibrium assumption the fluid magnetization vector, M , at any given instant is parallel to the vector of the magnetic field intensity, H , and the property of magnetization is determined by the fluid temperature, density and magnetic field intensity M=M(T,ρ,H). Although the equilibrium flow is an idealization for the physical behavior of the biomagnetic fluid, it provides a good insight to the biomagnetic fluid flow since the governing equations are much more simpler than the complete set of equations derived in the previous sections for non equilibrium case. The equations of motion for the equilibrium flow can be written now as: Continuity Equation ∇⋅ V= 0, (6) Linear Momentum DV 2 ρ =−∇ p+ρ F+η∇ V+µ oMo∇H, (7) Dt 1/2 2 2 where H= ⎡ ⎣Hx + H ⎤ y⎦ . 3.2 Saturation Magnetization Equations In equilibrium situation the magnetization property is generally determined by the fluid temperature, density and magnetic field intensity and various equations, describing the dependence of M o on these quantities, are given in bibliography [6], [7]. The simplest relation is the linear equation of state, given in [8]: M = K( T −T), (8) o c where K is a constant called pyromagnetic coefficient and T c is the Curie temperature. Above the Curie temperature the biofluid does not subjected to magnetization. Another equation for magnetization, below the Curie temperature is given in [9] M c o = 1⎜ ⎟ ⎝ T1 ⎠ ) β ⎛T − T⎞ M , (9) where β is the critical exponent for the spontaneous or saturation magnetization. For iron β=0.368, M 1 =54 Oe and T 1 =1.45 K. A linear equation involving the magnetic intensity H and temperature T is given in [10] M = KH T −T . (10) o ( c Finally, Higashi et. all [4], found that the magnetization process of red blood cells behaves like the following function, known as Langevin function, ⎡ ⎛µ omH ⎞ κT ⎤ Mo = mN⎢coth⎜ ⎟− ⎥ ⎣ ⎝ κ T ⎠ µ omH , (11) ⎦ where m is the particle magnetization, N is the number of particles per unit volume and κ the Boltzman’s constant. T c
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Page 5 and 6: of the magnetic source the friction

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