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MHD micropumping of viscoelastic fluids: an analytical solution

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Abstract

An analytical solution is found for examining the effect of a fluid’s elasticity on the performance of MHD micropumps. The test fluid is assumed to be an incompressible viscoelastic fluid obeying the Oldroyd-B model. The flow generated by the Lorentz force is assumed to be laminar, unidirectional, and two-dimensional. The effects of relaxation and retardation times are investigated on the volumetric flow rate. It is concluded that by a decrease in the relaxation time, the pulsatile nature of micropump can be eliminated in its transient phase. At sufficiently low relaxation times, the flow is predicted to monotonically reach its steady value at a much shorter time. By an increase in the retardation time, the pulsatile nature of micropump in its transient phase can also be eliminated and the flow will be more continuous in its steady conditions.

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Abbreviations

\(\overrightarrow B \) :

Magnetic field vector, T

B :

Magnitude of magnetic field, T

d :

Rate-of-deformation tensor, s−1

\(\overrightarrow E \) :

Electric field vector, V/m

E :

Magnitude of electric field, V/m

\(\overrightarrow F \) :

Body force vector, N

h :

Height of channel, m

Ha :

Hartman number, = \(LB\sqrt {\sigma /\mu } \)

J :

Current density, A/m2

L :

Channel length, m

L p :

Electrode length, m

\(L_p^*\) :

Dimensionless electrode length, = Lp/L

p :

Pressure, Pa

Q :

Volumetric flow rate, m3/s

Re:

Reynolds number, = ρu0L/μ

t :

Time, s

\(\overrightarrow u \) :

Velocity component vector, ms−1

u :

Magnitude of velocity component, ms−1

u 0 :

Characteristic velocity, ms−1

u* :

Dimensionless velocity, = u/u0

V :

Voltage differences, V

V* :

Dimensionless voltage differences, = \(V/{u_0}\sqrt {\sigma /\mu } \)

w :

Width of channel, m

We :

Weissenberg number, = λ1u0/L

x, y, z :

Coordinates, m

y*, z* :

Dimensionless Coordinates, = y/L, z/L

α :

Dimensionless height, = h/L

β :

Dimensionless width, = w/L

σ :

Electrical conductivity, S/m

μ :

Zero shear viscosity, N.s/m2

ρ :

Density, kgm−3

τ ij :

Shear stress, N/m2

τ :

Dimensionless time, = u0t/L

λ 1 :

Relaxation time, s

λ 2 :

Retardation time, s

λ *2 :

Dimensionless retardation time, = λ2u0/L

ω :

Frequency, Hz

ϕ :

Phase angle

Ψ(y, z):

Eigenfunctions

Ψ(y, z):

Eigenvalues

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Authors and Affiliations

  1. Department of Food Technology, College of Aburaihan, University of Tehran, Tehran, Iran

    Saied Moghaddam

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Moghaddam, S. MHD micropumping of viscoelastic fluids: an analytical solution. Korea-Aust. Rheol. J. 33, 93–104 (2021). https://doi.org/10.1007/s13367-021-0008-y

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