# (469e) A Sheared Suspension of Conducting Particles in a Magnetic Field: Particle Rotation, Antisymmetric Stress and Secondary Flows

#### AIChE Annual Meeting

#### 2019

#### 2019 AIChE Annual Meeting

#### Engineering Sciences and Fundamentals

#### Microfluidic and Nanoscale Flows: Multiphase Systems and External Fields

#### Wednesday, November 13, 2019 - 9:00am to 9:15am

^{p}= Ï‰ (Î·

_{c}

^{(1) }(

**Îµ:Ã´**) + Î·

_{c}

^{(2) }

**Îµ:**(

**Ã‚**- (

**Ã‚.Ã´**)

**Ã´**)/âˆš(1-(

**Ã‚.Ã´**)

^{2}) + Î·

_{c}

^{(3) }(

**Ã´**

**Ã‚**-

**Ã‚.Ã´**)/âˆš(1-(

**Ã‚.Ã´**)

^{2}), where Ï‰ is the vorticity,

**Ã‚**and

**Ã´**are the unit vectors in the direction of the magnetic field and the vorticity,

**Îµ**is the third order Levi-Civita antisymmetric tensor, and Î·

_{c}

^{(1)}, Î·

_{c}

^{(2)}, and Î·

_{c}

^{(3) }are called the first, second and third couple stress coefficients. The stress component proportional to Î·

_{c}

^{(1)}is perpendicular to the vorticity in the flow plane, that proportional to Î·

_{c}

^{(2)}is perpendicular to the unit normal to the vorticity in the

**Ã‚**-

**Ã´**plane and the component proportional to Î·

_{c}

^{(3)}is in the plane containing the vorticity and magnetic field. The couple stress coefficients are evaluated as a function of two parameters, âˆ‘ = (Î¼

_{0}H

_{0}

^{2}/ 4Ï€Î·Ï‰) the ratio of the magnetic and viscous torques, and Î² = (Ï‰ Î¼

_{0}R

^{2}/ 2 Ï) which is the product of the vorticity and the current relaxation time.Here, H

_{0}is the magnetic field strength, Î¼

_{0}and Ï are the magnetic permeability and electrical resistivity, and R is the particle diameter. Analytical expressions are derived for the three couple stress coefficients in the limits of low and high Î£ and Î², and numerical calculations are carried out in the intermediate regime. If the magnetic field is perpendicular to the particle angular velocity, the eddy current is opposite in direction to the hydrodynamic torque, thereby reducing the rotation rate and increasing the viscosity of the suspension. The maximum additional increase in the viscosity is (3/5) times that predicted by the Einstein relation for a suspension of particles in a shear flow. For high Î², there is the possibility of multiple steady states because the relation between the magnetic moment and the particle rotation rate is non-linear, leading to the possibility of spin banding in the system. A noteworthy feature of the constitutive relation is the third couple stress coefficient the magnetic field and the vorticity are not perpendicular to each other, due to the precession torque that acts in the direction perpendicular to the plane containing the vorticity and magnetic field. In a two-dimensional pressure-driven flow of interest in microfluidic applications, this antisymmetric stress results in a secondary flow in the plane of the channel cross-section, which could significantly enhance mixing across the channel.