I am modeling electroosmotic flow (EOF) in a microfluidic chip. The chip has 2 inlets and 1 outlet. The EOF part seems to be working well. I am using Electric Currents (ec) and Laminar Flow (spf) physics.
On top of the chip are fluid reservoirs. As fluid flows from inlet to outlet the fluid height in the inlet reservoirs decreases and the fluid height at the outlet reservoir increases. This should cause a change in pressure that I would like to model. I have attached a simplified sketch (not to scale) with one inlet and outlet that should clarify my problem.
I am new to fluid dynamics so I may be approaching this problem in the wrong way but here is what I have so far.
The initial pressure (P0) at each reservoir is the same. Once the problem starts (t=0) an electric field is applied between one of the inlets and the outlet. The second inlet is unbiased for now. At any time t>0 there will now be a pressure difference between the inlets and outlet.
I have derived an equation for the pressure in the reservoirs based on the equation P=rho*g*h, where P is pressure, rho is the fluid density, g is gravity, and h is the fluid height in a reservoir.
At a reservoir the pressure will be P = P0 + Pt, where Pt is the pressure change as a function of time. Which is to say the pressure change due to fluid emptying from the inlet or collecting at the outlet.
Therefore, Pt = rho*g*ht, where ht is the change in fluid height in a reservoir. Now volume (V) in the cylindrical reservoirs is given as V=pi*r^2*h, therefore, h = V/(pi*r^2). The change in volume in the reservoirs is equal to the volume flow rate Q multiplied by time. Q = A*v where A is the cross sectional area of the microfluidic channel and v is the flow velocity.
So Pt = rho*g*ht = rho*g*V/(pi*r^2) = rho*g*(Q*t)/(pi*r^2) = rho*g*(A*v*t)/(pi*r^2)
Therefore, at the inlet P = P0 - rho*g*(A*v*t)/(pi*r^2),
and at the outlet P = P0 + rho*g*(A*v*t)/(pi*r^2)
In Comsol for inlet 1, I am using the pressure boundary condition, suppress backflow is unchecked, and flow direction is normal flow. Under pressure condition the units are Pa and I have written the following:
101325-rho*g*((A*spf.U)*t)/(pi*rR^2)
where rR is the reservoir radius, and I have already identified the other variables.
I am leaving the second inlet electrically floating for the moment. The conditions are otherwise the same, so under the pressure condition I have simply left it as
101325
For the outlet, I am using the pressure boundary condition, suppress backflow and normal flow are unchecked. Under pressure condition the units are Pa and I have written the following:
101325+rho*g*((A*spf.U)*t)/(pi*rR^2)
I have tried solving the problem as segregated and as fully coupled, but either way I get the following error:
Attempt to evaluate negative power of zero.
- Function: ^
Failed to evaluate temporary symbolic derivative variable.
- Variable: comp1.spf.U@VDN$comp1.u
- Defined as: (0.5*(((comp1.w^2)+((comp1.v^2)+(comp1.u^2)))^(-0.5)))*(2*comp1.u)
Failed to evaluate expression.
- Expression: (0.5*(((comp1.w^2)+((comp1.v^2)+(comp1.u^2)))^(-0.5)))*(2*comp1.u)
Failed to evaluate Jacobian of expression.
- Expression: comp1.spf.U
Failed to evaluate Jacobian of operator.
- Operator: mean
- Geometry: geom1
- Boundary: 6
Failed to evaluate Jacobian of variable.
- Variable: comp1.spf.U
- Geometry: geom1
- Boundary: 6
Failed to evaluate Jacobian of expression.
- Expression: ((-comp1.spf.f0)*comp1.spf.nzmesh)*dvol
Failed to evaluate Jacobian of expression.
- Expression: (-comp1.spf.f0*(test(comp1.u)*comp1.spf.nxmesh+test(comp1.v)*comp1.spf.nymesh+test(comp1.w)*comp1.spf.nzmesh))*(dvol)
- Feature: Time-Dependent Solver 1 (sol1/t1)
I get no errors and the model seems to be working as expected when I set the pressure condition for the inlets and outlet to 101325 Pa. Therefore the problem seems to be with my equation/implementation.
Any help with how I can address this error or approach the problem differently is welcome.
Thanks.