Modeling Deformable Membrane and Flexing Walls
Our Development Note this issue highlights the deforming membrane and flexing wall models that will be introduced with the release of FLOW-3D Version 9.3.
Several years ago Flow Science introduced the General Moving Objects (GMO) model for general coupled motion of rigid objects in fluid. It can be considered the first step towards a full fluid-structure interaction (FSI) model where solid objects can also deform under external forces. This is one of the main development goals for FLOW-3D that will be completed in several phases. The next phase is to add a limited model for deforming walls and membranes, scheduled for release early in 2008.
The approach to modeling membranes and flexing walls is based on a user survey undertaken by Flow Science last year. The main application area for this capability is in microfluidics. Membranes serve as flow actuators in inkjet printer nozzles and micro-pumps, while pressure waves in fluid in long channels can cause deformations of the channel walls that in turn affect the flow. Some FLOW-3D users have already attempted modeling these types of problems applying their own customizations (see the accompanying article in this issue of the Newsletter).
The main assumptions for both the deforming membranes and flexing walls are:
- deformations are small compared to the length and width of the deforming surfaces.
- surface curvatures of the deformed objects are small compared to their reciprocal length and width.
Based on these premises, we can simplify the treatment of the moving boundary between the fluid and the deforming objects. The interface is assumed to be stationary, and the effect of its motion on the fluid is accounted for by volume sources and sinks distributed along its surface.
The dynamics of the deformations are computed as a result of external as well as fluid forces.
A membrane is represented by a solid component of a finite thickness and a flat surface facing the fluid. Currently, we plan to allow for circular and rectangular shapes. The deformation of a membrane is described by the motion of a single point at the center of the membrane, which will have the maximum displacement. Displacements at other points are described by an appropriate smooth function based on the shape of the membrane.
Figure 1. Schematic representation of a circular
membrane of radius a. Pressure in the
fluid is assumed to be uniform along the
surface of the membrane, while the actuator
force is concentrated at its center.
Figure 2. Displacement distribution on
a circular membrane
Figure 3. A membrane driven micro-pump.
Color represents pressure.
Fluid pressure is integrated over both surfaces of the membrane and is combined into a single force acting along the main axis at the center point. The difference between the forces is converted into the displacement using the stiffness coefficient of the membrane. An additional, time-dependent external pointwise force can also be taken into account to represent the action of a piezo-element or a push rod. The user will be able to define either an external force or a prescribed displacement of the membrane's center point. The initial model will allow the membrane to be oriented only normal to x, y or z directions.
The image in Figure 2 shows a result from an early test of the model where a circular membrane deforms under uniformly applied pressure of about 0.1 atm in the adjacent fluid. The membrane';s radius is 0.25 mm, thickness is 0.01 mm and the Young's modulus is 106 N/m2. The color represents the magnitude of the deformations on the surface of the membrane, while the vectors indicate the fluid motion along the surface of the membrane. The maximum displacement at the center is 0.0027 mm.
Figure 3 shows a model of a membrane-driven micro-pump. An oscillating circular membrane of the radius of 5 mm (colored in red) pushes the fluid through the two nozzles connect to constant (and equal) pressure boundaries. The motion of the membrane is prescribed by a sinusoidal function with the period of 0.0125 s and the displacement amplitude of 0.005 mm. Due to the specific shape of the nozzles, a net flow is generated through the evice over a period of the oscillation. The animation shows the fluid velocity distribution in the plane of the membrane over two periods.
A flexing wall is represented by a flat solid component. Each surface element in contact with fluid is represented by a "spring element." A spring deforms (compresses or expands) in response to the fluid pressure acting on its surface. Each spring element will deform independently of its neighbors (hence small deformations and curvatures). The spring coefficients are determined from the elastic modulus of the material. Each spring element is assumed to have the same cross-section area along the depth at its surface, i.e., has a shape of a rectangular rod, with the cross-section defined by the mesh size at the surface.
As with the membranes, the displacements are converted to volume sources and sinks and applied on the fixed surface of the wall. And like membranes, in the initial version of the model, the flexing walls will be oriented normal to a coordinate direction.
Figure 4 of shows a schematic for a simple two-dimensional test of the flexing wall model. The properties of the deforming materials are the same as for the membrane test. The fluid pressure pulse at the left boundary is 0.1 atm. The images in Figure 5 show a time series of wall deformations. The color represents the magnitude of the deformations on the wall surface, while the vectors indicate the fluid motion along the surface of the membrane.