# Magnetorheological Fluid-Based Haptic Feedback Damper
## 2 Theoretical Analysis and Simulation
### Fluid Mechanics Analysis  
As mentioned before, a magnetorheological fluid, under the influence of an applied magnetic field, transitions into a Bingham fluid state [21]. Conversely, in areas without an applied magnetic field, it behaves as a Newtonian fluid. In this study, the NdFeB magnet and the magnetorheological fluid demonstrated axisymmetric divisions. We therefore focused on one-quarter of the key area to investigate the distribution of magnetorheological fluid states under the influence of the magnetic field, as demonstrated in Figure 1.  
Upon pressing a button, the nitrile rubber film was compressed downward, triggering deformation and causing the fluid beneath the button to flow downward and laterally. Given the button's small, spherical base, we did not account for extrusion stress in our calculations. The keyboard box device confined the magnetorheological fluid between two plates. Assuming that the length (\(L\)) and width (\(W\)) of these two plates greatly surpassed the height (B), this flow could be considered a plane Poiseuille flow [22], with coordinates positioned at the midpoint between the two plates, as demonstrated in Figure 2.  
The fluid continuity equation is given by  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0\Rightarrow\frac{\partial u}{\partial x}=0\Rightarrow u=u(y) \tag{1}\]  
where \(u\), \(v\), and \(w\) are the flow velocities of the fluid on the \(x\)-, \(y\)-, and \(z\)-axes. The momentum equation on the \(x\)-axis is given by  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{\mu}{\rho} \bigg{(}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y ^{2}}\bigg{)} \tag{2}\]  
where \(\rho\) is the density of the fluid and \(p\) is the pressure acting on the cross-section of the fluid. If the fluid behaves as a Newtonian fluid in the absence of an applied magnetic field [23], substituting Equation (1) into Equation (2) yields  
\[\mu\frac{d^{2}u}{dy^{2}}=\frac{\partial p}{\partial x} \tag{3}\]  
Substituting the boundary condition when \(y=\pm B/2\) and \(u=0\) into the equation yields  
\[u=\frac{dp}{dx}\frac{1}{2\mu}\bigg{(}y^{2}-\bigg{(}\frac{B}{2}\bigg{)}^{2} \bigg{)},-\frac{B}{2}\leq y\leq\frac{B}{2} \tag{4}\]  
This represents the general solution for the planar Poiseuille flow of a Newtonian fluid induced by a pressure gradient, which exhibits a parabolic velocity profile, as demonstrated  
Figure 1: Distribution of the magnetorheological fluid states under the influence of a magnetic field. Figure 2: The plane Poiseuille flow distribution map. in Figure 3. Moreover, we need to calculate the maximum velocity corresponding to the vertex of the parabola:  
\[u_{\max}=-\frac{B^{2}}{8\mu}\frac{dp}{dx} \tag{5}\]  
From this, the flow rate of the Newtonian fluid per unit time is determined using  
Figure 2.0  
For a Bingham fluid with an applied magnetic field, and considering the boundary condition when \(y=\pm B/2\) and \(u=0\), we can calculate  
Figure 2.1  
Figure 2.2  
Figure 2.3  
Observably, there is a constant speed region within the Bingham fluid known as the plug flow region, denoted by \(\delta\), which is the height of this plug flow zone. The velocity distribution of a Bingham fluid is shown in Figure 2.4  
According to Equations (7)-(9), the flow rate of the Bingham fluid per unit time is calculated using  
Figure 2.5  
Assuming the fluid velocity is \(V\), the flow rate per unit time of the magnetorheological fluid between two plates is calculated using  
Figure 2.6  
Figure 2.7 Bingham fluid velocity distribution map. Figure 2.8 Newton fluid velocity distribution map.