Drag force acting on a neuromast in the fish lateral line trunk canal. II. Analytical modelling of parameter dependencies

In Part I of this two-part study, the coupled flows external and internal to the fish lateral line trunk canal were consecutively calculated by solving the Navier–Stokes (N–S) equations numerically in each domain. With the external flow known, the solution for the internal flow was obtained using a parallelepiped to simulate the neuromast cupula present between a pair of consecutive pores, allowing the calculation of the drag force acting on the neuromast cupula. While physically rigorous and accurate, the numerical approach is tedious and inefficient since it does not readily reveal the parameter dependencies of the drag force. In Part II of this work we present an analytically based physical–mathematical model for rapidly calculating the drag force acting on a neuromast cupula. The cupula is well approximated as an immobile sphere located inside a tube-shaped canal segment of circular cross section containing a constant property fluid in a steady-periodic oscillating state of motion. The analytical expression derived for the dimensionless drag force is of the form |FN|/(|PL−PR|π(D/2)2)=f(d/D,Lt/D,ωD*), where |F_N| is the amplitude of the drag force; |P_L−P_R| is the amplitude of the pressure difference driving the flow in the interpore tube segment; d/D is the ratio of sphere diameter to tube diameter; L_t/D is the ratio of interpore tube segment length to tube diameter; and ωD*=ω(D/2)2/ν is the oscillating flow kinetic Reynolds number (a dimensionless frequency). Present results show that the dimensionless drag force amplitude increases with decreasing L_t/D and maximizes in the range 0.65≤d/D≤0.85, depending on the values of L_t/D and ωD*. It is also found that in the biologically relevant range of dimensionless frequencies 1≤ωD*≤20 and segment lengths 4≤L_t/D≤16, the sphere tube (neuromast–canal) system acts as a low-pass filter for values d/D≤0.75, approximately. For larger values of d/D the system is equally sensitive to all frequencies, but the drag force is significantly decreased. Comparisons with N–S calculations of the drag force show good agreement with the analytical model results. By revealing the parameter dependencies of the drag force, the model serves to guide biological understanding and the optimized design of corresponding bioinspired artificial sensors.