Here, Kd and Cd are the direct stiffness and damping coefficients for the seal, respectively. Similarly, kc and cc are cross-coupled stiffness and damping coefficients for the seal. md  is the effective mass of the seal. These coefficients are functions of the flow as well as geometric properties of the seal. The commonly used models describing these dependencies are by Black and Jenssen, and by Childs. The Black and Jenssen model uses a constant inlet swirl ratio of 0.5, whereas in the Childs model the inlet swirl ratio can be varied. The inlet swirl ratio, α, is defined as the ratio of circumferential velocity of the flow to circumferential velocity of the rotor at a distance R from the center:

Here, uc is the circumferential velocity of the flow at the seal’s inlet, and Ω is the angular speed of the rotor. R is the radius of the seal.

where ΔP is axial pressure drop, V is axial flow velocity, ξ is entrance loss factor, σ is scaled friction factor given by λL/C. L is the seal length, and C is the seal clearance.

Since σ is a nonlinear function of the mean axial flow velocity (described below for different seal models), the pressure drop and the axial flow velocity relation need to be solved iteratively to determine the axial flow velocity for a given pressure drop across the seal.