We present a theoretical analysis of calendering of Ellis fluid based on lubrication approximation. The equations governing the flow are nondimensionalized and solved to get closed form expressions of velocity and pressure gradient. Runge–Kutta algorithm is employed to compute the pressure distribution. The operating variables which are used in the calendering process, i.e. roll-separating force, power input to the rolls and exiting sheet thickness are calculated. The influence of the material parameters on the velocity profile, pressure gradient, pressure distribution and operating variables is shown graphically and discussed in detail.

This paper presents the roll-coating analysis of couple stress fluid under lubrication approximation theory. The governing differential equation developed under the approximation is solved to get exact expressions for velocity and pressure gradient. The engineering quantities of interest such as pressure, roll-separating (load-carrying) force, and power input function are computed numerically. How the couple stress fluid material parameter affects velocity, pressure, pressure gradient, load-carrying force, and power input is shown graphically. One observes that couple stress model predicts higher pressure in the nip region when compared to the Newtonian model. The load-carrying force and power input exceed their Newtonian values for strong couple stress effects. Moreover, the separation point moves left of its Newtonian value with increasing couple stress effects.

Cilia serve as sensory organelles extending from cell surfaces, enabling the monitoring of intricate rheological surroundings. The objective of this study is to incorporate the governing equations of the Oldroyd 4-constant model into flows propelled by ciliary motion. Additionally, Maxwell's equations are employed to introduce a body force term within the classical Navier–Stokes equations. The problem is grounded in the assumptions of creeping flow and long wavelengths. The resulting differential equation is simulated using a robust finite difference method in MATLAB R2023a. The obtained solution exhibits convergence and is presented for fluid velocity, pressure rise, and contour lines. The solution is also validated via the shooting method. These results are beneficial in designing artificial magnetic cilia (with similar beating patterns) used for fluid manipulations in lab-on-chip devices.

This paper presents the exact solution for calendering a third-order fluid under lubrication approximation. The solution obtained is valid for all values of the third-order fluid parameter. This exact solution is compared to the perturbation solution. The results show that the perturbation solution is valid for very small values of the third-order fluid parameter. Therefore, no significant deviation from the corresponding results of the Newtonian fluid is observed with the perturbation solution. The interesting quantities for mechanically designing a calendering system such as the force separating the two rolls and total power input into both rolls are calculated and shown graphically for large third-order fluid parameter. The material’s rheological features modify the pressure, flow characteristics, and all other operating variables significantly. In fact, the nip region pressure increases with increasing third-order fluid parameter. The exiting sheet thickness, power input, and roll separating force also increase with increasing third-order fluid parameter.

The theoretical model of roll coating onto a moving sheet is developed based on micropolar fluid constitutive equations and lubrication approximation. Closed form expressions for velocity, microrotation and pressure gradient are obtained. Runge-Kutta method is used to calculate the engineering quantities of interest such as, pressure, roll-separating (load-carrying) force and power input. The separation point is numerically calculated using Newton’s iterative method together with generalized Leibniz rule. The effects of involved parameters on the pressure gradient, velocity, pressure and other mechanical quantities are displayed through various graphs. Extreme pressure is observed in the nip region for larger coupling numbers and microrotation parameters. This leads to increased load-carrying force and power input for micropolar fluid when compared to a Newtonian fluid. Moreover, the separation point decreases from its Newtonian value with increasing $N.$