Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates

Table 2

The analytic results of 𝑓(𝜂) at different orders of approximation compared with the numerical results obtained by the fourth-order Runge-Kutta for the two-dimensional case.

S,ℏ	𝜂	2nd-order	4th-order	6th-order	7th-order	8th-order	Numerical
−1.5,−1.3	0.2	0.332883	0.333591	0.333617	0.333618	0.333618	0.333618
	0.4	0.623190	0.624315	0.624358	0.624358	0.624358	0.624358
	0.6	0.838219	0.839284	0.839324	0.839325	0.839325	0.839325
	0.8	0.962441	0.962961	0.962983	0.962984	0.962984	0.962984
−0.5,−1	0.2	0.305436	0.305543	0.305545	0.305545	0.305545	0.305545
	0.4	0.582314	0.582468	0.582470	0.582470	0.582470	0.582470
	0.6	0.804271	0.804390	0.804392	0.804392	0.804392	0.804392
	0.8	0.949065	0.949107	0.949108	0.949108	0.949108	0.949108
0.5,−1	0.2	0.288347	0.288261	0.288260	0.288260	0.288260	0.288260
	0.4	0.556268	0.556145	0.556143	0.556143	0.556143	0.556143
	0.6	0.781768	0.781670	0.781671	0.781671	0.781671	0.781671
	0.8	0.939674	0.939641	0.939640	0.939640	0.939640	0.939640
1.5,−0.8	0.2	0.276526	0.276433	0.276432	0.276432	0.276432	0.276432
	0.4	0.537929	0.537754	0.537752	0.537752	0.537752	0.537752
	0.6	0.765463	0.765252	0.765249	0.765249	0.765249	0.765249
	0.8	0.932607	0.932474	0.932471	0.932471	0.932471	0.932471
2.5,−0.7	0.2	0.268041	0.267797	0.267791	0.267791	0.267791	0.267791
	0.4	0.524558	0.524057	0.524045	0.524045	0.524045	0.524045
	0.6	0.753285	0.752627	0.752605	0.752605	0.752605	0.752605
	0.8	0.927166	0.926724	0.926704	0.926703	0.926703	0.926703