Journal of the Egyptian Mathematical Society

The sole aim of this article is to examine the relative contribution of suction/injection parameter on Taylor–Dean flow in a composite annular gap partially filled with porous material. In the present setup, the Newtonian fluid flow is induced by the circumferential motion of both cylinders and pressure gradient imposed in the Azimuthal direction. The mathematical model governing the flow is rendered dimensionless using appropriate dimensionless quantities transformed using the Laplace transform technique. Using suitable Ansatz, the equation is reduced to the Bessel differential equations and solved. The solution of converted to the time domain using a well-known numerical scheme known as the Riemann-sum approximation. The variation of the Newtonian fluid for different flow parameters is presented graphically. The solution method is validated by obtaining the steady-state solution and also using the implicit finite different approach (IFD); comparison of the methods is depicted in tabular form (see Tables 1, 2). It is deduced generally that the Newtonian fluid is higher when injection at the outer cylinder except when Da is small also higher interfacial velocity can be achieved by taking positive value of $$\beta$$ β .

In this paper, an algorithm for the bi-objective median path (BMP) problem on a tree network is considered. The algorithm is based on the two-phase method which can compute all Pareto solutions for the BMP problem. The first phase is applied to find supported Pareto solutions by solving the uni-objective problem D(P,λ). The second phase is used to compute the unsupported Pareto solutions by applying a k-best algorithm which computes the k-best Pareto solutions in order of their objective values.

In the previous works, the limiting case for the motion of a rigid body about a fixed point in a Newtonian force field, which comes from a gravity center lies on Z-axis, is solved. The authors apply the small parameter technique which is achieved giving the body a sufficiently large angular velocity component r_o about the fixed z-axis of the body. The periodic solutions of motion are obtained in neighborhood r_o tends to $$\infty$$ ∞ . In our work, we aim to find periodic solutions to the problem of motion in the neighborhood of r₀ tends to $$0$$ 0 . So, we give a new assumption that: r_o is sufficiently small. Under this assumption, we must achieve a large parameter and search for another technique for solving this problem. This technique is named; a large parameter technique instead of the small one well known previously. We see the advantage of the new technique which appears in saving high energy used to begin the motion and give the solution of the problem in another domain. The obtained solutions by the new technique depend on r_o. We consider that the center of mass of this body does not necessarily coincide with the fixed point O. We reduce the six nonlinear differential equations of the body and their three first integrals to a quasilinear autonomous system of two degrees of freedom and one first integral. We solve the rational case when the frequencies of the generating system are rational except $$(\,\omega = \,1,\,2,1/2,3,1/3, \ldots )$$ ( ω = 1 , 2 , 1 / 2 , 3 , 1 / 3 , … ) under the condition $$\gamma^{\prime\prime}_{0} = \cos \theta_{o} \approx 0$$ γ 0 ″ = cos θ o ≈ 0 . We use the fourth-order Runge–Kutta method to find the periodic solutions in the closed interval of the time t and to compare the analytical method with the numerical one.