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The incompressible viscous steady flow through a rotating curved pipe of circular cross-section with magnetic field is investigated numerically to examine the combined effects of rotation (Coriolis force), magnetic field and curvature (centrifugal force) on the flow. The curvature of the pipe has been assumed to be small, that is, the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross section. Spectral method is applied as a main tool for the numerical technique, where Fourier series, Chebyshev polynomials, Collocation methods, and Iteration method are used as secondary tools. The flow depends on the Taylor number (
*Tr*), Dean Number (
*Dn*), Magnetic Parameter (
*M*) and the dimensionless curvature of the pipe δ. When
*Tr* > 0, the rotation is in the direction so that the Coriolis force enforces the curvature effect. When
*Tr* < 0, the rotation is in the direction so that the Coriolis force exhibits an opposite effect to that of the curvature. The calculations are carried out for
−1500 ≤
*Tr* ≤ 1500,
*Dn* ≥ 1000 (large Dean number),
*M* ≥ 0 and
* δ* = 0.01. Due to high magnetic field four-vortex solution is observed in a rotating curved pipe system. Visualization is attained with MAPLE software.

The flow through a curved pipe has attracted considerable attention not only because of its practical importance, but also because of the interesting physical properties under the action of the centrifugal force due to the curvature of the pipe. Curved pipe geometries are also in bio-fluid mechanics. It is widely used in the pipeline such as consumption in aircraft. All these important features occur as a result of the curvature of the pipe and the rotation. When the pipe is rotated about an axis normal to a plane containing the line, it originates Coriolis force, affecting the effect of the centrifugal force. Therefore, it reveals the most interesting features that expand the use of rotating curved pipe for industrial especially in chemical engineering and mechanical engineering.

All these interesting features drew the attention of a lot of scientist to different types of flows different types of ducts. Dean [

dent on a parameter termed as Dean number D_{n} given by

cosity,

The effects of the magnetic field on fluid flow have been studied primarily for straight pipes by Chang and Lundgren [

But if the curved pipe rotates, in addition to centrifugal force the fluid experiences Coriolis force. The rotation is considered to be positive if it is in such a direction that the Coriolis force results a positive effect to that of the centrifugal force and this case is known as co-rotating case. Otherwise the rotation is considered to be negative and is known as counter rotating case. But when the fluid flows through a curved pipe, the particles experience centrifugal force which brings about the secondary flow. This was first noticed by Williams et al. [

For a wide range of Dean number, the magnetic field, and the curvature, no work as regards the flow of curved pipe with the rotation has been made. In this regard, it is quite innovative to study the flow in a rotating curved pipe for a wide range of Dean number, magnetic field, and the curvature of understanding the behavior of the flow. In the present study, we apply a spectral collocation computation method (Beg et al. [

A curved pipe with magnetic field of uniform circular cross section has been considered. The pipe is rotated at a constant angular velocity about an axis passing through the center of curvature of the pipe and perpendicular to the plane containing the axis of the pipe. Now, consider the coordinate systems

the circumferential angle, θ is the axial variable and r is the radial variable. The non-dimensional velocity components along the radial, circumferential and axial direction are u', v', w' respectively. r' is non-dimensional radius, S' is the non-dimensional axial variable, δ is non-dimensional curvature and p' non-dimensional pressure.

Let us defined the following non-dimensional variables

We get following momentum equations,

Non-dimensional radial momentum equation:

Non-dimensional circumferential momentum equation:

Non-dimensional axial momentum equation:

and the non-dimensional continuity equation:

The other variables without primes are dimensional variables. Constant pressure gradient force is applied along the axial direction through the centre of cross section. With the help of the above dimensionless variables and the boundary conditions the equation of motion reduces to the following form:

and

Here,

where,

and

The spectral method is applied in the numerical calculation. The Fourier series and Chebyshev polynomials are used in circumferential and radial directions respectively. Assuming that steady solution is symmetric with respect to the horizontal line of the cross-section,

and

where N is the truncation number of the Fourier series. The collocation points are taken as

The obtained algebraic non-linear equations are solved by an iteration method with under relaxation. Convergence of the solution assured by taking

A rotating curved pipe with circular cross section has been considered which is rotating at a constant angular velocity. The main flow is forced by the magnetic field along the center line of the pipe and the corresponding Dean number. As the pipe is curved the flow is also subjected to centrifugal force, which results in a two-vortex secondary flow which remains symmetric about the horizontal line passing through the center of the cross section. If the effects of rotation and curvature are in the same direction, the rotation is called co-rotation in this case T_{r} is positive; if not, it is called counter rotation and in this case T_{r} is negative. Due to curved structure of the pipe magnetic force comes into action and various flow structures are found. In addition Coriolis force due to rotation makes it more complicated. In the present problem numerical calculations have been made at Dean number D_{n} = 2500 for two different magnetic parameter at curvature δ = 0.01.

The results have been shown through stream line, vector plots of the secondary flow and contour plots of the axial flow. The stream line, vector and contour plots of the flow development have been shown at different magnetic parameters and Taylor number for Dean number, D_{n} = 2500 at non-dimensional curvature δ = 0.01. The flow characteristics have been shown in _{r} in the first,

second and third column respectively. The increment of the axial velocity is taken as Δw = 10 whereas the increment of the constant ψ-lines is Δψ = 0.50. The arrow indicates the direction of motion of the particle and the length of the arrow indicates the relative secondary velocity. In these figures the inner side of the cross section is to the left and the outer side is to the right.

In _{r} for all cases. In _{r} > 0 the secondary flow gets stronger with the increase of T_{r}, as positive T_{r} enforces the effects of pressure gradient force.

But the axial flow decrease which results in a low flux. Again, for δ = 0.01 and T_{r} < 0 the Coriolis force oppose the effects of pressure gradient force and many interesting features are observed. The secondary flow gets weaker but the axial flow increase and as a result flux becomes higher. When −355.66 ≤ T_{r} ≤ 323.54, four-vor- tex secondary flow structures have been found for the _{r} = −350 highest flux occurs. After T_{r} goes below −350, flux again decrease and two-vortex flow pattern has been observed for magnetic parameter M = 500 and 1000.

Vector plots of the secondary flow have been shown in

particles get the radial velocity. Circumferential velocity is greater for the particles near the upper and lower boundary due to the friction with the wall. Also the velocity of the particles at the centre of the cross section is radially outward for the effect of the centrifugal force. As the flow downstream the secondary velocities of the particles near the centre of the cross section increase in the direction of the centrifugal force. At the same time, the particles near the upper and lower boundary experience high circumferential velocity in the direction opposite to the velocity of the particle at the core region. As a result two-vortex secondary flows is set up, which is symmetric about the horizontal plane passing through the centre of the cross section.

The contour plots of the axial velocity has been shown in _{n} = 2500, M = 500 and 1000 respectively. As the flow enters the pipe boundary layer develops. Boundary layer near the inner wall develops faster than that at the outer wall. And with the development of the flow the strength of the axial flow is increased and is shifted towards the outer wall of the cross-section, which is effect of the centrifugal force due to curvature. The axial flow is symmetric about the plane passing through the center of cross section at the absence of rotation. When the pipe rotates the symmetry breaks down. Due to Coriolis force the strength of the axial flow is shifted towards the lower portion of the cross section. Finally, the centrifugal force dominates the Coriolis force and symmetry is attained. But for higher rotation Coriolis force is too high to be dominated by the centrifugal force. As a result symmetry is not attained.

The spectral numerical solutions have been presented for the magnetohydrodynamic rotating curved flow geometry. The computations are visualized with MAPLE software and indicate the following:

1) The anti-clock wise rotating vortex in the upper portion is stronger than the clock wise rotating vortex in the lower portion for high T_{r}.

2) The two-vortex changes their position in anti-clock wise rotating manner with the decrease of T_{r} for the effect of magnetic field and rotation.

3) The maximum flux has been found when the effect of magnetic field and rotation tends to balance each other. At the maximum flux and high magnetic field the four-vortex solution has been observed.

The present study has aimed to further reveal the effects of magnetic field on Newtonian flow in rotating curved vessels, motivated by astronaut response to megnetospheric effects during spacecraft re-entry (Busby [