Magnetohydrodynamic viscous fluid flow and heat transfer in a circular pipe under an externally applied constant suction.

An analytical investigation of two-dimensional heat transfer behavior of an axisymmetric incompressible dissipative viscous fluid flow in a circular pipe is considered. The flow is subjected to an externally applied uniform suction over the pipe wall in the transverse direction and a constant magnetic field opposite to the wall. The reduced Navier-Stokes equations in the cylindrical system are applied for the velocity and temperature fields. Constant wall temperature is considered as the thermal boundary condition. The velocity components are expressed into stream function and its solution is acquired by the Homotopy analysis method (HAM). The effects of magnetic body force parameter(M), suction Reynolds number (Re), Prandtl number (Pr)and Eckert number (Ec) on velocity and temperature are examined and are presented in a graphical frame. Streamlines, isotherms and pressure contours are likewise pictured. It is observed that with increasing suction Reynold number decelerates axial flow, whereas it enhances the radial flow. The temperature distribution increases with an increase in Prandtl number, whereas it decreases with an increase in Eckert number (viscous dissipation effect).

Introduction

Viscous flows in channels and pipes possess large amounts of mechanical applications which may incorporate cooling frameworks, petrochemical transport (oil and petroleum gas) and biotechnology. Regularly such flows are going with heat transfer and a representative example is the removal of thermal energy from hydronic space heating framework [1] by means of circling water in the heater, after which it is transported to the individual areas through pipes. Different frameworks utilizing heat transfer in viscous pipe flow are space thermal control [2], solar collectors [3] and heat exchangers [4].

In ongoing decades, engineers have additionally explored the change of viscous flows by means of porosity of the pipe or channel. Injection or evacuation of fluid by means of pores is a powerful instrument for flow control [5]. This innovation finds vital potential in biomedical sciences (e.g. counterfeit dialysis, blood flows) and in other topics, for example, rocket transpiration cooling and food preservation. Scientific demonstrating of flows in channels/pipes with wall mass flux has, in this manner, invigorated some enthusiasm for the examination network. Berman [6] was the first to examine the steady Newtonian flow in a permeable (porous) straight channel with uniform suction/injection impacts. Later Bansal [7] stretched out this work to a steady viscous flow through a porous circular pipe with the suction and axial pressure gradient. He found that the velocity accomplishes the greatest incentive along the axis of the pipe. An analytical solution for the laminar flow through circular pipes with steady suction/injection at the wall was further discussed by Terril [8, 9]. Tsangaris and Kondaxakis [10] examined unsteady viscous laminar flow in a straight pipe. They got an exact solution for time-changing infusion/suction at the permeable wall. Cox and Hill [11] have inspected the Newtonian fluid through carbon nanotubes with a Navier slip at the boundary. Ramana Murthy et. al [12]. investigated micropolar flow generated by a porous cylinder showing rotatory motions. They found that drag diminishes numerically when the suction parameter increments. Srinivas and Ramana Murthy [13] have examined wall suction effects in immiscible couple stress flow between two homogeneous permeable walls. They recognized that fluid velocity increments with Darcy number.

Magnetohydrodynamic (MHD) is additionally a functioning zone of present day engineering applications and includes the interaction between the applied magnetic fields and electrically conducting fluids. MHD pipe flows emerge in ionized accelators, MHD flow control in atomic reactors, MHD bypass energy generators, fluid metal manufacture forms, bubble levitation and so on [14]. MHD flows including suction/injection wall impacts have accumulated significant consideration. Terrill and Shrestha [15] presented an analytical study of laminar flow between two parallel porous plates with a connected magnetic field. They had demonstrated that the surface friction increments with the rise in magnetic number. Attia [16] examined the unsteady flow through a circular pipe with axial pressure gradient applied for two-stage MHD non-Newtonian fluids. He found that with the stronger magnetic field, the velocity and temperature components for the two phases decrease. Attia and Ahmed [17] studied the unsteady flow through magnetized viscoplastic (Bingham) fluid in a circular pipe. They identified that skin friction enhances with particle-phase viscosity. El-Shahed [18] investigated the impacts of a transverse magnetic field and porous medium in second-grade fluid flow through a circular pipe. He obtained velocity solutions in terms of Fox's H-function. Ramana Murthy and Bahali [19]have examined the impacts of periodic suction/injection through magneto-micropolar flow in a permeable circular pipe. They identified that wall shear stress rises with the magnetic parameter. Ramana Murthy et al. [20] studied the effects of wall suction/blowing in hydromagnetic micropolar fluid through a rectangular channel. They found that the magnetic field reduces the flow rate sizes extensively. Mabood et al. [21]studied the impacts of MHD and radiation in chemically reactive nanofluid through a stretching surface. Mabood et al [22] investigated double-diffusive impacts in MHD, non-Darcian stretching sheet flow. Shateyi and Mabood [23] inspected mixed convection and stagnation point flow with radiation and viscous heating through the non-linear MHD stretching surface.

The above examinations overlooked viscous heating impacts which can apply a significant effect in numerous applications. It is realized that viscous dissipation adjusts the temperature distributions and acts as an energy source. This thusly particularly impacts heat transfer rates. The effect of viscous heating is firmly needy additionally on whether the pipe is being heated or cooled i.e. thermal boundary conditions at the pipe surface apply a important work in how much viscous heating changes the distribution of heat in viscous flows [24], In addition, viscous heating has been shown to play an enhanced role in fluids with low thermal conductivity and high viscosity. An essential investigation was conveyed by Gebhart [25] with regards to boundary layer flows. Historically, Brinkman started viscous heating [26] studies. Ou and Cheng [27] considered the impact of viscous dissipation on heat transfer at the inlet of a pipe with steady heat flux. Béget al. [28] obtained numerical solutions for the nonlinear flow and heat transfer of MHD Hartmann–Couette in a Darcian channel with porous medium and Ohmic dissipation. Other interesting and recent research into the thermofluid dynamics of pipes has been reported in [29, 30, 31, 32, 33, 34, 35, 36, 37].

To date, very few authors [27, 28, 29, 30, 31] have studied heat transfer in two-dimensional hydromagnetic convection flow in porous circular pipes/channels. Again, their results are not analytical and are obtained through FLUENT software or numerical methods. Therefore, in this case, analytical results are necessary and we aim to examine the analytical solution for the two-dimensional heat transfer effects in a circular pipe under externally applied uniform suction and magnetic field on the surface of the circular pipe. The transformed, non-dimensional boundary value problem is solved by the powerful Homotopy Analysis Method (HAM) [38], which offers excellent flexibility and convergence features for non-linear ODE resolution. The method was successfully applied to a number of interesting problems [39, 40, 41, 42, 43, 44]. The present study relates to MHD energy systems.

Model

Consider an endlessly long cylinder of radius a, in which an electrically conducting viscous fluid is streaming, is appearing in Fig. 1. The thickness of the pipe is ignored and the mechanism of heat transfer is investigated through uniform wall temperature on the surface of the pipe. Since the pipe has an infinite length, the flow is developed completely. The flow is subjected to a uniform external suction in the normal direction through the wall. There is a uniform magnetic field of solidarity B0 in the transverse direction (in contrast to the pipe wall) and no electric field outside is connected. The number of Magnetic Reynolds is sufficiently small to discredit incited magnetic field impacts. When modified for the magnetic body force effect, the primitive equations for the steady, axi-symmetric, incompressible Newtonian fluid flow (Bird et al. [45]) are shown to take the form:where (U, W) are the velocity parts in (R,Z) bearings separately, P is pressure, ?? is the density of the fluid, ?? is viscosity, σ is electrical conductivity, c is the specific heat at consistent pressure, T is the temperature of the liquid and kT is thermal conductivity. It is relevant to present the accompanying non-dimensional parameters, where capitalized letters signify physical (dimensional) amounts and lowercase letters speak to the relating non-dimensional amounts:

Fig. 1

Schematic diagram.

Here T1 is constant temperature at the surface.

Executing condition (5) into conditions (1)–(4), following dimensionless arrangement of coupled differential equations, develops:

Here the dissipation function φ is given by, the Laplacian operator defined as, , , and where Re is suction Reynolds number, M is the magnetic body force parameter (ratio of hydromagnetic and viscous body forces), Pr is Prandtl number, and Ec is the Eckert number (viscous heating parameter).

Since the flow is two dimensional, a stream function ψ may be defined to satisfy the mass conservation (continuity) Eq. (6):

Eliminating the pressure gradient termsfrom (7) and (8),yields:where is the stokes function operator

Following Terril and Shresta [15], ψ is assumed to take the following formWhere N= U0/v0, U0 is the entrance velocity, v0 is suction velocity (pipe surface lateral mass flux).where is a differential operator.

Eq. (13) is to be solved under the boundary conditions given by:

Methodology

The Eqs. (13) and (9) are fathomed utilizing HAM. This system was presented by Liao [38] during the 1990s. HAM depends on the homotopy of topology and is similarly adroit at explaining nonlinear ordinary and partial differential equation systems and has been utilized broadly in heat transfer, magnetic fluid dynamics, non-Newtonian and Newtonian flows. Late precedents incorporate [46, 47].

To create systematic arrangements by HAM, the differential Eqs. (13) and (9) are put in the accompanying homotopy shape, which is known as the zeroth request distortion equation, as

Wherewhere

With

In the above λ is the homotopy parameter, h, hare convergence control parameters, fhand θh are the homotopy functions which are taken as f and θ when λ = 0 and approaches f and θ as λ→1. The differential operatorsLand L are linear and can be chosen at our convenience and N(fh) =0 and N(θh)=0 are the original nonlinear problems for f and θ which can be obtained as λ→1. The initial functions f and θ when λ = 0 satisfy L(f)=0 and L(θ)=0 and the boundary conditions of the problem. Here, the auxiliary function H is taken as 1.

For our problem, the initial approximations and and auxiliary linear operators and are taken as follows:

With

Table 1

Correlation of the consequences of radial velocity by HAM and that arrangement given in [31].

Distance r	Radial velocity(f),Numerical solution by Ben-Mansour and Sahin [31]	Radial velocity(f),Present solution by HAM (15th order approximation)
0	0.0000046	0
0.2	0.057945	0.0581946
0.4	0.196226	0.206445
0.6	0.498711	0.498719
0.8	0.811455	0.811468
1	1.002766	1

To approve our results, we have contrasted them and the past non-magnetic (M = 0) arrangements of Ben-Mansour and Sahin [31] with the accompanying recommended information: Re = 5, N = 2, z = 0.5 and these are archived in Table 1. The relationship is close and thusly trust in the present arrangements is reasonably high. The table likewise unmistakably exhibits the noteworthy improvement in radial velocity with expanding distance.

The homotopy functions fh for fand θh for θ are assumed as follows;

These functions fh in (24) and θh in (25) are substituted in (15) and (17) and coefficient of λto get the viz.,Whereand

and .

The corresponding boundary conditions are

To solve Eqs. (26) and (27) with the conditions Eq. (30), we use the symbolic computation software Mathematica. We select and , properly in such a way that these series are convergent at , therefore we have the expressions as follows:

The dimensionless form for pressure(p), from Eqs. (7) and (8), is given by

Assume

The shear stressat the pipe wall is given by

Hence the coefficient of skin friction on r = 1is given bywhere Nu is the Nusselt number.

Results & discussions

A parametric investigation of the impact of the key thermophysical parameters on velocity and temperature capacities is led. To guarantee combination is accomplished we initially expand on this viewpoint. Calculations were done by setting up the parameters M = 5, Re = 10, Pr = 0.7 and Ec = 0.45 to examine the impacts of the rising parameters.

The velocity and temperature articulations contain the auxiliary parameters h1 and h2. As pointed out by Liao [38], the assembly and the rate of approximation for the HAM arrangement unequivocally rely upon the estimations of auxiliary parameter h. For this reason, h-curves are plotted for deciding the scopes of h1 and h2. The estimations of and from the h-curves are envisioned in Figs. 2 and 3 for the 10th-order of approximation. It is clear that the range for the allowable estimations of h1 from Fig. 2 is and from Fig. 3, the range for is . We have chosen and . Superb convergence is accomplished for the 10th order approximation which is embraced in every single consequent calculation and figures. Here highly precise arrangements are therefore achieved with .

Fig. 2

h curve for f(r).

Fig. 3

h curve for θ(r).

Figs. 4, 5, 6 demonstrates the impact of the magnetic parameter on radial velocity f, axial velocity and temperature. It tends to be seen from the Figs. 4 and 6 that the radial velocity and temperature profiles are both upgraded with an expansion in the parameter M, while the axial velocity changes are at first expanded for lower radial organize values and therefore decreased for higher radial facilitate values. The radial increasing speed and axial deceleration are because of the Lorentzian magnetic body powers, emerging in the changed energy preservation Eq. (13). These powers reliably quicken the radial flow(f) across the pipe cross-section, while their impact on the axial stream is subject to the area i.e. radial arrange. From Fig. 6, it is obvious that as magnetic parameter builds, the temperature likewise increments. The valuable work consumed in hauling the liquid against the activity of a magnetic field the axial way is disseminated as warm vitality. This warms the liquid and accordingly raises temperatures in the gooey liquid. The temperature profiles all join asymptotically to a most extreme at the pipe surface (divider). These patterns are steady with numerous different examinations e.g. Gardner [48] and Cunha and Sobral [49].

Fig. 4

Response of M on Radial velocity f.

Fig. 5

Response of M on axial velocity .

Fig. 6

Response of M on temperature θ.

From Fig. 7, obviously as Re expands, the radial velocity f likewise increments. From Fig. 8, we see that as Re expands, the most extreme estimations of (axial velocity) are lessened. This is perception is inverse to the impact of a magnetic parameter which improves the most extreme estimations of . This might be because of the way that more noteworthy Reynolds number suggests a more noteworthy inertial power in the routine in respect to thick power and this serves to quicken the radial stream. Reynolds number can't prompt a power the opposite way as on account of magnetic parameter M. It is additionally obvious from Fig. 9 that as Reynolds number increments, temperature(θ) profiles diminish definitely close to the pivot of the chamber. This shows as Re builds, warm dissemination in the laminar routine is repressed.

Fig. 7

Reynolds number effect on f.

Fig. 8

Reynolds number effect on .

Fig. 9

Reynolds number effect on θ.

From Figs. 10 and 11, we see that as Eckert number (Ec) builds temperature close to the pipe hub diminishes and as Prandtl number (Pr) expands, temperature increments. The clashing conduct of Prandtl and Eckert numbers is outstanding in heat transfer.

Fig. 10

Eckert number effect on θ.

Fig. 11

Prandtl number effect on θ.

In Fig. 12, it is discovered that Cf diminishes emphatically as suction Re increments, and accomplishes a steady an incentive after Re increments past 5. In Fig. 13 Cf diminishes altogether with more noteworthy magnetic parameter (M) until the point that Re increments past 5. In any case, for substantial qualities Re every one of the charts unite and consequent change in skin friction isn't found. The impact of magnetic power at lower Re esteems is plainly to restrain the flow i.e. prompt deceleration and decline skin friction.

Fig. 12

Skin friction effect on Re.

Fig. 13

Skin friction effect on M.

In Fig. 14 obviously Nu sizes diminish Pr increments. A solid decrease in Nusselt number is additionally joined by more noteworthy Eckert number (Ec). Fig. 15 demonstrates that numerically diminishes up to a basic Prandtl number (Pr) and from that point increments essentially towards the surface of the pipe with expanding estimations of Re.

Fig. 14

Nusselt number effect on Pr.

Fig. 15

Nusselt number effect on Re.

From Fig. 16, it is seen that the streamlines for are non-neative and negative for Streamlines are numerically symmetric about line. From Fig. 17, it is seen that temperature is symmetric about line. The temperature is most extreme close to pivot of the cylinder (since in that area progressive white shading is available). Most noteworthy temperature happens at and this greatest esteem increments from r = 1 to r = 0. From Fig. 18, we can see that presure is symmetric about and diminishes as and increments as . pressure is relatively consistent for z < 1 and r < 0.5 and again in the district z > 3, r ≤ 0.5. Pressure changes significantly just in the area 1 < z < 3.

Fig. 16

Stream lines for velocity.

Fig. 17

Contour graphs for temparature.

Fig. 18

Contour graphs for presure.

Conclusions

Analytical solutions for the thermal transfer of viscous magneto-hydrodynamic pipe flow using the homotopy analysis method (HAM) have been presented. Viscous heating, Mhd and suction/injection effects of the wall have been included. A HAM convergence study was also carried out to guarantee the robustness of the series solutions. The calculations showed that:

Increasing magnetic body force parameter quickens the radial flow while it will in general decelerate axial flow.

Increasing the magnetic parameter improves temperatures since it produces thermal energy dissemination attributable to the additional work required to drag the liquid against the magnetic field the pivotal way.

Increasing suction Reynolds number decelerates the axial flow and upgrades the radial flow.

With expanding Eckert number, the temperature along the outspread course is diminishing while with expanding Prandtl number it is raised.

Skin friction is almost constant after a certain range of suction Reynolds number for all magnetic parameter values i.e. for low Reynolds numbers, the effect of magnetic parameter is significant.

Declarations

Author contribution statement

G. Nagaraju: Performed the experiments; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Mahesh Garvandha: Conceived and designed the experiments; Analyzed and interpreted the data.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.