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On partially heated circular obstacle in a channel having heated rectangular ribs: Finite element outcomes
Journal Pre-proof On partially heated circular obstacle in a channel having heated rectangular ribs: Finite element outcomes Khalil Ur Rehman, Qasem M. Al-Mdallal PII:
S2214-157X(19)30372-7
DOI:
https://doi.org/10.1016/j.csite.2020.100597
Reference:
CSITE 100597
To appear in:
Case Studies in Thermal Engineering
Received Date: 12 September 2019 Revised Date:
12 January 2020
Accepted Date: 29 January 2020
Please cite this article as: K.U. Rehman, Q.M. Al-Mdallal, On partially heated circular obstacle in a channel having heated rectangular ribs: Finite element outcomes, Case Studies in Thermal Engineering (2020), doi: https://doi.org/10.1016/j.csite.2020.100597. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Author Contribution Statement
Khalil Ur Rehman: Conceptualization, Investigation, Writing- Original draft preparation, Reviewing, and Editing. Qasem M. Al-Mdallal: Conceptualization, Methodology, Investigation. Supervision.
On Partially Heated Circular Obstacle in a Channel Having Heated Rectangular Ribs: Finite Element Outcomes Khalil Ur Rehman1, and Qasem M. Al-Mdallal2,* 1 2
Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, Abu Dhabi, United Arab Emirates
Correspondence email address: [email protected] (Q. M. Al-Mdallal) [email protected] (K.U. Rehman),
Abstract. In this paper we investigate the fluid flow in rectangular domain having partially heated circular cylinder. To be more specific, the cold fluid is initiated at an inlet of rectangular channel with the parabolic velocity profile. The Neumann condition is carried at outlet of channel. The no-slip condition is implemented at both upper and lower walls. To increase novelty we have considered heat transfer aspects. In this direction, the partially heated circular cylinder is taken fixed in a rectangular channel. The heated rectangular shaped ribs are introduced case-wise. The left wall is taken at zero temperature while the outlet of channel is supported with an adiabatic condition. The whole physical design is controlled with the help of mathematical formulation. The obtained differential system is solved by finite element method. The outcomes are shared in terms of graphs and tables. The variation in drag force subject to different installation of rectangular ribs is debated. Such force is recorded by performing line integration around outer surface of an obstacle. Keywords: Heat transfer; Heated rectangular ribs; Partially heated obstacle; Hybrid meshing; Finite element method.
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1. Introduction The study of fluids which includes plasma, gases and liquids is termed as fluid mechanics. The subject of fluid mechanics claims numerous applications in biology, meteorology, astrophysics, geophysics, biomedical engineering, oceanography, chemical engineering, mechanical engineering and civil engineering to mention just a few. It seems very difficult to summarize the beauty of fluid mechanics but the credit goes to days of ancient Greece when the “Archimedes principle” was given by Archimedes by getting confidence on buoyancy and fluid statics properties. The said principle was declared as great work in the field of fluid mechanics and got published as “On Floating Bodies”. The later developments in fluid mechanics includes experiments and observations (Leonardo da Vinci), barometer invention (Evangelista Torricelli), viscosity examination (Isaac Newton), exploring hydrostatics and propose Pascal’s law (Blaise Pascal), and the role of mathematics in fluid dynamics debated in “Hydrodynamica” by Daniel Bernoulli . Owing the importance of mathematics towards fluid flow different mathematicians contributed their understanding in this direction like the inviscid flow field properties were examined by Siméon Denis Poisson, Pierre-Simon Laplace, Joseph Louis Lagrange and Jean le Rond d'Alembert. The viscous flow field characteristics were analyzed by prominent engineers namely Gotthilf Hagen and Jean Léonard Marie Poiseuille. To investigate compressible and incompressible fluid flows the outstanding mathematical formulation was concluded by ClaudeLouis Navier and George Gabriel Stokes with the named Navier–Stokes equations while the helping concept of boundary layer was introduced by Ludwig Prandtl. It is well consensus that the Navier-Stokes equations are the simplest and fruitful mathematical model to narrate the fluid flow field properties. Owing such mathematical frame as yet tremendous efforts are concluded in the literature like the classical viscous fluid flow problem was solved by Crane [1]. In this 2
problem, the viscous fluid due to stretching plate was mathematically modelled in two dimensional frame. The idea was extended by entertaining time dependent flow regime by Devi et al. [2]. The solution of energy equation to inspect the temperature variation in concerned medium is also a topic of great interest for the researchers. Particularly, in the field of fluid rheology when heat is transfer from one component to another component by way of gas or liquid (fluid) is known as heat transfer in fluid. Owing the importance of heat transfer in fluids researchers performed analysis by coupling Navier-Stokes equation with energy equation like Chaim [3] investigate temperature distribution in a fluid flow due to stretching surface under stagnation point region. In this attempt the novelty was increased by considering variable thermal conductivity. It is important to note that the linear variation of thermal conductivity with temperature was taken in this study. The two mathematical directories were proposed in this attempt towards variable and constant wall temperature. The closed form solutions were offered by using regular perturbation method. Further the shooting method was utilized as well for acceptable approximations. One can assessed the productive mathematical directories on heat transfer in fluids (Newtonian/non-Newtonian) subject to various geometric configurations in Refs. [4-41]. The present article contains evaluation on drag force experienced by partially heated circular shaped obstacle in rectangular domain. The heat transfer aspects are introduced by coupling the Navier-Stokes equations with energy equation. The pagination is organized in this way: the flow narration via Navier-Stokes equations is concluded as a literature survey in Section-1. The problem formulation is offered in Section-2. The benchmark quantity expression is added as Section-3. The adopted numerical scheme is directed in Section-4. The obtained outcomes are debated in Section-5. The key outcomes are reported in Section-6.
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2. Mathematical formulation The smooth rectangular channel is considered as a computational domain. The length of channel is taken 1m while height is taken 0.41m. The circular shaped cylinder is placed fixed in between channel as an obstacle with center (0.3, 0.2) m. The Neumann condition is carried at an outlet while the fluid enters from an inlet with the parabolic profile. The no slip condition is taken at outer surface of circular cylinder and upper/lower walls of channel. The main novelty of problem includes the heat transfer aspects. The outlet is manifested with an adiabatic condition. The walls are taken cold except the outer surface of circular cylinder and the region where the rectangular ribs are installed. To be more specific, the circular cylinder and rectangular ribs are taken uniformly heated. To narrate the heat transfer in fluid flow regime subject to rectangular computational domain one can conclude the following dimensionless differential system as follows:
∂u ∂v + = 0, ∂x ∂y
u u
∂u ∂x
+v
∂u ∂y
=−
∂p ∂x
(1)
1 ∂ u ∂ u 2 + 2 , Re ∂x ∂y 2
+
2
∂v ∂v ∂p 1 ∂ 2v ∂ 2v +v =− + + , ∂x ∂y ∂y Re ∂x 2 ∂y 2
u
∂θ ∂θ 1 ∂ 2θ ∂ 2θ +v = + ∂x ∂ y Re . Pr ∂ x 2 ∂ y 2
.
(2)
(3) (4)
The momentum boundary conditions are Inlet of channel
u = 4. umax . y ( H a − y ) , v = 0,
Outlet of channel
∂u ∂v = = p = 0, ∂x ∂x
(5)
Bottom wall u = 0, v = 0, Upper wall u = 0, v = 0, Outer surface of obstacle u = 0, v = 0. The heat transfer boundary constraints can be written as:
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Inlet of channel Outlet of channel
θ = 0, ∂θ = 0, ∂x
Bottom wall
θ = 0, for
(6)
0 ≤ x < 0.2, 0.4 < x ≤ 1, y = 0, 0.2 ≤ x ≤ 0.4, y = 0,
θ = 1, for Upper wall
0 ≤ x < 0.2, 0.4 < x ≤ 1, y = 0.41,
θ = 0, for
0.2 ≤ x ≤ 0.4, y = 0.41, the Eq. (6) corresponds the heat transfer conditions for the case when plane heaters are installed
θ = 1, for
at both lower and upper wall. Therefore, the heat transfer boundary constraints will change against position of rectangular heated ribs being installed at walls. 3. Drag coefficient The partially heated circular cylinder is considered as an obstacle and being placed fixed in between rectangular domain. The center is selected (0.3, 0.2). The incoming fluid strikes with partially heated circular cylinder and the drag face experienced by obstacle in terms of drag coefficient. The mathematical expression for the drag coefficient can be written as: CD =
2 DF
ρ um D 2
,
(7 )
here, C D stands for drag coefficient, DF for drag force, um denotes reference velocity and the diameter of circle is taken as characteristics length D . 4. Numerical Method The system given in Eqs. (1)-(4) along with the endpoint conditions provided as Eqs. (5)-(6) is coupled nonlinear. Therefore, to find exact solution seems difficult. To report best approximation we have used finite element method (FEM). For FEM [42-48] simulation the following 5
parametric values are carried: the fluid viscosity ( µ = 0.001), fluid density ( ρ = 1), the characteristics length ( D = 0.1), Reynolds number is reserved fixed (Re = 20), and the Prandtl number is considered (Pr = 0.71). Since the fluid flow is initiated with parabolic velocity profile. In this direction, the mean inflow velocity is attempted as reference velocity that is u m = ( 2 / 3)u max = 0.2,
where
u max = 0.3, is
the maximum inflow velocity of the parabolic profile.
5. Analysis
The present attempt contains case-wise study of evaluation of hydrodynamic force experienced by partially heated circular cylinder. In first case, we consider two line segments as a plane heaters. One is installed at lower wall (0.2 ≤ x ≤ 0.4, y = 0) and while the second one is placed at upper wall (0.2 ≤ x ≤ 0.4, y = 0.41) . The semi-circle (left face) is taken heated uniformly. To narrate flow field in a channel we have used FEM. For better approximation we have carried hybrid meshing of computational domain which includes triangular and rectangular elements. The five different meshing structure namely H-1, H-2, H-3, H-4 and H-5 are proposed. For first case in level H-1, for first case, the computational domain is discretized into 416 domain elements and 68 boundary elements. The drag force experienced by partially heated circular obstacle in a partially heated smooth channel is evaluated in terms of drag coefficient. It is observed that at this level the value of drag coefficient is C D = 5.5609.
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Fig. 1(a). Hybrid meshing of channel for the case of plane heaters.
Fig. 2(a). Velocity profile for the case of plane heaters.
Fig. 2(b). Temperature distribution for the case of plane heaters
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Table-1. Variation in drag coefficient for the case of plane heaters. Levels CD H-1 H-2 H-3 H-4 H-5
5.5609 5.3007 5.2149 5.2034 5.2070
The domain meshing is further improved by owing 700 domain elements and 92 boundary elements. Such level is H-2. The FEM simulation is performed and we observed that the value of drag coefficient is C D = 5.3007. The level H-3, contains 1130 domain elements and 120 boundary elements. For this level the value of drag coefficient is noticed C D = 5.2149. Further, in H-4, the computational domain contains 2104 domain elements and 170 boundary elements. The value of drag coefficient for this level is noticed C D = 5.2034. For better approximation the finest meshing is performed as a level H-5. In this level, the smooth rectangular channel is discretized into 2948 domain elements and 208 boundary elements. The trustful value of drag coefficient is noticed 5.2070. The conclusion subject to drag coefficient when two plane heaters are installed is provided in Table. 1. The Fig. 1(a) provides the geometric illustration of meshing H-5 when two plane heaters are installed at channel walls. The primitive variables namely, velocity and temperature are examined and the outcomes are offered in terms of contour plots. Such results are extracted at level H-5. To be more specific, the Fig. 2(a) is streamline plot. One can see that the fluid strikes with circular obstacle and the symmetric bifurcation occurs. The stagnation point region is develop at left face of partially heated obstacle. The corresponding temperature distribution in terms of isotherms is offered in Fig. 2(b). The uniformly heated rectangular rib is installed at bottom wall of channel along with one plane heater at upper wall (0.2 ≤ x ≤ 0.4, y = 0.41).
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Fig. 1(b). Hybrid meshing of channel when heated rectangular rib is installed at bottom wall.
Fig. 3(a). Velocity profile when heated rectangular rib is installed at bottom wall.
Fig. 3(b). Temperature distribution when heated rectangular rib is installed at bottom wall.
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Table-2. Variation in coefficient when heated rectangular rib is installed at bottom wall. Levels CD H-1 11.0410 H-2 10.8081 H-3 10.7788 H-4 10.1740 H-5 10.1000
The five various levels namely, H-1, H-2, H-3, H-4 and H-5 are structured for the case of heated rib introduced at lower wall. In detail, for level H-1, the computational domain divided into 526 domain elements and 80 boundary elements. The level H-2 contains 866 domain elements and 110 boundary elements while the level H-3 owns 1340 domain elements and 140 boundary elements. The meshing is improved further as a level H-4. In this level the computational domain own 2437 domain elements and 199 boundary elements. The extreme improvement is added in terms of level H-5. It contains of 3446 domain elements and 242 boundary elements. The drag force experienced by partially heated circular cylinder in the presence of uniformly heated rectangular rib at bottom wall of channel is evaluated in terms of drag coefficient. The value of drag coefficient is noted by performing the line integration around outer surface of obstacle. The Table. 2 reports the value of drag coefficient at five various meshing levels. The better approximation of drag coefficient at level H-5 is noticed C D = 10.1000. The drag force faced by partially heated cylinder is higher due to presence of heated rectangular rib at bottom wall. The Fig. 1(b) is the geometric description of meshing (H-5) when the heated rectangular rib is installed at bottom wall. For this case the velocity and temperature distribution are studied at level H-5. The Fig. 3(a) offers the velocity profile. One can see that the heated rectangular rib cause resistance to fluid flow as a result the bulk motion of fluid is observed from upper region of channel towards outlet of channel. The fluid strikes with partially heated obstacle and gain acceleration. 11
Fig. 1(c). Hybrid meshing of channel when heated rectangular rib is installed at upper wall.
Fig. 4(a). Velocity profile when heated rectangular rib is installed at upper wall.
Fig. 4(b). Temperature distribution with when heated rectangular rib is installed at upper wall.
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Table-3. Variation in drag coefficient when heated rectangular rib is installed at upper wall.. Levels CD H-1 11.694 H-2 10.943 H-3 10.889 H-4 10.874 H-5 10.871
The corresponding temperature distribution due to heated rectangular rib at bottom wall, plane heater at upper wall and partially heated circular cylinder is offered in Fig. 3(b). In third case, the heated rectangular rib is installed at upper wall of channel with plane heater at bottom wall (0.2 ≤ x ≤ 0.4, y = 0) . The five meshing structures are carried as follows: in level H-1, the
domain contains 528 domain elements and 80 boundary elements while level H-2 own 844 domain elements and 110 boundary elements. In level H-3, the channel is divided into 1348 domain elements and 142 boundary elements. The meshing is improved by adding 2427 domain elements and 199 boundary elements. Such level is termed as H-4. The meshing is improved further as a level H-5. In this level the computational domain contains 3472 domain elements and 244 boundary elements. The drag force experienced by partially heated circular cylinder is evaluated in terms of drag coefficient by adopting line integration around outer surface of cylinder. The value drag coefficient at various meshing levels is offered in Table. 3. The best approximation at level H-5 of drag coefficient is C D = 10.871. The level H-5 meshing geometric illustration of computational domain when uniformly heated rectangular rib is installed at upper wall is offered in Fig. 1(c). The velocity and temperature variations are inspected at level H-5. The Fig. 4(a) and Fig. 4(b) are offered in this direction.
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Fig. 1(d). Hybrid meshing of channel when heated rectangular ribs are installed at both upper and lower wall.
Fig. 5(a). Velocity profile when heated rectangular ribs are installed at both upper and lower wall.
Fig. 5(b). Temperature distribution when heated rectangular ribs are installed at both upper and lower wall.
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Table-4. Variation in drag coefficient when heated rectangular ribs are installed at both upper and lower wall. Levels CD H-1 39.969 H-2 37.767 H-3 37.750 H-4 37.721 H-5 37. 720
To be more specific, Fig. 4(a) offers the velocity distribution of fluid flow enters with parabolic velocity profile and strikes with partially heated circular cylinder placed being fixed in smooth channel having heated rectangular rib at upper wall and plane heater at bottom wall. One can observed that the heated rectangular rib offers resistance to fluid flow and hence the bulk motion of fluid is noticed towards lower region of channel. The stagnation point is noticed at left face of partially heated circular cylinder. The corresponding temperature distribution due to uniformly heated rectangular rib at upper wall, partially heated circular cylinder and plane heater at lower wall is offered in Fig. 4(b). In fourth case, we installed two heated rectangular ribs. The one heated rectangular rib is installed at bottom wall while the other one is taken at upper wall. In this case, to evaluate the drag force experience by partially heated circular cylinder we have carried five various meshed levels namely H-1, H-2, H-3, H-4 and H-5. In level H-1, the computational domain along with two rectangular ribs is divided into 632 domain elements and 92 boundary elements. The H-2 level own 975 domain elements and 127 boundary elements. The discretization is improved as a level H-3 by dividing the computational domain into 1589 domain elements and 163 boundary elements. The level H-4 claims 2774 domain elements and 278 boundary elements. For better approximation we take one more meshed level named H-5 and in this level the computational domain with two rectangular ribs is divided into 4020 domain elements and 278 boundary elements. In the presence of two heated rectangular ribs the partially 15
heated circular cylinder faced drag force and it is evaluated in terms of drag coefficient by carrying line integration around the outer surface of obstacle. The values are recorded for all five meshed level. The Table. 4 provided in this regard. It is noticed that the most refine value of drag coefficient at level H-5 is CD = 37. 720. The geometric illustration of level H-5 meshing for computational domain having two rectangular ribs is depicted in Fig. 1(d). The velocity and temperature profiles are evaluated at level H-5. In detail, the flow visualization in smooth rectangular channel with two heated rectangular ribs is offered in Fig. 5(a). One can see that the fluid has narrow region to pass on and after striking with partially heated circular cylinder the symmetric bifurcation occurs. The stagnation point region seems increases. The presence of heated rectangular ribs results the two vortices in open region of channel. The corresponding temperature distribution due to heated rectangular ribs and partially heated circular cylinder is offered in Fig. 5(b). Collectively, it is examine that the partially heated circular shaped obstacle experienced higher drag in the presence of rectangular heated ribs at both upper and lower walls. 6. Conclusion The fluid flow in a partially heated rectangular domain is investigated. The partially heated circular shaped cylinder is placed fixed in between channel as an obstacle. Problem is mathematically formulated and solved with the aid of finite element method. The analysis is made interesting by considering four different cases subject to installation of rectangular ribs. The hydrodynamic force namely drag force is evaluated in each case by adopting line integration around outer surface of obstacle. The rectification is shared up-to five meshing levels. It is noticed that in the presence of heated rectangular ribs at both upper and lower walls the drag force experience by circular obstacle is maximum.
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Nomenclature ( x, y ) r V = (u , v ) p Re Pr Ha
Dimensionless space variables Dimensionless velocity field
umax
Maximum velocity
θ DF
Dimensionless temperature Drag force
CD
Drag coefficient
um D
Reference velocity
Pressure Reynolds number Prandtl number Height of the channel
Characteristic length
Acknowledgement The authors would like to express their gratitude to Air University Group of Computational Mathematics (AUGOCM), Islamabad Pakistan and United Arab Emirates University, Al Ain, UAE for providing administrative and technical support. In addition, authors also would like to acknowledge and express their gratitude to the United Arab Emirates University, Al Ain, UAE for providing the financial support with Grant No. 31S363-UPAR (4) 2018.
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33. Rehman, Khalil Ur, M. Y. Malik, Waqar A. Khan, Ilyas Khan, and S. O. Alharbi. "Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk." Symmetry 11, no. 5 (2019): 699. 34. Bilal, S., Zubair Mustafa, Khalil Ur Rehman, and M. Y. Malik. "MHD Second Grade NanoFluid Flow Induced by a Rotatory Cone." Journal of Nanofluids 8, no. 4 (2019): 876884. 35. Rehman, Khalil Ur, Usman Ali, Iffat Zehra, M. Y. Malik, and Saleem Ullah. "Physical aspects of the Jeffery fluid inducing homogeneous–heterogeneous reactions in MHD flow: a Cattaneo–Christov approach." Canadian Journal of Physics 999 (2019): 1-7. 36. Rehman, Khalil Ur. "Magnetized and non-magnetized two layer liquids: A Lie symmetry analysis." Journal of Molecular Liquids 292 (2019): 111393. 37. Bibi, Madiha, A. Zeeshan, M. Y. Malik, and K. U. Rehman. "Numerical investigation of the unsteady solid-particle flow of a tangent hyperbolic fluid with variable thermal conductivity and convective boundary." The European Physical Journal Plus 134, no. 6 (2019): 298. 38. Rehman, Khalil Ur, M. Y. Malik, Qasem M. Al-Mdallal, and Mostafa Zahri. "On both magnetized and non-magnetized dual stratified medium via stream lines topologies: A generalized formulation." Scientific reports 9 (2019). 39. Ali, Usman, Ali S. Alqahtani, Khalil Ur Rehman, and M. Y. Malik. "On cattaneo-christov heat flux analysis with magneto-hydrodynamic and heat generation effects in a Carreau nanofluid over a stretching sheet." Revista Mexicana de Física 65, no. 5 Sept-Oct (2019): 479488. 40. Al-Mdallal, Qasem M., N. Indumathi, B. Ganga, and AK Abdul Hakeem. "Marangoni radiative effects of hybrid-nanofluids flow past a permeable surface with inclined magnetic field." Case Studies in Thermal Engineering 17 (2020): 100571. 41. Gokulavani, P., M. Muthtamilselvan, Qasem M. Al-Mdallal, and D. H. Doh. "Effects of orientation of the centrally placed heated baffle in an alternative configured ventilation cavity." The European Physical Journal Plus 135, no. 1 (2020): 1-16. 42. Rehman, Khalil Ur, Qasem M. Al-Mdallal, R. Mahmood, M. Y. Malik, and Ramzan Ali. "On inclined heated square obstacle in a liquid stream carried by partially heated channel: Finite element analysis." Case Studies in Thermal Engineering 15 (2019): 100532. 43. Rehman, Khalil Ur. "Communication on partially heated Aluminum 6063-T83 enclosure: Finite element visualization." Physica A: Statistical Mechanics and its Applications 535 (2019): 122503. 44. Rehman, Khalil Ur, M. Y. Malik, Mostafa Zahri, Qasem M. Al-Mdallal, Mohammed Jameel, and M. Imran Khan. "Finite element technique for the analysis of buoyantly convective multiply connected domain as a trapezium enclosure with heated circular obstacle." Journal of Molecular Liquids 286 (2019): 110892. 45. Mahmood, R., N. Kousar, Khalil Ur Rehman, and M. Mohasan. "Lid driven flow field statistics: A non-conforming finite element Simulation." Physica A: Statistical Mechanics and its Applications 528 (2019): 121198. 46. Rehman, Khalil Ur, M. S. Alqarni, R. Mahmood, N. Kousar, and M. Y. Malik. "A classical remark on the compatibility of inlet velocity and pressure singularities: Finite-element visualization." The European Physical Journal Plus 134, no. 5 (2019): 230. 20
47. Rehman, Khalil Ur, M. Y. Malik, R. Mahmood, N. Kousar, and Iffat Zehra. "A potential alternative CFD simulation for steady Carreau–Bird law-based shear thickening model: PartI." Journal of the Brazilian Society of Mechanical Sciences and Engineering 41, no. 4 (2019): 176. 48. Rehman, Khalil Ur, R. Mahmood, N. Kousar, S. Bilal, and I. Zehra. "On magnetized liquid stream statistics in grooved channel: A finite element visualization." Physica A: Statistical Mechanics and its Applications 535 (2019): 122463.
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Conflict of Interest Authors has no conflict of interest.
S2214-157X(19)30372-7
DOI:
https://doi.org/10.1016/j.csite.2020.100597
Reference:
CSITE 100597
To appear in:
Case Studies in Thermal Engineering
Received Date: 12 September 2019 Revised Date:
12 January 2020
Accepted Date: 29 January 2020
Please cite this article as: K.U. Rehman, Q.M. Al-Mdallal, On partially heated circular obstacle in a channel having heated rectangular ribs: Finite element outcomes, Case Studies in Thermal Engineering (2020), doi: https://doi.org/10.1016/j.csite.2020.100597. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Author Contribution Statement
Khalil Ur Rehman: Conceptualization, Investigation, Writing- Original draft preparation, Reviewing, and Editing. Qasem M. Al-Mdallal: Conceptualization, Methodology, Investigation. Supervision.
On Partially Heated Circular Obstacle in a Channel Having Heated Rectangular Ribs: Finite Element Outcomes Khalil Ur Rehman1, and Qasem M. Al-Mdallal2,* 1 2
Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, Abu Dhabi, United Arab Emirates
Correspondence email address: [email protected] (Q. M. Al-Mdallal) [email protected] (K.U. Rehman),
Abstract. In this paper we investigate the fluid flow in rectangular domain having partially heated circular cylinder. To be more specific, the cold fluid is initiated at an inlet of rectangular channel with the parabolic velocity profile. The Neumann condition is carried at outlet of channel. The no-slip condition is implemented at both upper and lower walls. To increase novelty we have considered heat transfer aspects. In this direction, the partially heated circular cylinder is taken fixed in a rectangular channel. The heated rectangular shaped ribs are introduced case-wise. The left wall is taken at zero temperature while the outlet of channel is supported with an adiabatic condition. The whole physical design is controlled with the help of mathematical formulation. The obtained differential system is solved by finite element method. The outcomes are shared in terms of graphs and tables. The variation in drag force subject to different installation of rectangular ribs is debated. Such force is recorded by performing line integration around outer surface of an obstacle. Keywords: Heat transfer; Heated rectangular ribs; Partially heated obstacle; Hybrid meshing; Finite element method.
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1. Introduction The study of fluids which includes plasma, gases and liquids is termed as fluid mechanics. The subject of fluid mechanics claims numerous applications in biology, meteorology, astrophysics, geophysics, biomedical engineering, oceanography, chemical engineering, mechanical engineering and civil engineering to mention just a few. It seems very difficult to summarize the beauty of fluid mechanics but the credit goes to days of ancient Greece when the “Archimedes principle” was given by Archimedes by getting confidence on buoyancy and fluid statics properties. The said principle was declared as great work in the field of fluid mechanics and got published as “On Floating Bodies”. The later developments in fluid mechanics includes experiments and observations (Leonardo da Vinci), barometer invention (Evangelista Torricelli), viscosity examination (Isaac Newton), exploring hydrostatics and propose Pascal’s law (Blaise Pascal), and the role of mathematics in fluid dynamics debated in “Hydrodynamica” by Daniel Bernoulli . Owing the importance of mathematics towards fluid flow different mathematicians contributed their understanding in this direction like the inviscid flow field properties were examined by Siméon Denis Poisson, Pierre-Simon Laplace, Joseph Louis Lagrange and Jean le Rond d'Alembert. The viscous flow field characteristics were analyzed by prominent engineers namely Gotthilf Hagen and Jean Léonard Marie Poiseuille. To investigate compressible and incompressible fluid flows the outstanding mathematical formulation was concluded by ClaudeLouis Navier and George Gabriel Stokes with the named Navier–Stokes equations while the helping concept of boundary layer was introduced by Ludwig Prandtl. It is well consensus that the Navier-Stokes equations are the simplest and fruitful mathematical model to narrate the fluid flow field properties. Owing such mathematical frame as yet tremendous efforts are concluded in the literature like the classical viscous fluid flow problem was solved by Crane [1]. In this 2
problem, the viscous fluid due to stretching plate was mathematically modelled in two dimensional frame. The idea was extended by entertaining time dependent flow regime by Devi et al. [2]. The solution of energy equation to inspect the temperature variation in concerned medium is also a topic of great interest for the researchers. Particularly, in the field of fluid rheology when heat is transfer from one component to another component by way of gas or liquid (fluid) is known as heat transfer in fluid. Owing the importance of heat transfer in fluids researchers performed analysis by coupling Navier-Stokes equation with energy equation like Chaim [3] investigate temperature distribution in a fluid flow due to stretching surface under stagnation point region. In this attempt the novelty was increased by considering variable thermal conductivity. It is important to note that the linear variation of thermal conductivity with temperature was taken in this study. The two mathematical directories were proposed in this attempt towards variable and constant wall temperature. The closed form solutions were offered by using regular perturbation method. Further the shooting method was utilized as well for acceptable approximations. One can assessed the productive mathematical directories on heat transfer in fluids (Newtonian/non-Newtonian) subject to various geometric configurations in Refs. [4-41]. The present article contains evaluation on drag force experienced by partially heated circular shaped obstacle in rectangular domain. The heat transfer aspects are introduced by coupling the Navier-Stokes equations with energy equation. The pagination is organized in this way: the flow narration via Navier-Stokes equations is concluded as a literature survey in Section-1. The problem formulation is offered in Section-2. The benchmark quantity expression is added as Section-3. The adopted numerical scheme is directed in Section-4. The obtained outcomes are debated in Section-5. The key outcomes are reported in Section-6.
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2. Mathematical formulation The smooth rectangular channel is considered as a computational domain. The length of channel is taken 1m while height is taken 0.41m. The circular shaped cylinder is placed fixed in between channel as an obstacle with center (0.3, 0.2) m. The Neumann condition is carried at an outlet while the fluid enters from an inlet with the parabolic profile. The no slip condition is taken at outer surface of circular cylinder and upper/lower walls of channel. The main novelty of problem includes the heat transfer aspects. The outlet is manifested with an adiabatic condition. The walls are taken cold except the outer surface of circular cylinder and the region where the rectangular ribs are installed. To be more specific, the circular cylinder and rectangular ribs are taken uniformly heated. To narrate the heat transfer in fluid flow regime subject to rectangular computational domain one can conclude the following dimensionless differential system as follows:
∂u ∂v + = 0, ∂x ∂y
u u
∂u ∂x
+v
∂u ∂y
=−
∂p ∂x
(1)
1 ∂ u ∂ u 2 + 2 , Re ∂x ∂y 2
+
2
∂v ∂v ∂p 1 ∂ 2v ∂ 2v +v =− + + , ∂x ∂y ∂y Re ∂x 2 ∂y 2
u
∂θ ∂θ 1 ∂ 2θ ∂ 2θ +v = + ∂x ∂ y Re . Pr ∂ x 2 ∂ y 2
.
(2)
(3) (4)
The momentum boundary conditions are Inlet of channel
u = 4. umax . y ( H a − y ) , v = 0,
Outlet of channel
∂u ∂v = = p = 0, ∂x ∂x
(5)
Bottom wall u = 0, v = 0, Upper wall u = 0, v = 0, Outer surface of obstacle u = 0, v = 0. The heat transfer boundary constraints can be written as:
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Inlet of channel Outlet of channel
θ = 0, ∂θ = 0, ∂x
Bottom wall
θ = 0, for
(6)
0 ≤ x < 0.2, 0.4 < x ≤ 1, y = 0, 0.2 ≤ x ≤ 0.4, y = 0,
θ = 1, for Upper wall
0 ≤ x < 0.2, 0.4 < x ≤ 1, y = 0.41,
θ = 0, for
0.2 ≤ x ≤ 0.4, y = 0.41, the Eq. (6) corresponds the heat transfer conditions for the case when plane heaters are installed
θ = 1, for
at both lower and upper wall. Therefore, the heat transfer boundary constraints will change against position of rectangular heated ribs being installed at walls. 3. Drag coefficient The partially heated circular cylinder is considered as an obstacle and being placed fixed in between rectangular domain. The center is selected (0.3, 0.2). The incoming fluid strikes with partially heated circular cylinder and the drag face experienced by obstacle in terms of drag coefficient. The mathematical expression for the drag coefficient can be written as: CD =
2 DF
ρ um D 2
,
(7 )
here, C D stands for drag coefficient, DF for drag force, um denotes reference velocity and the diameter of circle is taken as characteristics length D . 4. Numerical Method The system given in Eqs. (1)-(4) along with the endpoint conditions provided as Eqs. (5)-(6) is coupled nonlinear. Therefore, to find exact solution seems difficult. To report best approximation we have used finite element method (FEM). For FEM [42-48] simulation the following 5
parametric values are carried: the fluid viscosity ( µ = 0.001), fluid density ( ρ = 1), the characteristics length ( D = 0.1), Reynolds number is reserved fixed (Re = 20), and the Prandtl number is considered (Pr = 0.71). Since the fluid flow is initiated with parabolic velocity profile. In this direction, the mean inflow velocity is attempted as reference velocity that is u m = ( 2 / 3)u max = 0.2,
where
u max = 0.3, is
the maximum inflow velocity of the parabolic profile.
5. Analysis
The present attempt contains case-wise study of evaluation of hydrodynamic force experienced by partially heated circular cylinder. In first case, we consider two line segments as a plane heaters. One is installed at lower wall (0.2 ≤ x ≤ 0.4, y = 0) and while the second one is placed at upper wall (0.2 ≤ x ≤ 0.4, y = 0.41) . The semi-circle (left face) is taken heated uniformly. To narrate flow field in a channel we have used FEM. For better approximation we have carried hybrid meshing of computational domain which includes triangular and rectangular elements. The five different meshing structure namely H-1, H-2, H-3, H-4 and H-5 are proposed. For first case in level H-1, for first case, the computational domain is discretized into 416 domain elements and 68 boundary elements. The drag force experienced by partially heated circular obstacle in a partially heated smooth channel is evaluated in terms of drag coefficient. It is observed that at this level the value of drag coefficient is C D = 5.5609.
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Fig. 1(a). Hybrid meshing of channel for the case of plane heaters.
Fig. 2(a). Velocity profile for the case of plane heaters.
Fig. 2(b). Temperature distribution for the case of plane heaters
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Table-1. Variation in drag coefficient for the case of plane heaters. Levels CD H-1 H-2 H-3 H-4 H-5
5.5609 5.3007 5.2149 5.2034 5.2070
The domain meshing is further improved by owing 700 domain elements and 92 boundary elements. Such level is H-2. The FEM simulation is performed and we observed that the value of drag coefficient is C D = 5.3007. The level H-3, contains 1130 domain elements and 120 boundary elements. For this level the value of drag coefficient is noticed C D = 5.2149. Further, in H-4, the computational domain contains 2104 domain elements and 170 boundary elements. The value of drag coefficient for this level is noticed C D = 5.2034. For better approximation the finest meshing is performed as a level H-5. In this level, the smooth rectangular channel is discretized into 2948 domain elements and 208 boundary elements. The trustful value of drag coefficient is noticed 5.2070. The conclusion subject to drag coefficient when two plane heaters are installed is provided in Table. 1. The Fig. 1(a) provides the geometric illustration of meshing H-5 when two plane heaters are installed at channel walls. The primitive variables namely, velocity and temperature are examined and the outcomes are offered in terms of contour plots. Such results are extracted at level H-5. To be more specific, the Fig. 2(a) is streamline plot. One can see that the fluid strikes with circular obstacle and the symmetric bifurcation occurs. The stagnation point region is develop at left face of partially heated obstacle. The corresponding temperature distribution in terms of isotherms is offered in Fig. 2(b). The uniformly heated rectangular rib is installed at bottom wall of channel along with one plane heater at upper wall (0.2 ≤ x ≤ 0.4, y = 0.41).
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Fig. 1(b). Hybrid meshing of channel when heated rectangular rib is installed at bottom wall.
Fig. 3(a). Velocity profile when heated rectangular rib is installed at bottom wall.
Fig. 3(b). Temperature distribution when heated rectangular rib is installed at bottom wall.
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Table-2. Variation in coefficient when heated rectangular rib is installed at bottom wall. Levels CD H-1 11.0410 H-2 10.8081 H-3 10.7788 H-4 10.1740 H-5 10.1000
The five various levels namely, H-1, H-2, H-3, H-4 and H-5 are structured for the case of heated rib introduced at lower wall. In detail, for level H-1, the computational domain divided into 526 domain elements and 80 boundary elements. The level H-2 contains 866 domain elements and 110 boundary elements while the level H-3 owns 1340 domain elements and 140 boundary elements. The meshing is improved further as a level H-4. In this level the computational domain own 2437 domain elements and 199 boundary elements. The extreme improvement is added in terms of level H-5. It contains of 3446 domain elements and 242 boundary elements. The drag force experienced by partially heated circular cylinder in the presence of uniformly heated rectangular rib at bottom wall of channel is evaluated in terms of drag coefficient. The value of drag coefficient is noted by performing the line integration around outer surface of obstacle. The Table. 2 reports the value of drag coefficient at five various meshing levels. The better approximation of drag coefficient at level H-5 is noticed C D = 10.1000. The drag force faced by partially heated cylinder is higher due to presence of heated rectangular rib at bottom wall. The Fig. 1(b) is the geometric description of meshing (H-5) when the heated rectangular rib is installed at bottom wall. For this case the velocity and temperature distribution are studied at level H-5. The Fig. 3(a) offers the velocity profile. One can see that the heated rectangular rib cause resistance to fluid flow as a result the bulk motion of fluid is observed from upper region of channel towards outlet of channel. The fluid strikes with partially heated obstacle and gain acceleration. 11
Fig. 1(c). Hybrid meshing of channel when heated rectangular rib is installed at upper wall.
Fig. 4(a). Velocity profile when heated rectangular rib is installed at upper wall.
Fig. 4(b). Temperature distribution with when heated rectangular rib is installed at upper wall.
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Table-3. Variation in drag coefficient when heated rectangular rib is installed at upper wall.. Levels CD H-1 11.694 H-2 10.943 H-3 10.889 H-4 10.874 H-5 10.871
The corresponding temperature distribution due to heated rectangular rib at bottom wall, plane heater at upper wall and partially heated circular cylinder is offered in Fig. 3(b). In third case, the heated rectangular rib is installed at upper wall of channel with plane heater at bottom wall (0.2 ≤ x ≤ 0.4, y = 0) . The five meshing structures are carried as follows: in level H-1, the
domain contains 528 domain elements and 80 boundary elements while level H-2 own 844 domain elements and 110 boundary elements. In level H-3, the channel is divided into 1348 domain elements and 142 boundary elements. The meshing is improved by adding 2427 domain elements and 199 boundary elements. Such level is termed as H-4. The meshing is improved further as a level H-5. In this level the computational domain contains 3472 domain elements and 244 boundary elements. The drag force experienced by partially heated circular cylinder is evaluated in terms of drag coefficient by adopting line integration around outer surface of cylinder. The value drag coefficient at various meshing levels is offered in Table. 3. The best approximation at level H-5 of drag coefficient is C D = 10.871. The level H-5 meshing geometric illustration of computational domain when uniformly heated rectangular rib is installed at upper wall is offered in Fig. 1(c). The velocity and temperature variations are inspected at level H-5. The Fig. 4(a) and Fig. 4(b) are offered in this direction.
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Fig. 1(d). Hybrid meshing of channel when heated rectangular ribs are installed at both upper and lower wall.
Fig. 5(a). Velocity profile when heated rectangular ribs are installed at both upper and lower wall.
Fig. 5(b). Temperature distribution when heated rectangular ribs are installed at both upper and lower wall.
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Table-4. Variation in drag coefficient when heated rectangular ribs are installed at both upper and lower wall. Levels CD H-1 39.969 H-2 37.767 H-3 37.750 H-4 37.721 H-5 37. 720
To be more specific, Fig. 4(a) offers the velocity distribution of fluid flow enters with parabolic velocity profile and strikes with partially heated circular cylinder placed being fixed in smooth channel having heated rectangular rib at upper wall and plane heater at bottom wall. One can observed that the heated rectangular rib offers resistance to fluid flow and hence the bulk motion of fluid is noticed towards lower region of channel. The stagnation point is noticed at left face of partially heated circular cylinder. The corresponding temperature distribution due to uniformly heated rectangular rib at upper wall, partially heated circular cylinder and plane heater at lower wall is offered in Fig. 4(b). In fourth case, we installed two heated rectangular ribs. The one heated rectangular rib is installed at bottom wall while the other one is taken at upper wall. In this case, to evaluate the drag force experience by partially heated circular cylinder we have carried five various meshed levels namely H-1, H-2, H-3, H-4 and H-5. In level H-1, the computational domain along with two rectangular ribs is divided into 632 domain elements and 92 boundary elements. The H-2 level own 975 domain elements and 127 boundary elements. The discretization is improved as a level H-3 by dividing the computational domain into 1589 domain elements and 163 boundary elements. The level H-4 claims 2774 domain elements and 278 boundary elements. For better approximation we take one more meshed level named H-5 and in this level the computational domain with two rectangular ribs is divided into 4020 domain elements and 278 boundary elements. In the presence of two heated rectangular ribs the partially 15
heated circular cylinder faced drag force and it is evaluated in terms of drag coefficient by carrying line integration around the outer surface of obstacle. The values are recorded for all five meshed level. The Table. 4 provided in this regard. It is noticed that the most refine value of drag coefficient at level H-5 is CD = 37. 720. The geometric illustration of level H-5 meshing for computational domain having two rectangular ribs is depicted in Fig. 1(d). The velocity and temperature profiles are evaluated at level H-5. In detail, the flow visualization in smooth rectangular channel with two heated rectangular ribs is offered in Fig. 5(a). One can see that the fluid has narrow region to pass on and after striking with partially heated circular cylinder the symmetric bifurcation occurs. The stagnation point region seems increases. The presence of heated rectangular ribs results the two vortices in open region of channel. The corresponding temperature distribution due to heated rectangular ribs and partially heated circular cylinder is offered in Fig. 5(b). Collectively, it is examine that the partially heated circular shaped obstacle experienced higher drag in the presence of rectangular heated ribs at both upper and lower walls. 6. Conclusion The fluid flow in a partially heated rectangular domain is investigated. The partially heated circular shaped cylinder is placed fixed in between channel as an obstacle. Problem is mathematically formulated and solved with the aid of finite element method. The analysis is made interesting by considering four different cases subject to installation of rectangular ribs. The hydrodynamic force namely drag force is evaluated in each case by adopting line integration around outer surface of obstacle. The rectification is shared up-to five meshing levels. It is noticed that in the presence of heated rectangular ribs at both upper and lower walls the drag force experience by circular obstacle is maximum.
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Nomenclature ( x, y ) r V = (u , v ) p Re Pr Ha
Dimensionless space variables Dimensionless velocity field
umax
Maximum velocity
θ DF
Dimensionless temperature Drag force
CD
Drag coefficient
um D
Reference velocity
Pressure Reynolds number Prandtl number Height of the channel
Characteristic length
Acknowledgement The authors would like to express their gratitude to Air University Group of Computational Mathematics (AUGOCM), Islamabad Pakistan and United Arab Emirates University, Al Ain, UAE for providing administrative and technical support. In addition, authors also would like to acknowledge and express their gratitude to the United Arab Emirates University, Al Ain, UAE for providing the financial support with Grant No. 31S363-UPAR (4) 2018.
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19. Rehman, Khalil Ur, Abid Ali Khan, M. Y. Malik, and Usman Ali. "Mutual effects of stratification and mixed convection on Williamson fluid flow under stagnation region towards an inclined cylindrical surface." MethodsX 4 (2017): 429-444. 20. Rehman, Khalil Ur, A. A. Malik, M. Y. Malik, and T. Hayat. "Generalized Lie symmetry analysis for non-linear differential equations: A purely viscous fluid model." Results in physics 7 (2017): 3537-3542. 21. Awais, M., M. Y. Malik, Arif Hussain, and T. Salahuddin. "A computational analysis subject to thermophysical aspects of Sisko fluid flow over a cylindrical surface." The European Physical Journal Plus 132, no. 9 (2017): 392. 22. Rehman, Khalil Ur, Ali Saleh Alshomrani, M. Y. Malik, Iffat Zehra, and Muhammad Naseer. "Thermo-physical aspects in tangent hyperbolic fluid flow regime: a short communication." Case studies in thermal engineering 12 (2018): 203-212. 23. Jahan, Shah, Hamzah Sakidin, Roslinda Nazar, and Ioan Pop. "Analysis of heat transfer in nanofluid past a convectively heated permeable stretching/shrinking sheet with regression and stability analyses." Results in Physics 10 (2018): 395-405. 24. Ali, Usman, Khalil Ur Rehman, Ali Saleh Alshomrani, and M. Y. Malik. "Thermal and concentration aspects in Carreau viscosity model via wedge." Case studies in thermal engineering 12 (2018): 126-133. 25. Bibi, Madiha, M. Y. Malik, and M. Tahir. "Numerical study of unsteady Williamson fluid flow and heat transfer in the presence of MHD through a permeable stretching surface." The European Physical Journal Plus 133, no. 4 (2018): 154. 26. Bilal, S., Ali Saleh Alshomrani, M. Y. Malik, Nabeela Kausar, and Farzana Khan. "Analysis of Carreau fluid in the presence of thermal stratification and magnetic field effect." Results in Physics 10 (2018): 118-125. 27. Ur Rehman, Khalil, M. Y. Malik, S. Bilal, Iffat Zehra, and S. Abdul Gaffar. "On Both Magnetohydrodynamics Thermal Stratified and Dual Convection Flow Field Features: A Computational Study." Journal of Nanofluids 8, no. 2 (2019): 460-465. 28. Ali, Usman, Khalil Ur Rehman, and M. Y. Malik. "The influence of MHD and heat generation/absorption in a Newtonian flow field manifested with a Cattaneo–Christov heat flux model." Physica Scripta 94, no. 8 (2019): 085217. 29. Rehman, Khalil Ur, Iqra Shahzadi, M. Y. Malik, Qasem M. Al-Mdallal, and Mostafa Zahri. "On heat transfer in the presence of nano-sized particles suspended in a magnetized rotatory flow field." Case Studies in Thermal Engineering 14 (2019): 100457. 30. Rehman, Khalil Ur, Qasem M. Al-Mdallal, and M. Y. Malik. "Symmetry analysis on thermally magnetized fluid flow regime with heat source/sink." Case Studies in Thermal Engineering 14 (2019): 100452. 31. Rehman, Khalil Ur, and M. Y. Malik. "On Lie symmetry mechanics for Navier–Stokes equations unified with non-Newtonian fluid model: A classical directory." Physica A: Statistical Mechanics and its Applications 535 (2019): 122469. 32. Bilal, S., Khalil Ur Rehman, Zubair Mustafa, and M. Y. Malik. "Maxwell Nanofluid Flow Individualities by Way of Rotating Cone." Journal of Nanofluids 8, no. 3 (2019): 596-603.
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33. Rehman, Khalil Ur, M. Y. Malik, Waqar A. Khan, Ilyas Khan, and S. O. Alharbi. "Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk." Symmetry 11, no. 5 (2019): 699. 34. Bilal, S., Zubair Mustafa, Khalil Ur Rehman, and M. Y. Malik. "MHD Second Grade NanoFluid Flow Induced by a Rotatory Cone." Journal of Nanofluids 8, no. 4 (2019): 876884. 35. Rehman, Khalil Ur, Usman Ali, Iffat Zehra, M. Y. Malik, and Saleem Ullah. "Physical aspects of the Jeffery fluid inducing homogeneous–heterogeneous reactions in MHD flow: a Cattaneo–Christov approach." Canadian Journal of Physics 999 (2019): 1-7. 36. Rehman, Khalil Ur. "Magnetized and non-magnetized two layer liquids: A Lie symmetry analysis." Journal of Molecular Liquids 292 (2019): 111393. 37. Bibi, Madiha, A. Zeeshan, M. Y. Malik, and K. U. Rehman. "Numerical investigation of the unsteady solid-particle flow of a tangent hyperbolic fluid with variable thermal conductivity and convective boundary." The European Physical Journal Plus 134, no. 6 (2019): 298. 38. Rehman, Khalil Ur, M. Y. Malik, Qasem M. Al-Mdallal, and Mostafa Zahri. "On both magnetized and non-magnetized dual stratified medium via stream lines topologies: A generalized formulation." Scientific reports 9 (2019). 39. Ali, Usman, Ali S. Alqahtani, Khalil Ur Rehman, and M. Y. Malik. "On cattaneo-christov heat flux analysis with magneto-hydrodynamic and heat generation effects in a Carreau nanofluid over a stretching sheet." Revista Mexicana de Física 65, no. 5 Sept-Oct (2019): 479488. 40. Al-Mdallal, Qasem M., N. Indumathi, B. Ganga, and AK Abdul Hakeem. "Marangoni radiative effects of hybrid-nanofluids flow past a permeable surface with inclined magnetic field." Case Studies in Thermal Engineering 17 (2020): 100571. 41. Gokulavani, P., M. Muthtamilselvan, Qasem M. Al-Mdallal, and D. H. Doh. "Effects of orientation of the centrally placed heated baffle in an alternative configured ventilation cavity." The European Physical Journal Plus 135, no. 1 (2020): 1-16. 42. Rehman, Khalil Ur, Qasem M. Al-Mdallal, R. Mahmood, M. Y. Malik, and Ramzan Ali. "On inclined heated square obstacle in a liquid stream carried by partially heated channel: Finite element analysis." Case Studies in Thermal Engineering 15 (2019): 100532. 43. Rehman, Khalil Ur. "Communication on partially heated Aluminum 6063-T83 enclosure: Finite element visualization." Physica A: Statistical Mechanics and its Applications 535 (2019): 122503. 44. Rehman, Khalil Ur, M. Y. Malik, Mostafa Zahri, Qasem M. Al-Mdallal, Mohammed Jameel, and M. Imran Khan. "Finite element technique for the analysis of buoyantly convective multiply connected domain as a trapezium enclosure with heated circular obstacle." Journal of Molecular Liquids 286 (2019): 110892. 45. Mahmood, R., N. Kousar, Khalil Ur Rehman, and M. Mohasan. "Lid driven flow field statistics: A non-conforming finite element Simulation." Physica A: Statistical Mechanics and its Applications 528 (2019): 121198. 46. Rehman, Khalil Ur, M. S. Alqarni, R. Mahmood, N. Kousar, and M. Y. Malik. "A classical remark on the compatibility of inlet velocity and pressure singularities: Finite-element visualization." The European Physical Journal Plus 134, no. 5 (2019): 230. 20
47. Rehman, Khalil Ur, M. Y. Malik, R. Mahmood, N. Kousar, and Iffat Zehra. "A potential alternative CFD simulation for steady Carreau–Bird law-based shear thickening model: PartI." Journal of the Brazilian Society of Mechanical Sciences and Engineering 41, no. 4 (2019): 176. 48. Rehman, Khalil Ur, R. Mahmood, N. Kousar, S. Bilal, and I. Zehra. "On magnetized liquid stream statistics in grooved channel: A finite element visualization." Physica A: Statistical Mechanics and its Applications 535 (2019): 122463.
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Conflict of Interest Authors has no conflict of interest.