www.nature.com/scientificreports OPEN MHD Flow of Sodium Alginate- Based Casson Type Nanofluid Passing Through A Porous Medium Received: 24 October 2017 With Newtonian Heating Accepted: 21 March 2018 Published: xx xx xxxx 1 2 3 4,6 2 5 Arshad Khan , Dolat Khan , Ilyas Khan , Farhad Ali , Faizan ul Karim & Muhammad Imran Casson nanofluid, unsteady flow over an isothermal vertical plate with Newtonian heating (NH) is investigated. Sodium alginate (base fluid)is taken as counter example of Casson fluid. MHD and porosity effects are considered. Effects of thermal radiation along with heat generation are examined. Sodium alginate with Silver, Titanium oxide, Copper and Aluminum oxide are added as nano particles. Initial value problem with physical boundary condition is solved by using Laplace transform method. Exact results are obtained for temperature and velocity fields. Skin-friction and Nusselt number are calculated. The obtained results are analyzed graphically for emerging flow parameters and discussed. It is bring into being that temperature and velocity profile are decreasing with increasing nano particles volume fraction. The fluid is a particular kind of matter which have no fixed shape and deforms easily due to external pressure . Fluids are mainly of two type’s i.e Newtonian and non-Newtonian. Non-Newtonian fluids have numerous indus- 2,3 trial applications . Furthermore, its application with magnetohydrodynamic (MHD) flow in a porous medium can widely be seen in irrigation problem, biological system, petroleum, textile, polymer industries. More investi- 4–7 gations have been published on numerous aspects of MHD non-Newtonian fluid passes over a porous medium . The entropy analysis for nanoui fl d with different type of nano particles and water type base fluid for unsteady MHD flow was studied by . The impact of magnetic field on free convection of nanouid fl in a porous medium 9 10 is presented by . The effects of heat transfer on MHD nanouid in a p fl orous semi annulus has investigated by using numerical methods. Sheikholeslami et al. examined the influence of free convection in a semi annulus enclosure for ferrouid flo fl w in the presence of magnetic source with the consideration of thermal radiation. The observation of non-uniform magnetic field and variable magnetic field on forced convection heat is investigated 12,13 14–16 17,18 by . The observation of MHD on fluid flow with heat transfer is studded by . Recently investigated the nanouid t fl ransportation in a in the presence of magnetic source and porous cavity using CuO nano particles. The influence of external magnetic field for nanouid a fl s water is a base fluid of free convection flow is studied in . Sheikholeslami and Ganji have investigated the effect of convective heat transfer for the nanouid b fl y semi ana- lytical and numerical approaches. The same author has also investigated the influence of heat transfer for nano- uid b fl etween parallel plates in . The influence of Lorentz forces and convection nanouid flo fl w is investigated 22–24 25,26 by . Dissimilar types of nano particles with water based fluid are studied by . The influence of melting heat 27 28 for nanouid i fl s studied by . The transportation of nanouid in p fl orous media is investigated by . The influence 29–31 of magnetic field for nanouid w fl ith entropy generation is analysed by . Nanotechnology is that kind of technology which provides the materials with size less than 100 nm called nanomaterials. On the basis of the structure and their properties, nanomaterials are divided into four categories . Carbon based nano materials, metal based nano materials, Dendrimers and composite. The terminology of Institute of Business and Management Sciences, The University of Agriculture, Peshawar, Khyber Pakhtunkhwa, Pakistan. Department of Mathematics, City University of Science and Information Technology, Peshawar, 25000, Pakistan. Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah, 11952, Saudi Arabia. Computational Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam. Department of Mathematical Sciences, United Arab Emirates University, P. O. Box, 15551, Al Ain, United Arab Emirates. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. Correspondence and requests for materials should be addressed to F.A. (email: farhad.ali@tdt.edu.vn) SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 1 www.nature.com/scientificreports/ nanou fl id was first investigated by Choi . He defined that the fluids occupying the sizes of particles less than 100 nm is called nanouid fl . The categorieswith different attitude of nano particles are particle material, Base fluid, size and concentration, of the nanouid fl . Suspend these nano particles into any type of conventional uid li fl ke oil, water, ethylene glycol to make nanouid fl s. The reason why nano size particles are preferred over micro size parti- cles has been explained by . Nano particles over micro particles, good improvement have seen in thermo physical properties. Nanouid fl s have various applications such as in air conditioning cooling, automotive, power plant cooling, improving diesel generator efficiency etc. . Usually water, ethylene glycol are utilized as heat transfer base fluids. Different substances are used for the production of nanoparticles, which are generally divided into 36 37 38 metallic i.e. copper , metal-oxide i.e. CuO , chalcogenides sulphides, selenides and telluride’s, mentioned and 39 40 different particles, such like carbon nanotubes . In literature the size of one particle is in between 20 nm and 100 nm . Casson fluid model was first presented by Casson in 1959. Casson fluids in tubes was first studied by Oka . Examples of Casson fluids are honey, blood, soup, jelly, stuffs, slurries, artificial fibers etc. Cassonnanouid flo fl w 43 44 with Newtonian heatingpresented by . Sarojamma et al. investigated Casson nanouid p fl ast over perpendicular cylinder in the occurrence of a transverse magnetic field with internal heat generation or absorption. Khalid et al. examined unsteady MHD Casson fluid withfree convection flow in a porous medium. Bhattacharyya et al. studied systematically magnetohydrodynamic Casson fluid flow over a stretching shrink- ing sheet with wall mass transfer. Arthur et al. studied Casson fluid flow in excess of a perpendicular porous surface, chemical reaction in the existence of magnetic field. Recently, Fetecau et al . has investigated fractional nanouid fl s for natural convection flow over an isothermal perpendicular plate with thermal radiation. Hussanan et al. investigates the unsteady heat transfer flow of a non-Newtonian Casson fluid over an oscillating perpen- dicular plate with Newtonian heating. Recently, Imran et al. analyzed the effect of Newtonian heating with slip condition on MHD flow of Casson fluid. MHD flow of Casson fluid with heat transfer and Newtonian heating is 51 43,52 analyzed by Hussanan et al. . The effect of Newtonian heating for nanouid i fl s recently investigated by . But no work is done until now on heat transfer enhancement in Sodium alginate fluid with additional ee ff cts of NH, MHD, porosity, heat generation, and thermal radiation. Silver (Ag), Titanium oxide (TiO ), Copper (Cu) and Aluminum oxide (Al O ) are nano particles suspended in base fluid. Problem is solved and interpreted graphi- 2 3 cally with some conclusions. Mathematical Modeling and solution of the Problem Sodium alginate with Silver (Ag), Titanium oxide (TiO ), Copper (Cu) and Aluminum oxide (Al O ) nano par- 2 2 3 ticles is considered. Heat transfer, thermal radiation and heat generation are taken. Unsteady flow is over an infinite vertical plate (ξ > 0) embedded in a saturated porous medium. MHD effect with uniform magnetic field B of strength B and small magnetic Reynolds number. Initially both the plate and fluid are at rest with constant temperature Θ . At time t = 0 the plate originates oscillation in its plane ξ = 0 according to condition uU == Ht ()cos(ωω ti ); or us Ut in()it ;0 > (1) Aer s ft ome time, plate temperature is raised to Θ . The fluid is electrically conducting. Therefore, by Maxwell equations ∂B div, B0 == CurlE − ,CurlBJ =. μ (2) ∂t By using Ohm’s law JE = σ () +× VB , nf (3) e Th quantities ρ , μ and σ are assumed constants. Magnetic field B is normal to V. e Th Reynolds number is so nf e small that flow is laminar. Hence, σ σ B V 1 nf nf 0 JB ×= [(VB ××= )] B −. ρ ρ ρ nf nf nf (4) 53–55 Equation for an incompressible Casson fluid flow ττ =+ μγ (5) Or p 2 μ + e , ππ > ab c 2π τ = , ab p λ 2μ + e , ππ > ab c 2π c (6) ah where π = e e and e is the (a, b) factor of the deformation rate, π is represent the product of the factor of ab ab ab deformation rate with itself, π is represent the critical value of this product based on the non-Newtonian model, μ is represent the plastic dynamic viscosity of the non- Newtonian fluid and P is yield stress of fluid. Under these η λ SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 2 www.nature.com/scientificreports/ −3 −1 −1 −1 −1 −5 −1 ρ (kgm ) c (kg k ) k (Wm k ) β × 10 (k ) C H NaO (SA) 989 4175 0.613 0.99 6 9 7 Al O 3970 765 40 0.85 2 3 Cu 8933 385 401 1.67 TiO 4250 686.2 8.9528 0.9 Ag 10500 235 429 1.89 58–60 Table 1. er Th mophysical properties of nanouid fl s . conditions alongside with the assumption that the viscous dissipation term in the energy equation is neglected, we get the following system : μψ 1 1 2 nf ρ () uu = 1 + μσ () − B + 1 + ug +Θ () ρβ [] −Θ ;, t ξ > 0, t ξξ nf 0 nf ∞ nf nf γ γ k (7) ⁎ 3 16σ Θ () ρck Θ= 1 + TQ +Θ () −Θ ;, ξ t > 0, pnft nf ξξ 0 ∞ ⁎ 3kk nf (8) ut =Θ 0, =Θ ;0 ξ ≥< ,0 ∂Θ uU == Ht () cos(ωω tu )ors Ut in(),; =− ht Θ≥ 0, ξ = 0 s ∂ξ u →Θ 0, →Θ as ξ →∞ ∞ (9) * * where k is absorption coefficient and σ is Stefan-Boltzmann constant. Where Q is the heat generation term, ρ 0 nf is the density of nanouid fl s, μ is the dynamic viscosity, u is the fluid velocity in the x-axis perpendicular direc- nf tion, γ is the Casson fluid parameter, ψ (0 < ψ < 1), K > 0, ψ is the porous medium and K is the permeability of porous medium, h is a constant heat transfer coefficient, Θ is the constant plate temperature (Θ < Θ , s w w ∞ Θ > Θ due to the cooled or heated plate, respectively), g is the acceleration due to gravity, and β is the thermal w ∞ nf expansion coefficient of the nanouid fl . Expressions for (ρc ) , (ρβ) , μ , ρ , σ , k are given by : p nf nf nf nf nf nf ff ρφ =− (1 ), ρφ += ρμ ,, σ = nf fs nf 25 . (1 − φ) σ () ρβ =− (1 φρ )( βφ )( += ρβ), () ρφ cc (1 −+ )(ρφ )(ρc ), nf fs pnfp fp s kk +− 22φ() kk − 3(σφ − 1) sf fs kk = ,1 σσ = + , nf f nf f kk ++ 2( φ kk − ) (2 σσ +− )( − 1)φ sf fs (10) where φ the volume fraction of nano particles, ρ and ρ is represent the density of base fluid and particle respec- f s tively, and c is specific heat on constant pressure. k , k , and k are the thermal conductivities of the nanouid fl , the p nf f s base-fluid, and the solid particles, respectively. The expressions of Eq. (10) are classified to nano particles . For 58–60 supplementary nano particles with unlike thermal conductivity, dynamic viscosity, see to Table 1 . the dimensionless variables are , u U U Θ− Θ ⁎⁎ ⁎ 00 ∞ u = ,, ξ === ξ t t,, θ U ν ν Θ− Θ 0 w ∞ (11) Into Eqs (7–9), we get uc =− uHuG +> rt θξ ,0, t 20 ξξ (12) θθ =+ cc θξ ;, t > 0 t 45 ξξ (13) ut == 0, θξ 0; ≥< 0, 0 uH == () tt cos(ωω )ors ut in(),( θλ =− 1) +≥ θξ ;0 t ,0 = u →→ 0, θξ 0as →∞ (14) where SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 3 www.nature.com/scientificreports/ ρ 3(σφ − 1) 11 25 . ϕφ =− (1 )1 −+ φφ ,1 cc =+ ,1 =+ , 1 1 2 ρ (2 σσ +− )( − 1)φγ () ϕ f 1 () (1 ) ρc λ + Nr k ps nf 0 νQ nf cc =− 1, φφ += ,, c == λ , 3 4 5 2 nf () ρc ν Pr c k Uk pf 3 f 0 f 2 2 cB σν νρ g () β νψ 10 f f 1 f M MG == ,( rH Θ− Θ= ), ,, =+ 2 3 w ∞ 2 K ϕϕ K ρ U U ρ kU 0 0 0 2 1 f f ⁎ 3 ρ () ρc ϕ νρ g () β 16σ Θ h ν pf f s ∞ s 3 ϕφ =− 1, += φλ Nr ,Pr, == ,( Gr Θ− Θ ) 2 0 3 w ∞ ρ 3kk k U ϕ U ρ f f f 0 20 ⁎ 3 () ρβ nf 16σ Θ s ∞ ϕφ =− 1, += φ Nr 3 0 () ρβ kk 3 k f nf f where is permeability of pours medium, M is the magnetic parameter, Gr is thermal Grashof number, Pr is Prandtl number, Nr is radiation parameter, and λ is Newtonian heating parameter. Laplace Transform Solution Laplace transforms of Eqs (12, 13) gives: cu −+ () qH uG =− r θ , 20 ξξ (15) cq θθ −− () c = 0, 45 ξξ (16) uq == 0, θξ 0; ≥< 0, 0 ω 1 u = , θλ =− + θξ ;0 q ≥= ,0 22 q + ω q u →→ 0, θξ 0as →∞ (17) Eq. (16) using Eq. (17) gives: qc − λ c −ξ 4 θξ (, q) = e 4 . q qc −− λ c 54 (18) Aer t ft aking the inverse Laplace of Eq. (18): ct () −− ττ ct () e +− λλ ce erfc ct − τ ⁎ 4 () 4 −c πτ t − 5 t 1 −ξ θξ (, tc ) = λ e 4 dτ. 4 ∫ 2 cc τξ −c5 2 cc τξ 2 0 −− −+ 54 2y 54 2 − erfc + e c 4 erfc 2 c τ 2 c τ 4 4 (19) Solution of Eq. (15) is: uq (, ξξ )[ =+ uq (, )( uq ξξ ,) ++ uq (, )( uq ξ,)]. (20) ab cd Arranging Eq. (20) as: qH + −ξ uq (, ξ ), = e q + ω qc qH + −ξξ − c c uq (, ξ ), = ee − qc − + qH −ξξ − B 1 c c 42 uq (, ξ ), = ee − cc q + qc − qH + −ξξ − c c uq (, ξ ), = ee 42 − d qc −− λ c 54 (21) where SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 4 www.nature.com/scientificreports/ c cc c 8 86 8 AB == ,, C = , 2 2 cc − λ c c 75 () 4 7 λλ cc ++ cc cc + 45{ 6() 45 7} cc λ −− c 74 5 c B == ,, cc −= cc cH += cc ,, cc Gr λ c 1 64 27 42 58 40 4 Upon inversion: ut (, ξξ )[ =+ ut (, )( ut ξξ ,) ++ ut (, )( ut ξ,)], (22) ab cd where Hw +i 1 ξ −ξ itw c ut (, ξ ) = ee erfc −+ tH (iw) 4i 2 ct 2 Hw +i + e erfc ++ tH (iw) 2 ct 2 Hw −i 1 −ξ ξ −itw − ee 2 erfc −− tH (iw) 4i 2 ct 2 Hw −i ξ ξ c + e 2 erfc +− tH (iw) 2 ct 2 (23) c c 5 5 2 −− cc t ξ 2 −+ cc t ξ −− ξ 2 − ξ 45 45 c c e2 4 − erfc + e 4 erfc 2 ct 2 ct 4 4 1 ut (, ξ ) = A , H H 2 cH t −+ ξ 2 cH t ξ − ξ 2 ξ 2 2 c c −e2 2 − erfc + e 2 erfc ct 2 2 ct 2 2 (24) c c 7 7 ++ c c 5 c 5 c 6 6 −ξi ξi ξ c ξ c 7 7 c c 4 4 eerfci −+ ct + eerfci ++ ct c 5 5 () () 7 2 tc c 2 tc c 4 6 4 6 c Be 6 ut (, ξ ) = , c c 7 7 H−− H c c 6 6 −ξ ξ ξ c ξ c 7 7 c c −ee 2 rfcH −− te − 2 erfc +− Ht () () 22 tc c tc c 2 6 2 6 (25) t c τ 2 5 4c τ ξC 1 e 4 ct () −τ ct () −τ 56 ut (, ξτ )e = +− ce e[ rfcc td − ] ⁎ τ d 6 6 3/ 2 ∫ () 2 c π πτ t − 4 τ 0 t −− Hτ 4c τ ξC 1 e 2 ct () −τ ct () −τ 56 − ee +− cerfcc [] td − ττ ⁎ . 6 6 3/ 2 ∫ () 2 c π πτ t − 2 τ 0 (26) Particular Cases In order to link our found solutions with published literature, the following particular cases are examined by taking some parameters absent. Making Gr = γ = 0 and Re = 1 in Eq. (22), reduces to: 1 ξ ii tw −+ ξ Hw ut (, ξ ) = ee erfc −+ tH (iw) 4i 2 t ξ Hw +i + e erfc ++ tH (iw) 2 t 1 ξ −− itwH ξ −iw − ee erfc −− tH (iw) 4i 2 t ξ Hw −i + e erfc +− tH (iw), 2 t (27) which is identical to results of , Eq. (24). Taking in the above relation, we get: M == 0 SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 5 www.nature.com/scientificreports/ 1 ξξ ii tw −ξξ ww i ut (, ξ ) = ee erfc − iwt +eerfc + iwt 4i 2 t 2 t 1 ξξ −− itww ξξ iiw − ee erfc −−iwt +eerfc +−iwt 4i 2 t 2 t (28) Which is in accordance with , Eq. (25). Taking GrM = === 0, in Eq. (22), it moderates to: k γ ξ Re ξ Re 1 iR tw −ξ eiw ξ Reiw ut (, ξ )e = ei erfc −+ wt eerfcw + i t {} {} 4i 2 t 2 t 1 ξ Re ξ Re −− itww ξ −− Rei ξ Reiw − ee erfc −−ii wt ++ eerfcw − t {} {} 4i 2 t 2 t (29) Identical to , Eq. (35). Skin friction and Nusselt Number 1 1 ∂ut (, ξ ) C = 1 + , f 25 . (1 − φ) γ ∂ξ ξ=0 (30) 1 1 1 −itw C = ie 1 + f 25 . 2 γ (1 − φ) −− tH (iw) ei Hw − 2itw 1−− e − erft [(Hw − i) ] ct π c −+ tH (iw) ei Hw + 2itw +e + erft [(Hw − i) ] ct π c −Ht e H A erfc Ht − + () ct π c c c 7 7 c c 7 7 c + H − c ct +− Ht − () 5 c 5 () c 7 6 6 c c ee 6 6 − Be 6 + i −− ct ππ cc t c 4 42 2 C C + − 22 c ππ c 42 ct () −τ t 5 e1 ct () −τ 6 × +− ce e[ rfcc td − ττ ] . 6 6 3/2 0 πτ t − (31) ∂θξ (, t) Nu =− λ , nf ∂ξ ξ=0 (32) 1 ct () −− ττ ct () 56 e +− λλ ce erfc ct − τ 1 4 () 4 Nu =− λλ c πτ t − dτ. nf 4 ⁎ 2 −− erfc c τ { ()} 5 (33) Discussion In this section different parameters including γ , φ, Gr, M, K, Pr, Nr Figs 2–11 are plotted. Geometry of problem is shown in Fig. 1. The influence of γ on u(y, t) which shows oscillatory behavior increasing first then decreasing is highlighted in Fig. 2. Figures 3 and 4 show effects of φ on u(ξ, t) and θ(ξ, t).ϕ is take in between 0 ≤ φ ≤ 0.04 due to sedimenta- tion when the range goes above from 0.08. It is observed in both cases if the nano particles volume fraction φ is increased it leads to the decreasing of temperature and velocity profile. Figure 5 highlights the effect of Gr for Sodium alginate -based, Casson nanouid fl s on velocity profile. It is found that with increasing Gr, velocity increases. Because increasing effect in Gr , due to increase of buoyancy forces and decrease of viscous forces. Figure 6 the effect of M = 0, 1, 2 on the velocity profile. u (ξ, t) decreases due to increasing dragging force. M = 0, shows absence of MHD. Figure 7 shows K effect of on u(ξ, t). Velocity decrease due to decreasing friction. Figure 8 highlights that profile of velocity is increased with increasing radia- tion parameter Nr. The effect is studied for TiO nano particle. SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 6 www.nature.com/scientificreports/ Figure 1. Geometry of the flow. Figure 2. Effects of Casson fluid parameter γ on the velocity profile of Sodium alginate based Casson nanouid fl when Pr = 0.7, Gr = 2 and ϕ = 0.04. Figure 3. Effects of nano particles volume fraction parameter ϕ on the velocity profile of Sodium alginate based nano fluid when Gr = 0.2, Nr = 0.2 and t = 1. SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 7 www.nature.com/scientificreports/ Figure 4. Effects of nano particles volume fraction parameter ϕ on the temperature profile of Sodium alginate based nano fluid when Pr = 5 and t = 1. Figure 5. Effects of thermal Grashof number Gr on the velocity profile of Sodium alginate based Casson nano uid w fl hen Pr = 0.7, Nr = 2, ϕ = 0.04 and t = 1. Figure 6. Effects of magnetic parameter M on the velocity profile of Sodium alginate based Casson nano fluid when Pr = 0.7, Nr = 2, Gr = 10, k = 2 and t = 1. Figure 7. Effects of permeability of porous medium k on the velocity profile of Sodium alginate based nano uid w fl hen Pr = 10, Gr = 10, Nr = 8, ϕ = 0.04 and t = 1. SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 8 www.nature.com/scientificreports/ Figure 8. Effects of radiation parameter Nr for TiO on the velocity profile of Sodium alginate based nano fluid when Pr = 0.7, Gr = 8, ϕ = 0.04 and t = 1. Figure 9. Comparison of velocities profiles for different types of nano particles for Casson nanouid fl s when Pr = 0.71, Gr = 10, Nr = 2, ϕ = 0.04 and t = 1. Figure 10. Comparison of velocities profiles of Cu and Ag Casson nanouid fl s when Pr = 0.71, Gr = 10, Nr = 2, ϕ = 0.04 and t = 1. Figure 11. Comparison of velocities profiles of Al O and TiO for Casson nanouid fl s when Pr = 0.71, Gr = 10, 2 3 3 Nr = 2, ϕ = 0.04 and t = 1. SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 9 www.nature.com/scientificreports/ The impact of two different types of nano particles (Al O Sodium alginate -based Casson nanofluid and 2 3 Ag-Sodium alginate -based nanouid) o fl n profile of velocity is studied in Fig. 9. The profile of velocity is greater for Al O Sodium alginate -based Casson nanouid a fl nd lower profile velocity for Ag -Sodium alginate -based 2 3 nanouid i fl s observed. Figure 10 highlights the comparison of both (Cu Sodium alginate -based Casson nanou fl id and Ag-Sodium alginate -based nanofluid) on u (ξ, t). Velocity of Ag-Sodium alginate -based nanofluid is lower than copper Sodium alginate -based nanou fl id. This shows that Cu nano particles have more thermal diffusivity compare to Ag which is physically true. 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Heat transfer and flow analysis for SA-TiO2 non-Newtonian nanouid p fl assing through the porous media between two coaxial cylinders. Journal of molecular liquids. 188, 155–161 (2013). 60. Hatami, M. & Ganji, D. D. Natural convection of sodium alginate (SA) non-Newtonian nanou fl id flow between two vertical flat plates by analytical and numerical methods. Case Studies in Thermal Engineering 2, 14–22 (2014). 61. Farhad, A., Khan, I. & Shafie, S. A Note on New Exact Solutions for some Unsteady Flows of Brinkman-Type Fluids over a Plane Wall. Zeitschrif ft ür Naturforschung A 67(6-7), 377–380 (2012). Author Contributions A.K. and I.K. designed the study; D.K. and F.K. conducted the experiments with technical assistance from F.A. and M.I.D.K. analyzed the data and wrote the paper; A.K., I.K. and M.I. provided general assistance. All authors have read and approved the final submission. Additional Information Competing Interests: The authors declare no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 11 www.nature.com/scientificreports/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Cre- ative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not per- mitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. © The Author(s) 2018 SCIEnTIFIC RePo R TS | (2018) 8:8645 | DOI:10.1038/s41598-018-26994-1 12
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