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Thermal radiation impact in mixed convective peristaltic flow of third grade nanofluid
Results in Physics 7 (2017) 3687–3695
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Thermal radiation impact in mixed convective peristaltic flow of third grade nanofluid Sadia Ayub a,⇑, T. Hayat a,b, S. Asghar c, B. Ahmad b a
Department of Mathematics, Quaid-I-Azam University, 45320 Islamabad 44000, Pakistan NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Department of Mathematics, CIIT, Chak Shahzad, Islamabad, Pakistan b
a r t i c l e
i n f o
Article history: Received 1 August 2017 Received in revised form 12 September 2017 Accepted 14 September 2017 Available online 23 September 2017 Keywords: Peristalsis Third grade nanofluid Curved channel Mixed convection Thermal radiation Chemical reaction Flexible walls Numerical solutions
a b s t r a c t This paper models the peristaltic transport of magnetohydrodynamic (MHD) third grade nanofluid in a curved channel with wall properties. Combined effects of heat and mass transfer are retained via mixed convection. The present analysis is made in the presence of thermal radiation and chemical reaction. Noslip effect is maintained at the boundary for the velocity, temperature and nanoparticle volume fraction. Resulting formulation is simplified by employing the assumptions of long wavelength and low Reynolds number approximations. Results of axial velocity, temperature, nanoparticle mass transfer and heat transfer are studied graphically. Results reveal increment in fluid velocity for larger values of heat transfer Grashof number. There is reduction in nanoparticle mass transfer with the increase in thermophoresis parameter. Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction In recent years the scientists have developed various techniques to enhance thermo physical properties of fluids. One of such techniques is to add small solid particles of higher thermal conductivity in fluids. Fluid suspensions with micro or above sized particles lack stability while adding flow resistance. Introducing such fluids in human body can result in abrasion and clogging. However suspensions with particles (generally a metal or metal oxide) of nano size (1–100 nm) cover more surface area while increasing conduction and convection coefficients. Term ‘nanofluid’ was first introduced by Choi [1] in 1995. Nanofluids have been extensively used in various industrial and engineering applications. Cancer therapy involves nanofluids for targeted drug delivery. Nanofluids act as coolants in nuclear reactors as well as in automobile and in industrial cooling. Advance in technology allowed us to decrease the size of computer chips while increasing the storage capacity. However a crucial limitation on developing smaller microchips is unrequited heat generation. Use of nanofluids provides liquid cooling for computer processors due to their high thermal conductivity. Such ⇑ Corresponding author. E-mail address: [email protected] (S. Ayub).
usefulness of nanofluids in several industrial and medical applications motivated the researchers to carry out various investigations (see Refs. [2–12]). Fast growing interest in peristalsis is attributed due to its contribution in medical and engineering applications such as finger pumps, bile transportation through bile duct, fluid mechanics in perivascular space of brain, chyme movement through intestine and food transport through esophagus. Peristalsis is the manifestation of two reflexes generated by the enteric nervous system. Human physiology recognizes peristalsis as the involuntary contraction and relaxation of muscles initiated by presence of food bolus in esophagus. Bypass surgery utilizes peristaltic pumps for the circulation of blood in heart lung machine. Other medical applications include dialysis machine, stethoscope, heart lung machine and ventilator machine. Peristaltic activity gained interest among the researchers in recent years after the pioneer work of Latham [13] followed by Shapiro et al. [14]. They studied peristaltic activity under the assumptions of long wavelength and low Reynolds number in the functioning of ureter and gastrointestinal tracts. After that extensive research has been carried out to study peristaltic activity under various effects (see Refs. [15–23]). Effect of magnetic field on peristalsis has been the center of attention because of its contribution to blood regulation in human body.
https://doi.org/10.1016/j.rinp.2017.09.029 2211-3797/Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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MHD is also used in MRI for diagnosis of tumors, infections and magnetotherapy. Peristaltic activity in presence of MHD is discussed in the investigations [24–30] and several studies therein. Excessive heat generation is a problem in engineering applications like nuclear power plants, concrete industry, computer processors and inside of earth. Effective heat transfer can dramatically enhance the efficiency in such cases. There is no doubt that the heat transfer is quite useful in hyperthermia, dilution technique, oxygenation and dialysis. Tissue engineering uses thermal excursion to selectively destroy cells and tissues. All these applications require heat transfer in the most efficient way using both free and forced convection which often results in transfer of mass as well. Heat and mass transfer via mixed convection is most prominent in processes like diffusion of nutrients, food processing, solidification, float glass production cell separation, reverse osmosis, cooling of nuclear reactors, chemical waste management, cooling of combustion chamber wall in a gas turbine and defroster system. Keeping aforementioned uses in mind some relevant studies in this direction have been carried out via Refs. [31–36]. Propulsion gadgets like aircraft missiles, nuclear plants, space vehicles and satellites maintain heat transfer with the help of thermal radiation. In general radiation along with the free and forced convective flows is of crucial importance in space technology and high temperature procedures. Human body maintains appropriate temperature by utilizing these two processes. Few studies for thermal radiation in the presence of mixed convection are given in Refs. [37–40]. In addition heat and mass transfer is most prominent in the production of hydrometallurgical material, polymer production and food processing. These processes cannot be carried out at industrial level without the help of chemical reactions (see Refs. [41–43]). Motivated by the investigations and applications mentioned above, the aim of this article is to study the impact of radiation on mixed convection peristaltic flow of third grade nanofluid in a curved channel. Interestingly even in steady case the third grade fluid flow explores both shear thinning and thickening effects. Although some of the literature includes third grade fluid (see Refs. [44–47]) but third grade nanofluid is scarcely discussed. The current study investigates impact of thermal radiation on peristaltic flow of third grade nanofluid which is not discussed previously. Third grade fluids represent viscoelastic fluids which are important in industrial processes particularly in polymer processing simulations and in biofluid dynamics. Along with thermal radiation and mixed convection, third grade fluids are involved in wire coating, in hemodialysis and skin vasodilation. The current study also includes chemical reaction. Geometry of the problem includes wall compliance properties. The relevant formulation is made and graphical solutions are computed and analyzed for various parameters. The main findings are summarized in the conclusions. Problems formulation We investigate the peristaltic flow of electrically conducting incompressible third grade nano fluid in a channel having width 2d . The channel is molded into a circle with radius R and center O . Coordinates of the channel are considered such that x -axis along the channel and r -axis is perpendicular to x (see Fig. 1). Sinusoidal waves of wavelength k and amplitude b propagate along the channel walls with constant speed c . Flow is generated due to these sinusoidal waves. Geometry of the channel walls is:
2p g ðx ; t Þ ¼ d þ b sin ðx c t Þ ; k
B ¼
where g used above represents the displacements of the upper and lower walls respectively. Fluid flow is under the influence of radially applied magnetic field of strength B0 of the form:
B0 ; 0; 0 : r þ R
ð2Þ
Flow configuration neglects induced magnetic field under the consideration of small magnetic Reynolds number. Thus the Lorentz force F ð¼ J B Þ is incorporated in the system as follows [48,49,50,51,52]:
F ¼
0;
r B2 0 u
!
;0 : 2 ðr þ R Þ
ð3Þ
In above equation J ¼ r ðV B Þ represents the current density, r the electrical conductivity and u be the axial component of velocity field V ¼ ½v ðr ; x ; t Þ; u ðr ; x ; t Þ; 0. Conservation of mass takes the following form:
@v R @u v þ þ ¼ 0: @r r þ R @x r þ R
ð4Þ
Components of momentum equation with mixed convection and Lorentz force are given by [40]:
qf
@v R u @ v u2 @v þ þv @t @r r þ R @x r þR @p 1 @ R @ ðr þ R ÞSr r þ S ¼ þ r þ R @r @r r þ R @x r x RS x x ; r þR
ð5Þ
@u R u @u u v @u þ v þ þ @t @r r þ R @x r þ R R @p 1 @ 2 þ fðr þ R Þ Sr x g ¼ r þ R @x ðr þ R Þ2 @r
qf
þ
r
R @ r B2 0 u Sx x þ qf g bT ðT T 0 Þ 2 þ R @x ðr þ R Þ
þ ðqp qf Þg bC ðC C 0 Þ:
ð6Þ
Energy equation with viscous dissipation and radiation is:
@T R u @T @T þ v þ @t @r r þ R @x 2 2 ! 2 1 @T R @ T @ T þ þ ¼K @r 2 r þ R @r r þ R @x2 @v @u R @ v u þ Sr x þ þ Sr r Sx x @r @r r þ R @x r þR ! 2 @T @C R @T @C þ þ s1 qp cp DB @r @r r þ R @x @x ! s1 qp cp Dt @T 2 R 2 @T 2 þ þ T m @r 2 r þ R @x2 ! @ 16r1 T 3 0 @T : @r @r 3l
qp cp
ð7Þ
Nanoparticle mass transfer equation in the presence of chemical reaction is:
@C @C R u @C þ v þ @t @r r þ R @x ¼
ð1Þ
DB
þ
2 2 ! @2C 1 @C R @ C þ þ @r2 r þ R @r r þ R @x2
2 2 ! Dt @ 2 T 1 @T R @ T M ðC C 0 Þ: ð8Þ þ þ T m @r 2 r þ R @r r þ R @x2
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Fig. 1. Physical sketch of problem.
here qf represents the density of fluid, p the pressure, Sr r ; Sr x and
Sx x the components of extra stress tensor of third grade fluid, qp the density of the nanoparticles, g the gravitational acceleration, T the temperature of fluid, T 0 and C 0 the temperature and concentration at upper and lower channel walls, C the nanoparticle concentration, bT and bC the thermal and concentration expansion coefficients, cp the specific heat at constant temperature, K the q c thermal conductivity of fluid, s1 ¼ qp cp the effective heat capacity f f
of nanoparticles, DB the Brownian diffusivity coefficient, Dt thermophoresis diffusion coefficients, r1 the Stefan–Boltzmann con stant, l the mean absorption coefficient, T m the mean temperature of fluid and M the chemical reaction parameter. The stress tensor for third grade fluid is given by [40]:
S ¼ l A1 þ a1 A2 þ a2 A21 þ b ðtrA21 ÞA1 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a þ a 6 24l b ; 1 2
b P 0;
ð10Þ
T
A1 ¼ ðgrad V Þþðgrad V Þ ; A2 ¼
dA1 T þ A1 ðgrad V Þþðgrad V Þ A1 : dt
ð11Þ
here l denotes the dynamic viscosity and Ai ði ¼ 1; 2Þ are the first and second Rivlin-Ericksen tensors. The subjected conditions can be structured in the form:
u ¼ 0;
at r ¼ g ; C ¼ C 0
T ¼ T 0 ;
ð12Þ
at r ¼ g ;
ð13Þ
" # 2 2 R @ @ @ s þ m1 2 þ d1 g @t r þ R @x2 @t @u @u R u @u u v 1 þ ¼ qf þ þ þv 2 @t @r r þ R @x r þ R ðr þ R Þ 2 n o @ R @ r B0 u 2 ðr þ R Þ Sr x þ S 2 @r r þ R @x x x ð r þ R Þ þ qf g bT ðT T 0 Þ þ ðqp qf Þg bC ðC C 0 Þ; at r ¼ g ;
ð14Þ
m1
where s denotes the tension in the membrane, the mass per unit area and d1 the coefficient of viscous damping. We consider
v ¼
R @w ; r þ R @x
x r c t d w g ; r ¼ ; t ¼ ; d ¼ ; w ¼ ; g ¼ ; k k k d c d d
R¼
R q c d ; ; Re ¼ l d
p¼
d p T T 0 C C 0 d ; C¼ ; Sij ¼ Si j ; ; T ¼ klc T0 C 0 cl
¼
a ; d
2
Pr ¼
cp l c2 ; Ec ¼ ; K T 0 cp
Br ¼ EcPr; M ¼
r d2 B2 l 16r1 T 3 0 0 ; Sc ¼ ; Rd ¼ ; l q DB 3l l cf
2
Gr ¼
where fluid parameters a1 ; a2 and b should satisfy
@w ; @r
x¼
ð9Þ
a1 P 0;
u ¼
b¼
d qf g bT ðT T 0 Þ
l c
b c2 2
Nb ¼
ð15Þ 2
; Qr ¼
ðqp qf Þd g bC ðC C 0 Þ
l s1 qp l Dt ðT T 0 Þ ; Nt ¼ q T m d
;
l c
;
3 3 s1 l DB s d3 m c d d d1 ; E2 ¼ 13 ; E1 ¼ 3 ; E3 ¼ ; q k lc k l l k3
where d is the wave number, w the stream function, Re the Reynolds number, the amplitude ratio parameter, Pr the Prandtl number, Ec the Eckert number, Br the Brinkman number, M the Hartman number, Sc the Schmidt number, Rd the thermal radiation parameter, Gr the local temperature Grashof number, b the Deborah number, Qr the local nanoparticle Grashof number, Nt the thermophoresis parameter, Nbthe Brownian motion parameter and Ei ði ¼ 1 3Þ the wall elastance parameters. Further analysis is taken under the assumption that channel width is small compared to wavelength of peristaltic waves otherwise known as long wavelength approximation ðd ! 0Þ. Such approximation is used in studying peristalsis in ureter, spermatozoa in ductus efferentes and chyme movement in intestine. The Reynolds number is taken low. The Eqs. (5)–(8) subject to these approximations are
@p ¼ 0; @r
ð16Þ
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R @p 1 @ ¼ r þ R @x ðr þ RÞ2 @r " ( ðr þ RÞ þ
2
Axial velocity
! !3 93 @w 1 @2w @w 1 @ 2 w =5 þ 2b @r r þ R @r 2 @r r þ R @r 2 ;
M2
@w þ GrT þ QrC; ðr þ RÞ2 @r
ð17Þ
Eq. (16) depicts that p – pðrÞ. Further combining Eqs. (16) and (17) result in the following expressions: 8 0 2 ! !3 931 < @w 1 @@ 1 @4 @2w @w 1 @ 2 w =5A ðr þ RÞ2 2 þ 2b 2 : @r r þ R @r ; @r ðr þ RÞ @r @r r þ R @r ! 2 @ M @w @ þ þ fðr þ RÞðGrT þ QrC Þg ¼ 0; ð18Þ @r ðr þ RÞ @r @r
@2T 1 @T @2T @T @C þ PrNt 2 þ PrNb þ 2 @r r þ R @r 8 @r @r @r !2 !2 9 < 2 @w 1 @ w @w 1 @2w = þ Br 1 þ 2b ¼ 0; ð19Þ @r r þ R @r 2 : @r r þ R @r2 ;
ð1 þ PrRdÞ
@2C 1 @C Nt @ 2 T 1 @T þ þ þ @r2 r þ R @r Nb @r 2 r þ R @r
This subsection is prepared to examine the effect of various parameters of interest on axial velocity u. Fig. 2 discusses the influence of fluid parameter b on axial velocity u. Increase in Deborah number results in increasing viscosity and so velocity profile weakens. Fig. 3 shows opposite behavior for different values of wall elastance parameters (E1 ; E2 and E3 Þ. Larger wall elastance parameters E1 and E2 allow fluid to flow more freely whereas reverse effect is seen for an increase in damping parameter E3 . This result is in accordance with [14]. Fig. 4 depicts a slight uplift in u via heat transfer Grashof number Gr. Increase in Gr causes viscosity to reduce. Similar effect is seen through [33]. Reverse effect is seen for increasing Hartman number M in Fig. 5. It is observed that larger magnetic field decreases axial velocity as Lorentz force acts like a retarding force for fluid flow. Magnetic field is found useful in medicine i.e., to avoid clotting and regulate blood flow. Fig. 6 indicates decrease in velocity profile for larger values of Qr as it enhances the fluid concentration. This result is found compatible with [11]. Channels with higher curvature reduce velocity profile u. As curvature of the channel offers no clear path for fluid flow therefore the fluid velocity decreases. This effect can be seen in Fig. 7.
! SccC ¼ 0;
ð20Þ
with
g ¼ 1 þ sin 2pðx tÞ;
ð21Þ
@w ¼ 0; T ¼ 0; @r
at r ¼ g;
ð22Þ
@w ¼ 0; T ¼ 0; @r
at r ¼ g;
ð23Þ
" # R @3 @3 @2 M 2 @w E1 3 þ E2 þ GrT þ QrC þ E g¼ 3 2 rþR @x @t@x @x@t ðr þ RÞ2 @r 8 2 ! !3 9 3 < @w 1 1 @4 @2 w @w 1 @ 2 w =5 2 ; at r ¼ g: þ 2b ðr þ RÞ þ : @r r þ R @r2 @r r þ R @r 2 ; ðr þ RÞ2 @r ð24Þ
Temperature distribution Figs. 8–12 are prepared to discuss the graphical illustrations of embedded parameters on temperature distribution T. Fig. 8 displays the behavior of T in response to b. Increment in Deborah number creates an increasing response from temperature profile. Fig. 9 depicts inverse relation between thermophoresis parameter Nt and temperature T, which is in accordance with [11]. Opposite behavior is displayed by T in Fig. 10 for increasing Nb. The reason being kinetic energy is raised as the Brownian motion parameter is enhanced which transforms it into internal energy and so temperature inside the channel rises. Same result is seen in [11]. Larger value of curvature parameter R strengthens the temperature profile as temperature is directly related to velocity via kinetic energy. Hence increase in R is followed by larger T. This effect is seen via Fig. 11. Fig. 12 demonstrates that with larger values of Rd the temperature profile for the fluid decreases. Increase in Rd decreases mean absorption parameter ðl Þ which refers to less energy absorption by the fluid. Thermal radiation in present case may be useful in metal coating, which is a process for the supplying insulation,
Non-dimensional heat transfer coefficient Z is
@T
Z ¼ gx
: @r r¼g
ð25Þ
Results and discussion The objective of this section is to examine the influences of physical quantities via different parameters of interest. The above nonlinear system of equations subject to boundary condition seems very difficult to solve analytically. Hence we compute the graphical solutions by NDSolve in mathematica. The subsequent subsections deal with plots of velocity u, temperature T, volume fraction for nanoparticles C and heat transfer profiles Z with the variations in fluid parameter b, wall flexibility parameters Ei ði ¼ 1 3Þ, heat transfer Grashof number Gr, radial magnetic field M, mass transfer Grashof number Qr, curvature parameter R, thermophoresis parameter Nt, Brownian motion parameter Nb, Brinkman number Br, radiation parameter Rd and chemical reaction parameter c.
Fig. 2. Effect of b on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3, M = 3, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
S. Ayub et al. / Results in Physics 7 (2017) 3687–3695
Fig. 3. Effect of Ei ði ¼ 1 3Þ on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc = 3, b = 0.0001, M = 0.3, Gr = 0.5, Qr = 0.7, Nt = 7 and Nb = 7.
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Fig. 6. Effect of Qr on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3; b = 0.0001, M = 3, Gr = 0.5, Nt = 3, Nb = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 4. Effect of Gr on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3; b = 0.0001, M = 3, Qr = 0.5, Nt = 3, Nb = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 7. Effect of R on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, Sc ¼ 3; b = 0.0001, M = 3, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 5. Effect of M on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3; b = 0.0001, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 8. Effect of bon T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc ¼ 3, M = 3, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
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Fig. 9. Effect of Nt on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, M = 3, Gr = 1.7, Qr = 1.7, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 12. Effect of Rd on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, M = 3, Gr = 0.7, Qr = 0.6, Nt = 3, Nb = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
mechanical damage, environmental safety and protection against signal attenuation. Nanoparticle mass transfer distribution This subsection deals with impact of fluid parameter b, Hartman number M, thermophoresis parameter Nt, Brownian motion parameter Nb and chemical reaction parameter c on nanoparticle mass transfer profile C. Fig. 13 shows that nanoparticle mass transfer decreases with increasing fluid parameter b. Reverse behavior is depicted by C for larger value of M via Fig. 14. Fig. 15 elucidates fall in mass transfer of nanoparticle for larger values of Nt (as thermophoresis encourages the diffusion process). Opposite effect is displayed in Fig. 16 for Brownian motion parameter Nb. Since Nb is indicator of density of nanoparticles in the fluid so an increase in Nb leads to stronger nanoparticle mass transfer profile. Fig. 17 shows substantial increase in C as chemical reaction parameter c attains higher values. Fig. 10. Effect of Nb on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, M = 3, Gr = 0.7, Qr = 0.7, Nt = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
Heat transfer coefficient Impact of various pertinent parameters on heat transfer coefficient Z is discussed in the following subsection. The graphical results are oscillatory in nature due to contraction and expansion of peristaltic waves across the channel walls. Fig. 18 anticipates a
Fig. 11. Effect of R on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, Sc ¼ 3; b = 0.0001, M = 3, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 13. Effect of bon C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3, M = 3, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
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Fig. 14. Effect of M on C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 17. Effect of con C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 15. Effect of Nt on C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 18. Effect of bon Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 16. Effect of Nb on C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
minimal decrease in heat transfer rate as fluid parameter b is enhanced. Decrease in heat transfer rate is captured for higher values of Nt (see Figs. 19). Increase in Nt leads to reduction in magnitude of heat transfer. Same effect is captured for larger thermopherosis parameter as shown via Fig. 20. This result corresponds to study presented by Hayat et al. [40]. As shown by Fig. 21
Fig. 19. Effect of Nt on Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
the magnitude of heat transfer coefficient Z near centerline decays for radiation parameter. Conclusions Channel flow of an incompressible electrically conducting MHD third grade nanofluid has been modeled with curvature, compliant
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Fig. 20. Effect of Nb on Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 21. Effect of Rd on Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
properties, mixed convection, thermal radiation and chemical reaction. The conclusions for this study are: Velocity is decreased by Deborah number. Velocity increases for larger heat transfer Grashof number however reverse effect is noted for increasing mass transfer Grashof number. Both velocity and temperature increase for higher curvature parameter. Nanoparticle mass transfer profile shows opposite behavior for thermophoresis and Brownian motion. Heat transfer rate decreases with the increase in thermophoresis.
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[6] Sandeep N, Chamkha AJ, Animasaun L. Numerical exploration of magnetohydrodynamic nanofluid flow suspended with magnetite nanoparticles. J Braz Soc Mech Sci Eng 2017;39:3635–44. [7] Kumaran G, Sandeep N. Thermophoresis and Brownian moment effects on parabolic flow of MHD Casson and Williamson fluids with cross diffusion. J Mol Liq 2017;233:262–9. [8] Sajid M, Iqbal SA, Naveed M, Abbas Z. Effect of homogeneous-heterogeneous reactions and magnetohydrodynamics on Fe3 O4 nanofluid for the Blasius flow with thermal radiation. J Mol Liq 2017;233:115–21. [9] Das M, Mahathaa BK, Nandkeolyar R. Mixed convection and nonlinear radiation in the stagnation point nanofluid flow towards a stretching sheet with homogenous-heterogeneous reactions effects. Proc Eng 2015;127:1018–25. [10] Narla VK, Prasad KM, Ramanamurdhy JV. Peristaltic transport of Jeffery nanofluid in curved channels. Int Conf Comput Heat Mass Transf 2015;127:869–76. [11] Tanveer A, Hayat T, Alsaadi F, Alsaedi A. Mixed convection peristaltic flow of Eyring-Powell nanofluid in a curved channel with compliant walls. Comput Bio Med 2017;82:71–9. [12] Hayat T, Farooq S, Alsaedi A. Mixed convection peristaltic motion of copperwater nanomaterial with velocity slip effects in a curved channel. Comput Methods Programs Biomed 2017;142:117–28. [13] Latham TW. Fluid motion in peristaltic pump M.S. Thesis. MIT; 1966. [14] Shapiro AH, Jaffrin MY, Weinberg SL. Peristaltic pumping with long wavelengths at low Reynolds number. J Fluid Mech 1969;37:799–825. [15] Hayat T, Bibi S, Alsaadi F, Rafiq M. Peristaltic transport of Prandtl-Eyring liquid in a convectively heated curved channel. PLoS One 2016;11. https://doi.org/ 10.1371/journal.pone.0156995. [16] Bhatti MM, Zeeshan A, Ijaz N. Slip effects and endoscopy analysis on blood flow of particle-fluid suspension induced by peristaltic wave. J Mol Liq 2016;218:240–5. [17] Ali N, Abbasi A, Ahmad I. Channel flow of Ellis fluid due to peristalsis. AIP Adv 2015;5. https://doi.org/10.1063/1.4932042. [18] Hayat T, Ayub S, Alsaedi A, Tanveer A, Ahmad B. Numerical simulation for peristaltic activity of Sutterby fluid with modified Darcy’s law. Results Phys 2017;7:762–8. [19] Ramesh K. Effects of slip and convective conditions on the peristaltic flow of couple stress fluid in an asymmetric channel through porous medium. Comput Methods Programs Biomed 2016;135:1–14. [20] Hayat T, Ali N, Asghar S. Hall effects on peristaltic flow of a Maxwell fluid in a porous medium. Phys Lett A 2007;363:397–403. [21] Sankad GC, Nagathan PS. Influence of the wall properties on the peristaltic transport of a couple stress fluid with slip effect in porous medium. Proc Eng 2015;727:862–8. [22] Hayat T, Bibi A, Yasmin H, Ahmad B. Simultaneous effects of Hall current and homogeneous/heterogeneous reactions on peristalsis. J Taiwan Inst Chem Eng 2016;58:28–38. [23] Hayat T, Tanveer A, Yasmin H, Alsaadi F. Effects of convective conditions and chemical reaction on peristaltic flow of Eyring-Powell fluid. Appl Bion Biomech 2014;11:221–33. [24] Hayat T, Tanveer A, Alsaadi F, Mousa G. Impact of radial magnetic field on peristalsis in curved channel with convective boundary conditions. J Magn Magn Mater 2016;403:47–59. [25] Ellahi R, Hussain F. Simultaneous effects of MHD and partial slip on peristaltic flow of Jeffery fluid in a rectangular duct. J Magn Magn Mater 2015;393:284–92. [26] Ali N, Sajid M, Abbas Z, Javed T. Non-Newtonian fluid flow induced by peristaltic waves in a curved channel. Eur J Mech B/Fluids 2010;29:387–94. [27] Reddy MG, Reddy KV, Makinde OD. Hydromagnetic peristaltic motion of a reacting and radiating couple stress fluid in an inclined asymmetric channel filled with a porous medium. Alex Eng J 2016;55:1841–53. [28] Ali ME, Sandeep N. Cattaneo-Christov model for radiative heat transfer of magnetohydrodynamic Casson-ferrofluid: a numerical study. Results Phys 2017;7:21–30. [29] Hayat T, Rafiq M, Ahmad B. Soret and Dufour effects on MHD peristaltic flow of Jeffrey fluid in a rotating system with porous medium. PLoS One 2016;11: e0145525. [30] Sinha A, Shit GC, Ranjit NK. Peristaltic transport of MHD flow and heat transfer in an asymmetric channel: Effects of variable viscosity, velocity slip and temperature jump. Alex Eng J 2015;54:691–704. [31] Ramesh K. Influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel. J Mol Liq 2016;219:256–71. [32] Tanveer A, Hayat T, Alsaedi A, Ahmad B. Mixed convective peristaltic flow of Sisko fluid in curved channel with homogeneous-heterogeneous reaction effects. J Mol Liq 2017;233:131–8. [33] Tanveer A, Hayat T, Alsaadi F, Alsaedi A. Mixed convection peristaltic flow of Eyring-Powell nanofluid in a curved channel with compliant walls. Comput Bio Med 2017;82:71–9. [34] Das S, Jana RN, Makinde OD. Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating. Alex Eng J 2015;54:251–61. [35] Hayat T, Iqbal R, Tanveer A, Alsaedi A. Mixed convective peristaltic transport of Carreau-Yasuda nanofluid in a tapered asymmetric channel. J Mol Liq 2016;223:1100–13.
S. Ayub et al. / Results in Physics 7 (2017) 3687–3695 [36] Hayat T, Ahmed B, Abbasi FM, Ahmad B. Mixed convective peristaltic flow of carbon nanotubes submerged in water using different thermal conductivity models. Comput Meth Prog Biomed 2016;135:141–50. [37] Dulal P, Hiranmoy M. Non-Darcian buoyancy driven heat and mass transfer over a stretching sheet in a porous medium with radiation and Ohmic heating. Int J Nonlinear Sci 2012;14:115–23. [38] Zheng L, Zhang C, Zhang X, Zhang J. Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium. J Franklin Inst 2013;350:990–1007. [39] Sheikholeslami M, Ganji DD, Javed MY, Ellahi R. Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model. J Magn Magn Mater 2015;374:36–43. [40] Hayat T, Tanveer A, Alsaedi A. Numerical analysis of partial slip on peristalsis of MHD Jeffery nanofluid in curved channel with porous space. J Mol Liq 2016;224:944–53. [41] Hayat T, Tanveer A, Alsaedi A. Mixed convective peristaltic flow of CarreauYasuda fluid with thermal deposition and chemical reaction. Int J Heat Mass Transf 2016;96:474–81. [42] Srinivasacharya D, Reddy GS. Chemical reaction and radiation effects on mixed convection heat and mass transfer over a vertical plate in power-law fluid saturated porous medium. J Egypt Math Soc 2016;24:108–15. [43] Singh K, Kumar M. Influence of chemical reaction on heat and mass transfer flow of a micropolar fluid over a permeable channel with radiation and heat generation. J Thermodynamics 2016;2016. https://doi.org/10.1155/2016/ 8307980.
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Contents lists available at ScienceDirect
Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
Thermal radiation impact in mixed convective peristaltic flow of third grade nanofluid Sadia Ayub a,⇑, T. Hayat a,b, S. Asghar c, B. Ahmad b a
Department of Mathematics, Quaid-I-Azam University, 45320 Islamabad 44000, Pakistan NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Department of Mathematics, CIIT, Chak Shahzad, Islamabad, Pakistan b
a r t i c l e
i n f o
Article history: Received 1 August 2017 Received in revised form 12 September 2017 Accepted 14 September 2017 Available online 23 September 2017 Keywords: Peristalsis Third grade nanofluid Curved channel Mixed convection Thermal radiation Chemical reaction Flexible walls Numerical solutions
a b s t r a c t This paper models the peristaltic transport of magnetohydrodynamic (MHD) third grade nanofluid in a curved channel with wall properties. Combined effects of heat and mass transfer are retained via mixed convection. The present analysis is made in the presence of thermal radiation and chemical reaction. Noslip effect is maintained at the boundary for the velocity, temperature and nanoparticle volume fraction. Resulting formulation is simplified by employing the assumptions of long wavelength and low Reynolds number approximations. Results of axial velocity, temperature, nanoparticle mass transfer and heat transfer are studied graphically. Results reveal increment in fluid velocity for larger values of heat transfer Grashof number. There is reduction in nanoparticle mass transfer with the increase in thermophoresis parameter. Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction In recent years the scientists have developed various techniques to enhance thermo physical properties of fluids. One of such techniques is to add small solid particles of higher thermal conductivity in fluids. Fluid suspensions with micro or above sized particles lack stability while adding flow resistance. Introducing such fluids in human body can result in abrasion and clogging. However suspensions with particles (generally a metal or metal oxide) of nano size (1–100 nm) cover more surface area while increasing conduction and convection coefficients. Term ‘nanofluid’ was first introduced by Choi [1] in 1995. Nanofluids have been extensively used in various industrial and engineering applications. Cancer therapy involves nanofluids for targeted drug delivery. Nanofluids act as coolants in nuclear reactors as well as in automobile and in industrial cooling. Advance in technology allowed us to decrease the size of computer chips while increasing the storage capacity. However a crucial limitation on developing smaller microchips is unrequited heat generation. Use of nanofluids provides liquid cooling for computer processors due to their high thermal conductivity. Such ⇑ Corresponding author. E-mail address: [email protected] (S. Ayub).
usefulness of nanofluids in several industrial and medical applications motivated the researchers to carry out various investigations (see Refs. [2–12]). Fast growing interest in peristalsis is attributed due to its contribution in medical and engineering applications such as finger pumps, bile transportation through bile duct, fluid mechanics in perivascular space of brain, chyme movement through intestine and food transport through esophagus. Peristalsis is the manifestation of two reflexes generated by the enteric nervous system. Human physiology recognizes peristalsis as the involuntary contraction and relaxation of muscles initiated by presence of food bolus in esophagus. Bypass surgery utilizes peristaltic pumps for the circulation of blood in heart lung machine. Other medical applications include dialysis machine, stethoscope, heart lung machine and ventilator machine. Peristaltic activity gained interest among the researchers in recent years after the pioneer work of Latham [13] followed by Shapiro et al. [14]. They studied peristaltic activity under the assumptions of long wavelength and low Reynolds number in the functioning of ureter and gastrointestinal tracts. After that extensive research has been carried out to study peristaltic activity under various effects (see Refs. [15–23]). Effect of magnetic field on peristalsis has been the center of attention because of its contribution to blood regulation in human body.
https://doi.org/10.1016/j.rinp.2017.09.029 2211-3797/Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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MHD is also used in MRI for diagnosis of tumors, infections and magnetotherapy. Peristaltic activity in presence of MHD is discussed in the investigations [24–30] and several studies therein. Excessive heat generation is a problem in engineering applications like nuclear power plants, concrete industry, computer processors and inside of earth. Effective heat transfer can dramatically enhance the efficiency in such cases. There is no doubt that the heat transfer is quite useful in hyperthermia, dilution technique, oxygenation and dialysis. Tissue engineering uses thermal excursion to selectively destroy cells and tissues. All these applications require heat transfer in the most efficient way using both free and forced convection which often results in transfer of mass as well. Heat and mass transfer via mixed convection is most prominent in processes like diffusion of nutrients, food processing, solidification, float glass production cell separation, reverse osmosis, cooling of nuclear reactors, chemical waste management, cooling of combustion chamber wall in a gas turbine and defroster system. Keeping aforementioned uses in mind some relevant studies in this direction have been carried out via Refs. [31–36]. Propulsion gadgets like aircraft missiles, nuclear plants, space vehicles and satellites maintain heat transfer with the help of thermal radiation. In general radiation along with the free and forced convective flows is of crucial importance in space technology and high temperature procedures. Human body maintains appropriate temperature by utilizing these two processes. Few studies for thermal radiation in the presence of mixed convection are given in Refs. [37–40]. In addition heat and mass transfer is most prominent in the production of hydrometallurgical material, polymer production and food processing. These processes cannot be carried out at industrial level without the help of chemical reactions (see Refs. [41–43]). Motivated by the investigations and applications mentioned above, the aim of this article is to study the impact of radiation on mixed convection peristaltic flow of third grade nanofluid in a curved channel. Interestingly even in steady case the third grade fluid flow explores both shear thinning and thickening effects. Although some of the literature includes third grade fluid (see Refs. [44–47]) but third grade nanofluid is scarcely discussed. The current study investigates impact of thermal radiation on peristaltic flow of third grade nanofluid which is not discussed previously. Third grade fluids represent viscoelastic fluids which are important in industrial processes particularly in polymer processing simulations and in biofluid dynamics. Along with thermal radiation and mixed convection, third grade fluids are involved in wire coating, in hemodialysis and skin vasodilation. The current study also includes chemical reaction. Geometry of the problem includes wall compliance properties. The relevant formulation is made and graphical solutions are computed and analyzed for various parameters. The main findings are summarized in the conclusions. Problems formulation We investigate the peristaltic flow of electrically conducting incompressible third grade nano fluid in a channel having width 2d . The channel is molded into a circle with radius R and center O . Coordinates of the channel are considered such that x -axis along the channel and r -axis is perpendicular to x (see Fig. 1). Sinusoidal waves of wavelength k and amplitude b propagate along the channel walls with constant speed c . Flow is generated due to these sinusoidal waves. Geometry of the channel walls is:
2p g ðx ; t Þ ¼ d þ b sin ðx c t Þ ; k
B ¼
where g used above represents the displacements of the upper and lower walls respectively. Fluid flow is under the influence of radially applied magnetic field of strength B0 of the form:
B0 ; 0; 0 : r þ R
ð2Þ
Flow configuration neglects induced magnetic field under the consideration of small magnetic Reynolds number. Thus the Lorentz force F ð¼ J B Þ is incorporated in the system as follows [48,49,50,51,52]:
F ¼
0;
r B2 0 u
!
;0 : 2 ðr þ R Þ
ð3Þ
In above equation J ¼ r ðV B Þ represents the current density, r the electrical conductivity and u be the axial component of velocity field V ¼ ½v ðr ; x ; t Þ; u ðr ; x ; t Þ; 0. Conservation of mass takes the following form:
@v R @u v þ þ ¼ 0: @r r þ R @x r þ R
ð4Þ
Components of momentum equation with mixed convection and Lorentz force are given by [40]:
qf
@v R u @ v u2 @v þ þv @t @r r þ R @x r þR @p 1 @ R @ ðr þ R ÞSr r þ S ¼ þ r þ R @r @r r þ R @x r x RS x x ; r þR
ð5Þ
@u R u @u u v @u þ v þ þ @t @r r þ R @x r þ R R @p 1 @ 2 þ fðr þ R Þ Sr x g ¼ r þ R @x ðr þ R Þ2 @r
qf
þ
r
R @ r B2 0 u Sx x þ qf g bT ðT T 0 Þ 2 þ R @x ðr þ R Þ
þ ðqp qf Þg bC ðC C 0 Þ:
ð6Þ
Energy equation with viscous dissipation and radiation is:
@T R u @T @T þ v þ @t @r r þ R @x 2 2 ! 2 1 @T R @ T @ T þ þ ¼K @r 2 r þ R @r r þ R @x2 @v @u R @ v u þ Sr x þ þ Sr r Sx x @r @r r þ R @x r þR ! 2 @T @C R @T @C þ þ s1 qp cp DB @r @r r þ R @x @x ! s1 qp cp Dt @T 2 R 2 @T 2 þ þ T m @r 2 r þ R @x2 ! @ 16r1 T 3 0 @T : @r @r 3l
qp cp
ð7Þ
Nanoparticle mass transfer equation in the presence of chemical reaction is:
@C @C R u @C þ v þ @t @r r þ R @x ¼
ð1Þ
DB
þ
2 2 ! @2C 1 @C R @ C þ þ @r2 r þ R @r r þ R @x2
2 2 ! Dt @ 2 T 1 @T R @ T M ðC C 0 Þ: ð8Þ þ þ T m @r 2 r þ R @r r þ R @x2
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S. Ayub et al. / Results in Physics 7 (2017) 3687–3695
Fig. 1. Physical sketch of problem.
here qf represents the density of fluid, p the pressure, Sr r ; Sr x and
Sx x the components of extra stress tensor of third grade fluid, qp the density of the nanoparticles, g the gravitational acceleration, T the temperature of fluid, T 0 and C 0 the temperature and concentration at upper and lower channel walls, C the nanoparticle concentration, bT and bC the thermal and concentration expansion coefficients, cp the specific heat at constant temperature, K the q c thermal conductivity of fluid, s1 ¼ qp cp the effective heat capacity f f
of nanoparticles, DB the Brownian diffusivity coefficient, Dt thermophoresis diffusion coefficients, r1 the Stefan–Boltzmann con stant, l the mean absorption coefficient, T m the mean temperature of fluid and M the chemical reaction parameter. The stress tensor for third grade fluid is given by [40]:
S ¼ l A1 þ a1 A2 þ a2 A21 þ b ðtrA21 ÞA1 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a þ a 6 24l b ; 1 2
b P 0;
ð10Þ
T
A1 ¼ ðgrad V Þþðgrad V Þ ; A2 ¼
dA1 T þ A1 ðgrad V Þþðgrad V Þ A1 : dt
ð11Þ
here l denotes the dynamic viscosity and Ai ði ¼ 1; 2Þ are the first and second Rivlin-Ericksen tensors. The subjected conditions can be structured in the form:
u ¼ 0;
at r ¼ g ; C ¼ C 0
T ¼ T 0 ;
ð12Þ
at r ¼ g ;
ð13Þ
" # 2 2 R @ @ @ s þ m1 2 þ d1 g @t r þ R @x2 @t @u @u R u @u u v 1 þ ¼ qf þ þ þv 2 @t @r r þ R @x r þ R ðr þ R Þ 2 n o @ R @ r B0 u 2 ðr þ R Þ Sr x þ S 2 @r r þ R @x x x ð r þ R Þ þ qf g bT ðT T 0 Þ þ ðqp qf Þg bC ðC C 0 Þ; at r ¼ g ;
ð14Þ
m1
where s denotes the tension in the membrane, the mass per unit area and d1 the coefficient of viscous damping. We consider
v ¼
R @w ; r þ R @x
x r c t d w g ; r ¼ ; t ¼ ; d ¼ ; w ¼ ; g ¼ ; k k k d c d d
R¼
R q c d ; ; Re ¼ l d
p¼
d p T T 0 C C 0 d ; C¼ ; Sij ¼ Si j ; ; T ¼ klc T0 C 0 cl
¼
a ; d
2
Pr ¼
cp l c2 ; Ec ¼ ; K T 0 cp
Br ¼ EcPr; M ¼
r d2 B2 l 16r1 T 3 0 0 ; Sc ¼ ; Rd ¼ ; l q DB 3l l cf
2
Gr ¼
where fluid parameters a1 ; a2 and b should satisfy
@w ; @r
x¼
ð9Þ
a1 P 0;
u ¼
b¼
d qf g bT ðT T 0 Þ
l c
b c2 2
Nb ¼
ð15Þ 2
; Qr ¼
ðqp qf Þd g bC ðC C 0 Þ
l s1 qp l Dt ðT T 0 Þ ; Nt ¼ q T m d
;
l c
;
3 3 s1 l DB s d3 m c d d d1 ; E2 ¼ 13 ; E1 ¼ 3 ; E3 ¼ ; q k lc k l l k3
where d is the wave number, w the stream function, Re the Reynolds number, the amplitude ratio parameter, Pr the Prandtl number, Ec the Eckert number, Br the Brinkman number, M the Hartman number, Sc the Schmidt number, Rd the thermal radiation parameter, Gr the local temperature Grashof number, b the Deborah number, Qr the local nanoparticle Grashof number, Nt the thermophoresis parameter, Nbthe Brownian motion parameter and Ei ði ¼ 1 3Þ the wall elastance parameters. Further analysis is taken under the assumption that channel width is small compared to wavelength of peristaltic waves otherwise known as long wavelength approximation ðd ! 0Þ. Such approximation is used in studying peristalsis in ureter, spermatozoa in ductus efferentes and chyme movement in intestine. The Reynolds number is taken low. The Eqs. (5)–(8) subject to these approximations are
@p ¼ 0; @r
ð16Þ
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R @p 1 @ ¼ r þ R @x ðr þ RÞ2 @r " ( ðr þ RÞ þ
2
Axial velocity
! !3 93 @w 1 @2w @w 1 @ 2 w =5 þ 2b @r r þ R @r 2 @r r þ R @r 2 ;
M2
@w þ GrT þ QrC; ðr þ RÞ2 @r
ð17Þ
Eq. (16) depicts that p – pðrÞ. Further combining Eqs. (16) and (17) result in the following expressions: 8 0 2 ! !3 931 < @w 1 @@ 1 @4 @2w @w 1 @ 2 w =5A ðr þ RÞ2 2 þ 2b 2 : @r r þ R @r ; @r ðr þ RÞ @r @r r þ R @r ! 2 @ M @w @ þ þ fðr þ RÞðGrT þ QrC Þg ¼ 0; ð18Þ @r ðr þ RÞ @r @r
@2T 1 @T @2T @T @C þ PrNt 2 þ PrNb þ 2 @r r þ R @r 8 @r @r @r !2 !2 9 < 2 @w 1 @ w @w 1 @2w = þ Br 1 þ 2b ¼ 0; ð19Þ @r r þ R @r 2 : @r r þ R @r2 ;
ð1 þ PrRdÞ
@2C 1 @C Nt @ 2 T 1 @T þ þ þ @r2 r þ R @r Nb @r 2 r þ R @r
This subsection is prepared to examine the effect of various parameters of interest on axial velocity u. Fig. 2 discusses the influence of fluid parameter b on axial velocity u. Increase in Deborah number results in increasing viscosity and so velocity profile weakens. Fig. 3 shows opposite behavior for different values of wall elastance parameters (E1 ; E2 and E3 Þ. Larger wall elastance parameters E1 and E2 allow fluid to flow more freely whereas reverse effect is seen for an increase in damping parameter E3 . This result is in accordance with [14]. Fig. 4 depicts a slight uplift in u via heat transfer Grashof number Gr. Increase in Gr causes viscosity to reduce. Similar effect is seen through [33]. Reverse effect is seen for increasing Hartman number M in Fig. 5. It is observed that larger magnetic field decreases axial velocity as Lorentz force acts like a retarding force for fluid flow. Magnetic field is found useful in medicine i.e., to avoid clotting and regulate blood flow. Fig. 6 indicates decrease in velocity profile for larger values of Qr as it enhances the fluid concentration. This result is found compatible with [11]. Channels with higher curvature reduce velocity profile u. As curvature of the channel offers no clear path for fluid flow therefore the fluid velocity decreases. This effect can be seen in Fig. 7.
! SccC ¼ 0;
ð20Þ
with
g ¼ 1 þ sin 2pðx tÞ;
ð21Þ
@w ¼ 0; T ¼ 0; @r
at r ¼ g;
ð22Þ
@w ¼ 0; T ¼ 0; @r
at r ¼ g;
ð23Þ
" # R @3 @3 @2 M 2 @w E1 3 þ E2 þ GrT þ QrC þ E g¼ 3 2 rþR @x @t@x @x@t ðr þ RÞ2 @r 8 2 ! !3 9 3 < @w 1 1 @4 @2 w @w 1 @ 2 w =5 2 ; at r ¼ g: þ 2b ðr þ RÞ þ : @r r þ R @r2 @r r þ R @r 2 ; ðr þ RÞ2 @r ð24Þ
Temperature distribution Figs. 8–12 are prepared to discuss the graphical illustrations of embedded parameters on temperature distribution T. Fig. 8 displays the behavior of T in response to b. Increment in Deborah number creates an increasing response from temperature profile. Fig. 9 depicts inverse relation between thermophoresis parameter Nt and temperature T, which is in accordance with [11]. Opposite behavior is displayed by T in Fig. 10 for increasing Nb. The reason being kinetic energy is raised as the Brownian motion parameter is enhanced which transforms it into internal energy and so temperature inside the channel rises. Same result is seen in [11]. Larger value of curvature parameter R strengthens the temperature profile as temperature is directly related to velocity via kinetic energy. Hence increase in R is followed by larger T. This effect is seen via Fig. 11. Fig. 12 demonstrates that with larger values of Rd the temperature profile for the fluid decreases. Increase in Rd decreases mean absorption parameter ðl Þ which refers to less energy absorption by the fluid. Thermal radiation in present case may be useful in metal coating, which is a process for the supplying insulation,
Non-dimensional heat transfer coefficient Z is
@T
Z ¼ gx
: @r r¼g
ð25Þ
Results and discussion The objective of this section is to examine the influences of physical quantities via different parameters of interest. The above nonlinear system of equations subject to boundary condition seems very difficult to solve analytically. Hence we compute the graphical solutions by NDSolve in mathematica. The subsequent subsections deal with plots of velocity u, temperature T, volume fraction for nanoparticles C and heat transfer profiles Z with the variations in fluid parameter b, wall flexibility parameters Ei ði ¼ 1 3Þ, heat transfer Grashof number Gr, radial magnetic field M, mass transfer Grashof number Qr, curvature parameter R, thermophoresis parameter Nt, Brownian motion parameter Nb, Brinkman number Br, radiation parameter Rd and chemical reaction parameter c.
Fig. 2. Effect of b on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3, M = 3, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
S. Ayub et al. / Results in Physics 7 (2017) 3687–3695
Fig. 3. Effect of Ei ði ¼ 1 3Þ on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc = 3, b = 0.0001, M = 0.3, Gr = 0.5, Qr = 0.7, Nt = 7 and Nb = 7.
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Fig. 6. Effect of Qr on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3; b = 0.0001, M = 3, Gr = 0.5, Nt = 3, Nb = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 4. Effect of Gr on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3; b = 0.0001, M = 3, Qr = 0.5, Nt = 3, Nb = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 7. Effect of R on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, Sc ¼ 3; b = 0.0001, M = 3, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 5. Effect of M on u when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.3, Sc ¼ 3; b = 0.0001, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 8. Effect of bon T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc ¼ 3, M = 3, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
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Fig. 9. Effect of Nt on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, M = 3, Gr = 1.7, Qr = 1.7, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 12. Effect of Rd on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, M = 3, Gr = 0.7, Qr = 0.6, Nt = 3, Nb = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
mechanical damage, environmental safety and protection against signal attenuation. Nanoparticle mass transfer distribution This subsection deals with impact of fluid parameter b, Hartman number M, thermophoresis parameter Nt, Brownian motion parameter Nb and chemical reaction parameter c on nanoparticle mass transfer profile C. Fig. 13 shows that nanoparticle mass transfer decreases with increasing fluid parameter b. Reverse behavior is depicted by C for larger value of M via Fig. 14. Fig. 15 elucidates fall in mass transfer of nanoparticle for larger values of Nt (as thermophoresis encourages the diffusion process). Opposite effect is displayed in Fig. 16 for Brownian motion parameter Nb. Since Nb is indicator of density of nanoparticles in the fluid so an increase in Nb leads to stronger nanoparticle mass transfer profile. Fig. 17 shows substantial increase in C as chemical reaction parameter c attains higher values. Fig. 10. Effect of Nb on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, M = 3, Gr = 0.7, Qr = 0.7, Nt = 3, E1 = 7/10, E2 = 2/10 and E3 = 1.
Heat transfer coefficient Impact of various pertinent parameters on heat transfer coefficient Z is discussed in the following subsection. The graphical results are oscillatory in nature due to contraction and expansion of peristaltic waves across the channel walls. Fig. 18 anticipates a
Fig. 11. Effect of R on T when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 1.5, c = 1, Sc ¼ 3; b = 0.0001, M = 3, Gr = 1.7, Qr = 1.7, Nt = 3, Nb = 3, E1 = 7/10, E2 = 1/10 and E3 = 1.
Fig. 13. Effect of bon C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3, M = 3, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
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Fig. 14. Effect of M on C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3; b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 17. Effect of con C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 15. Effect of Nt on C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 18. Effect of bon Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 16. Effect of Nb on C when x = 0.2, t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, c = 1, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
minimal decrease in heat transfer rate as fluid parameter b is enhanced. Decrease in heat transfer rate is captured for higher values of Nt (see Figs. 19). Increase in Nt leads to reduction in magnitude of heat transfer. Same effect is captured for larger thermopherosis parameter as shown via Fig. 20. This result corresponds to study presented by Hayat et al. [40]. As shown by Fig. 21
Fig. 19. Effect of Nt on Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
the magnitude of heat transfer coefficient Z near centerline decays for radiation parameter. Conclusions Channel flow of an incompressible electrically conducting MHD third grade nanofluid has been modeled with curvature, compliant
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Fig. 20. Effect of Nb on Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, Rd = 2, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
Fig. 21. Effect of Rd on Z when t = 0.1, = 0.2, Pr = 1.5, Br = 2.5, R = 1.5, Sc ¼ 3, M = 3, b = 0.0001, Gr = 0.7, Qr = 0.7, Nt = 2, Nb = 2, E1 = 7/10, E2 = 2/10 and E3 = 1.
properties, mixed convection, thermal radiation and chemical reaction. The conclusions for this study are: Velocity is decreased by Deborah number. Velocity increases for larger heat transfer Grashof number however reverse effect is noted for increasing mass transfer Grashof number. Both velocity and temperature increase for higher curvature parameter. Nanoparticle mass transfer profile shows opposite behavior for thermophoresis and Brownian motion. Heat transfer rate decreases with the increase in thermophoresis.
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