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(2) IAENG International Journal of Applied Mathematics, 49:2, IJAM_49_2_14 ______________________________________________________________________________________ in boundary condition. Rest of the paper is organized successfully. Section 2 is concerned with theproblem formulation. The convergence and stability of the scheme for the MHD flow and heat transfer is discussed in section 3. Section 4 gives a detailed account regarding impact of several physical parameters on fluid flow with graphical representation. Section 5 concludes the paper. II. CONSTRUCTION OF PROBLEM The stress tensor P for an incompressible homogeneous and thermodynamically compatible third grade fluid by Fosdick and Rajagopal (1980) is. ππ = ππ =. (2.1). πΌ1 , πΌ2 = ππππππ π π‘πππ π  ππππ’ππ. π. ,πΌ =. ππ ππ π. πΌ1 πππ. , πΈ=. Here lower plate is taken along x'-axis and y'-axis is normal to it. The upper plate isspecified by y'=1. Let both plates can extends to infinite in either sides of the x'-axis, where the upper plate is assumed to be fixed and lower plate moves witha time varying velocity F(t). Let V > 0 indicate the suction velocity at the plate. A transverse uniform magnetic field with strength π΅0 is applied at the lower plate.. Porous Plate. ππ€. With the above physical assumptions and the stress components given in eq.(2.1) the governing third grade flow equation becomes. πΌ1 π. ππ¦ ′3. −. =π. ππ¦ ′ π π΅02. π€. π. π2π€ ′ ππ¦ ′. 2. + πΌ1. π 3 π€′ 2. ππ¦ ′ ππ‘ ′. ππ€ ′ 2 π 2 π€′ ) ππ¦ ′ ππ¦ ′2. + 6π½3 (. +. ′. (2.2). With subjected initial and boundary conditions are π‘ ′ = 0: π€ ′ = 0,. ∀π¦′. π‘ ′ > 0: π€ ′ = πΉ(π‘) ππππ¦ = 0 ′. ππ¦ 2. π3π€. π3π€. (2.3). ′. π€ = 0 ππππ¦ = 1. πππΌ ππ¦ 3 − π2 π€. (2.5). πππ πΌ1 π. ππ′ ππ‘ ′. +π. ππ ′ ππ¦ ′. π2π€ ′. ππ€ ′. π π¦ ′2. ππ¦ ′. =π. π2π ′ 2 ππ¦ ′. + 2π½3. π¦′ ππ. +π ππ€ ′ 4 ππ¦ ′. ∀π¦. 2. ππ€ ′ ππ¦ ′. + πΌ1. + ππ΅02 π€ ′. π2π€ ′. ππ€ ′. ππ¦ ′ ππ‘ ′. ππ¦ ′. +. (2.7). With the initial and boundary condition which are subjected to the equation (2.7) are π‘′ = 0 βΆ π′ = 0. ∀π¦ ′. π‘ ′ > 0 βΆ π ′ = π2 ππππ¦ ′ = 0. (2.8). π′ = 0ππππ¦ ′ = 1. +πΌπππΈ. π2π€. ππ€. ππ¦ 2. ππ’. +. πΎπΈ. π‘′. π€′. π. π΄. ,π‘ = ,π€ =. , π=. π ′ −π1 π2 −π1. 3. ππ€ 4 ππ¦. 2. + πΌπΈ. +πΈπ2 π€. π2 π€ ππ¦ππ‘. ππ€ ππ¦ (2.9). With the initial and boundary conditions: π‘ = 0 βΆ π = 0,. πππππππ¦. π‘ > 0 βΆ π = 1,. ππππ¦ = 0. π = 0,. (2.10). ππππ¦ = 1. The above system of non-linear coupled differential equations is solved under the relevant initial and boundary conditions with the help of implicit finite difference scheme. Then the numerical solution of the system is obtained using above said method described in Conte, De Boor (1980).. Where πΉ π‘ = π΄sin(π€′π‘′) π¦=. π2π€. ππ€. + πΌ ππ¦ 2 ππ‘ + πΎ( ππ¦ )2 ππ¦ 2 +. ππ ππ 1 π2π ππ€ + ππ = +πΈ ππ‘ ππ¦ ππ ππ¦ 2 ππ¦. Porous Plate Fig. 0, Fluid flow between porous plates. ππ‘ ′ π 3 π€′. π2π€. ππ€. + ππ ππ¦ =. Non-dimensionalized field equation of energy with initial and boundary condition is secure as. B0. ππ€ ′. ,. ρ = densityoffluid, σ = electrical conductivity, μ = viscosity, cp = specificheat, k = thermalconductivity andθ′ = temperature .. Third grade Fluid. +π. π. π‘ > 0 βΆ π€ = sin ππ‘ , ππππ¦ = 0 (2.6)π€ = 0, ππππ¦ = 1 The thermal field equation for the thermodynamically compatible third grade fluid with viscous dissipation. π΄1 , π΄2 = πππππππ‘πππππ‘πππ ππ, defined by Rivlin and Erickson (1995). ππ€ ′. π π΅02 π. .. π‘ =0 βΆπ€ = 0,. πΌ = πππππ‘ππ‘π¦π‘πππ ππ. π. π΄2 ππ (π2 −π1 ). ,π2 =. Along with the initial and boundary condition as follows:. π = ππππ π π’ππ. Y=0. π π 2π. Then the non-dimensionalized field equation of velocity can be obtained as. π½3 = ππππππ πππππ π£ππ πππ ππ‘π¦. Y=1. 6π½ 3 π΄2. ,πΎ =. Where Re – local Reynolds number, ο‘ - visco-elastic parameter, ο§ - third grade elastic parameter, m- Hartmann number, Pr – Prandtl number, E –viscous dissipation parameter, π – kinematic coefficient of viscosity. A= constant.. ππ‘. π = −ππΌ + ππ΄1 + πΌ1 π΄2 + πΌ2 π΄12 + π½3 (π‘ππ΄12 )π΄1 π€ππππ π = π£ππ πππ ππ‘π¦. π π. , π = π′π(2.4). (Advance online publication: 27 May 2019).

(3) IAENG International Journal of Applied Mathematics, 49:2, IJAM_49_2_14 ______________________________________________________________________________________ π€π0 = 0, π = 0,1, … , π + 1. III. NUMERICAL SOLUTION METHOD. π. The equation (2.5), (2.6), (2.9) and (2.10) are solved by finite difference scheme of crank- Nicolson type. Taking a uniform mesh of step h and time step βt the grid points generated are π¦π , π‘π = ππ , πβπ‘ ,. π. π. π. π. =. ′′π π3π. π. π. π 10π€π. +. π π€π−3. =. π. −. π 12π€π+1. π 6π€π−2. −. +. +. π 6π€π+2. π 12π€π−1. −. −. π 10π€π. π +1. ππ¦. π2π€. +. π. ππ¦ 2. π. π +1. (3.1). 2π 2. π. π 3 π€ π3π + π3π ≈ ππ¦ 3 2π3 π +1. π. π3π€ π −π ≈ 2π 2 2π 2 ππ¦ ππ‘ π βπ‘ At the point (1, jβπ‘) and (N, πβπ‘ ) the third grade derivative π3π€ ππ¦ 3 ′π +1. π3π. is replaced respectively by ′′π +1. ′π. +π3π. 2π 3. and. π3π. ′′π. +π3π. πΎπΈ. 3×16 2 ×π 4 πΌπππΈ π π1π + 3 8π. π. π +1. (π1π. π. π +1 π1π. π. π. πΈπΌ π +1 π 8π 2 βπ‘ 1π π +1 π π2π + π2π +. + π1π )4 + +. ). π π1π. +. +. (3.4). π = 0,1,2, … , π + 1 (3.5). To impose the boundary conditions at infinity, we choose N so that the absolute difference of two solutions obtained by assuming the boundary conditions at∞ to hold at π + 1 π , πβπ‘ πππ ( π + 2 π , πβπ‘) successively, so that the difference between the two consecutive solutions obtained is less than a prescribed error tolerance∈ given in Jain (1984).. Then the system of non-linear equations for velocity (3.2) with its initial and boundary conditions (3.3) is solved by damped-Newton method. For implementation of this method the residuals (R i , i = 0,1,2, … , N) and the non-zero elements of the Jacobian matrix. .. 2π 3. π +1. + ππ−1 + ππ+1 − 2ππ + ππ−1. A tri-diagonal system of equations is obtained after rearrangement of energy equation. Then it is solved using special form of Gaussian elimination method with the help of initial velocities which are obtained again by solving the tri-diagonal system for velocity formed by making the parameter m, αandγbecome zero in constitutive equation of velocity given in eq.(3.2). Similarly initial choice of temperature is made converting the energy equation into a tri-diagonal system by making the parameter Eand α become zero in eq.(3.4) and then it is solved using the exponentially fitted scheme described in Morton(1996).. π. +π2π. π +1. 2. π. π. 4π π2π. π +1. − 2ππ. ) + −. π. ππ+1 = 0, π = 1,2, … , π.. π1π +π1π. ≈. 4h π +1 π1π. π +1. π. ππ€ π€π − π€π ≈ ππ‘ βπ‘ π +1. π +1 π π1π +π1π 2. π +1. ππ+1 − ππ−1 + (ππ+1 − ππ−1 ) =. π0 = 1 πππ. π 3π€π+1. π = 0,1, … , π + 1arewritten as. ≈. π π1π. 4π π +1 ππ+1. ππ0 = 0,. π π€π+3. The difference approximation to the derivatives at nodes ih , jβt , i = 1,2, … , N + 1 and. ππ€. πΈ(. ππ. With initial and boundary conditions on θ in discretised form. π. π3π = −π€π−2 + 2π€π−1 − 2π€π+1 + π€π+2 π −3π€π−1. π π 2π 2. +. πΈπ (. π. π2π = π€π+1 − 2π€π + π€π−1. ′π π3π. π. π +1 π 2 π€ π +π€ π. π1π = π€π+1 − π€π−1 π. = 0, π = 1, … , π. −ππ. βπ‘ 1. Notations:. (3.3). The governing equation of energy in discretized form is written as ππ. Following notations and difference approximations to the derivative are taken at different nodes asfij ’s. π. π π€π+1. π +1. π = 0,1,2, … , π + 1,. π = 0,1,2, … , π − 1. π. π€0 = sin ππβπ‘and. ∂R i ∂w j. , i = 1,2, … , Nandj = 1,2, … , M. For i = 2,3,...,N-1, j=0,1,2,...,M.. of the system are computed. The above method isquadratically convergent and a new approximation π₯ π +1 being accepted as x m + h i for some suitable i, 2 only when it satisfies π π₯ π + π π < 2 2 π π π₯ 2 ,otherwise the failure occur. This ensuresthere must be decrease in residual error after every iteration. Hence it establishes the convergence of the iteration scheme. The MATLAB coding for the above numerical scheme is verified with the existing result given in Conte, De Boor (1980) and the results are found to be correct up to 10−5 .. The initial and boundary conditions in discretized form as follows:. Hence the aforementioned iteration scheme is stable and convergent. Finally procure results are presented through. Here the difference scheme is seen to be unconditionally stable and has second order convergence in time and space. With the implementation of above said notations in (3.1), the governing equation of velocity is discretized as π +1. π€π. πππΌ. π. −π€ π. βπ‘. + ππ. π +1 π π3π +π3π 3 2π. π +1. π1π. +. π. +π1π. 4π πΎ 32π 4. =. π +1. (π1π. π +1. π2π. π. +π2π. 2π 2. π. +πΌ. π +1. π2π. π +1. + π1π )2 (π2π. π. −π2π. π 2 βπ‘. π. +. + π2π )−π2 (. π +1. π€π. π. +π€ π. 2. ). (3.2). (Advance online publication: 27 May 2019).

(4) IAENG International Journal of Applied Mathematics, 49:2, IJAM_49_2_14 ______________________________________________________________________________________ graphs and its physical concepts are described in following section. IV. RESULTS& DISCUSSION The current investigation shows the unsteady MHD flow and heat transfer characteristics of third grade fluid placed between two porous plates subjected to a uniform magnetic field. Results are plotted graphically in fig-1 to fig-24 for different physical parameters.. hike of velocity revealed near the upper plate and then velocity decreases towards the lower plate. Whereas a significant increase of temperature profile is observed throughout the flow field and the maximum effect is seen at the center of the plates. Fig-5, 6(πΌ < 1) illustrates a similar impact as to previous case for both the fields, which is obtained with the reduction of non-Newton parametric value γ to1 and magnetic parameter value m to 5 than the previous case. Fig-7, 8 (γ>1) Forlarge values of third grade elastic parameter γ, it is observed that velocity profile increases and temperature profile decreases with increase of the nonNewtonian parameter γ.. Fig. 1, Effect of ο‘ on Velocity field (ο§ = 3,Re=7, w=1, pr=.3, E=.7, m=7, n=100). Fig. 4,Effect of ο‘ on Temp.field (ο§ = 5,Re=10, w=1, pr=.3, E=.7, m=5, n=100). Fig. 2, Effect of ο‘ on Temp. field (ο§ = 3,Re=7, w=1, pr=.3, E=.7, m=7, n=100). Fig. 5, Effect of n=100). πΌon Velocity field (ο§ = 1,Re=7, w=1, pr=.3, E=.7, m=5,. Fig. 3, Effect of ο‘ on Velocity field (ο§ = 5,Re=10, w=1, pr=.3, E=.7, m=5, n=100). Fig-1, 2, 3, 4 (πΌ > 1) it is observed that with increasing the second grade elastic parameter πΌvalues, elasticity of the fluid increases that reduces fluid velocity profile. A sudden. (Advance online publication: 27 May 2019).

(5) IAENG International Journal of Applied Mathematics, 49:2, IJAM_49_2_14 ______________________________________________________________________________________ Fig. 6, Effect of n=100). πΌ on Temp. field (ο§ = 1,Re=7, w=1, pr=.3, E=.7, m=5,. .. Fig. 7, Effect of ο§ on Velocity field (πΌ = 3, Re=8, w=1, pr=.3, E=.7, m=7, n=100). Fig. 11, Effect of ο§ on Velocity field (ο‘ = 3, Re = 8, w = 1, pr =.3, E =.7, m= 7, n = 100). Fig. 12,Effect of ο§ on Temp. (ο‘ = 3, Re = 8, w = 1, pr =.3, E =.7, m= 7, n = 100 Fig. 8, Effect of ο§ on Temp (πΌ = 3 ,Re=8, w=1, pr=.3, E=.7, m=7, n=100). Fig. 13, Effect of Re on Vel. (ο‘ = 3, ο§ = 3, w = 1, pr =.3, E=.7,m=5, n = 100) Fig. 9, Effect of ο§ on Velocity (ο‘ = 4, Re=8, w = 1, pr =.3, E =.7, m= 7, n = 100). Fig. 10,Effect ofο§on Temp. (ο‘ = 4, Re = 8, w = 1, pr =.3, E =.7, m= 7, n = 100). Fig. 14, Effect of Re on Temp. (ο‘ = 3, ο§ = 3, w = 1, pr =.3, E =.7, m=5, n = 100). (Advance online publication: 27 May 2019).

(6) IAENG International Journal of Applied Mathematics, 49:2, IJAM_49_2_14 ______________________________________________________________________________________. Fig. 15, Effect of Re on velocity field (ο‘ = .01, ο§ = 1, w = 1, pr =.3, E =.7,m=0.5, n = 100). Fig. 17, Effect of m on velocity field (ο‘ = 3, ο§ = 3, Re = 7, w = 1, pr =.3,E=.7,n = 100). Fig-9, 10 But when second grade elastic parameter value of πΌ raised to 4 keeping rest parameters fixed, a decreasing behavior of velocity profile is observed with increasing the values ofγ. Also it slow down the velocity as well as temperature profile and conforms that influence of second grade elastic parameter ο‘ is more on velocity profile than third grade elastic parameter γ.. Fig-17, 18(π > 1) Show that velocity gradually decreases when magnetic parameter m values increases. Due to increase in magnetic field strength the transverse Lorentz force increases on flow field, which in turns reduces the fluid velocity. But a substantial increase is seen on temperature field. Fig-11, 12 (ο§<1) Again for small values of third grade elastic parameter ο§ the momentum boundary layer thickness gradually decreases with increase of ο§ values. This might occur, because when ο§ values are small, the influence of second grade elastic parameter is more on flow field and that causes gradual deceleration of velocity profile. The effect is not pronounced near the boundary, but a noticeable effect is seen away from boundary. The temperature field increases with increase of the parameter valuesο§.. Fig.18, Effect of m on Temp. (ο‘ = 3, ο§ = 3, Re = 7, w = 1, pr =.3,E=.7, n = 100). Fig. 16,Effect of Re on Temp. (ο‘ = .01, ο§ = 1, w = 1, pr =.3, E =.7,m=.5, n = 100). Fig-13, 14(Re >1) Depict the impact of Reynolds number Re on velocity and temperature field. It shows when values of Re increases, viscous force of the fluid gradually increases which causes decrease in both velocity as well as temperature profile at all points of the domain of fluid flow. Fig-15, 16(Re <1) Similar effect is seen on velocity and temperature profile when Reynolds number values are less than 1 obtained along with a reduction inthe values of elastic parameters ο‘, ο§and magnetic parameter m. Here the momentum and thermal boundary layer influence near the plates observed to be insignificant.. Fig. 19, Effect of m on vel. (ο‘ = 3, ο§ = 3, Re = 7, w = 1,pr=.3, E =.7, n = 100). Fig-19, 20 (m< 1) Portray the effect of smaller magnitude magnetic field effect on velocity and temperature profile. It is observed here that the effect of Lorentz force become negligible and that provides less amount of resistance to the fluid flow, so a profound increase of velocity profile is seen. But the temperature field observed to be decreasing.. (Advance online publication: 27 May 2019).

(7) IAENG International Journal of Applied Mathematics, 49:2, IJAM_49_2_14 ______________________________________________________________________________________ Fig-21, 22 recite the effect of viscous dissipation parameter E on temperature profile. It is seen that temperature profile increases with increase of every value of viscous dissipation parameter E.. ο·. So when E increases viscous dissipation heat generated is more which causes rise in the temperature field.. for valuesof ο§become 4 or more and strengthening second grade elastic parameter ο‘ to 4. Velocity decreases and temperature increases with increase of the magnetic parameter m values but on reduction of parametric values of m, a reversed effect is seen on both the fields.. Fig. 20,Effect of m on Temp. (ο‘ = 3, ο§ = 3, Re = 7, w = 1,pr=.3, E =.7, n = 100) Fig. 22, Effect of E on Temp. (ο‘ = 2, ο§ = 3, Re = 7, w = 1, pr =.3,m = 7, n = 100). Fig. 21,Effect of E on Temp. (ο‘ = 2, ο§ = 3, Re = 7, w = 1, pr =.3,m = 7, n = 100). Finally fig-23, 24 Depicts when ππ < 1 temperature profile decreases with increase of Prandtl number. Where the effect is noticeable near the boundary and gradually diminishes away from the boundary. For ππ > 1 it is observed that when the parameter values increases, the temperature field decreases near the surface of the plate and then gradually increases away from the plate and that causes a cross over temperature profile.. Fig. 23,Effect of Pr on Temp. (ο‘ = 3, ο§ = 3, Re = 9, w = 1, E =.7, m= 7, n = 100). V. CONCLUSION This study analyses unsteady magneto-hydrodynamic flow and heat transfer of third grade fluid in a porous channel. The results are pictorially presented through several graphs and a comparison study is made between small and large values of physical parameters on velocity and temperature field. The imperative finding of the present study are as follows ο· Velocity profile decreases and temperature profile increases with increase of low as well as high values of second grade elastic parameter ο‘. ο· Velocity profile increases and temperature decreases with increase of low as well as high value of third grade elastic parameter ο§ whenο‘ = 3.But the effect become opposite on both the fields. Fig. 24Effect of Pr on Temp. (ο‘ = 3, ο§ = 3, Re = 9, w = 1, E =.7, m= 7, n = 100). ο·. Moreover when viscous dissipation factor E increases, temperature profile increases whatever the Eckert number E is large or small.. (Advance online publication: 27 May 2019).

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Whereas in ii below, two modal values within the same information units will be compositional and the dominant value will prevail: ii GIL: // I think that it has to be these In ii there

For the triplet dataset, the model yields a main effect of verb type, and produces the lowest thematic fit for the metonymic condition corresponding to the longest reading times and eye

We included patients who received therapeutic hypothermia using overall surface or endovascular cooling devices and compared the neurological outcomes, efficacies and adverse events of

Although the amount of data obtained for each question is still relatively small the largest amount of answers given to a reasonably sized amount of questions was 6, the results on

information incorporate this information in ways that are not always efficient We used restricted cubic splines to assess how admission vital signs data predict mortality in patients

This requires representations of interpretation that explicitly link discourse entities with grounded symbols and track the heterogeneous prominence that these entities get in virtue of