In this article, the velocity profile of squeezing flow of an incompressible viscous fluid in a p... more In this article, the velocity profile of squeezing flow of an incompressible viscous fluid in a porous medium under the influence of uniform transverse magnetic field is argued. Both the continuity and momentum equations, with the help of vorticity and stream functions, are simultaneously transformed to an ordinary differential equation. Boundary conditions are also transformed with the help of transformation ψ(r, z) = rT (z). Homotopy Perturbation Method (HPM) is used to solve the boundary value problem obtained. Efficiency of the proposed scheme is examined with the help of residual. Effect of different parameters on the velocity profile is discussed through graphs. It is observed that both imposed magnetic field and electro conductivity are directly proportional to the velocity of fluid.
Advances in Materials Science and Engineering, 2017
This article presents a scheme for the analysis of an unsteady axisymmetric flow of incompressibl... more This article presents a scheme for the analysis of an unsteady axisymmetric flow of incompressible Newtonian material in the form of liquid squeezed between two circular plates. The scheme combines traditional perturbation technique with homotopy using an adaptation of the Laplace Transform. The proposed method is tested against other schemes such as the Regular Perturbation Method (RPM), Homotopy Perturbation Method (HPM), Optimal Homotopy Asymptotic Method (OHAM), and the fourth-order Explicit Runge-Kutta Method (ERK4). Comparison of the solutions along with absolute residual errors confirms that the proposed scheme surpasses HPM, OHAM, RPM, and ERK4 in terms of accuracy. The article also investigates the effect of Reynolds number on the velocity profile and pressure variation graphically.
An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing th... more An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing through porous medium channel is modeled and investigated. Similarity transformations are used to convert the partial differential equations (PDEs) of non-Newtonian fluid to a highly nonlinear fourth-order ordinary differential equation (ODE). The obtained boundary value problem is solved analytically by Homotopy Perturbation Method (HPM) and numerically by explicit Runge-Kutta method of order 4. For validity purpose, we compare the analytical and numerical results which show excellent agreement. Furthermore, comprehensive graphical analysis has been made to investigate the effects of various fluid parameters on the velocity profile. Analysis shows that positive and negative squeeze numberSqhave opposite effect on the velocity profile. It is also observed that Casson parameterβshows opposite effect on the velocity profile in case of positive and negative squeeze numberSq. MHD parameterMgan...
In this work, we present the approximate solutions of higher order nonlinear boundary value probl... more In this work, we present the approximate solutions of higher order nonlinear boundary value problems by an efficient numerical algorithm. The New Iterative Method (NIM) will by used to find such solutions. The solutions thus obtained by NIM, are in the form of rapidly convergent series. The approach developed is tested through examples, which gives the stability and efficiency of the proposed algorithm.
We investigate the unsteady flow of a viscous fluid near a vertical heated plate. The momentum an... more We investigate the unsteady flow of a viscous fluid near a vertical heated plate. The momentum and energy equations are considered as fractional differential equations with respect to the time t. Solutions of the initial-boundary values problem are determined by means of the Laplace transform technique and are represented by means of the Wright functions. The fundamental solution for the temperature field is obtained. This allows obtaining the temperature field for different conditions on the wall temperature. A numerical case is analyzed in order to obtain information regarding the influence of the fractional parameters on the temperature and velocity fields. Some physical aspects of the fluid behavior are presented by graphical illustrations.
The aim of this article is to model and analyze an unsteady axisymmetric flow of nonconducting, N... more The aim of this article is to model and analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.
Objective: The objective of the present investigation is to design formulate and characterized th... more Objective: The objective of the present investigation is to design formulate and characterized the bilayer tablet containing Diclofenac sodium and Aloe Vera gel powder. In which diclofenac sodium is sustained release and Aloe Vera gel powder is immediate release. In order to produce a single dosage form containing two different classes, drug are widely prescribed by the physician to have better patient compliance. Methods: Bilayer tablet was prepared by direct compression, The immediate release layer of Aloe Vera gel powder was prepared by using different excipients such as starch, sodium starch glycolate, lactose, talc etc. sustained release layer of diclofenac sodium was prepared by using HPMC K4M, lactose, Talc Magnesium stearate, talc etc. for preparation of bilayer tablet sodium starch glycolate are use as super disintegrants in immediate release tablet and HPMC K4M are use as controlled release polymer. Various Preformulation parameter i.e. Identification, melting point, compatibility study, solubility are checked. Micromeritics properties of powder blend such as bulk density, tapped density, hausner's ratio, Carr's index, angle of repose are performed. Post-compression parameter was done such as hardness, friability, weight variation, drug content uniformity, thickness, in vitro drug release. Results: Result was found within the limit of the standard of optimized formulation. The drug release of the tablet was in the range of 82 to 92%in 8 h. Conclusion: Bilayer tablet was prepared by optimized batches of both layers. The prepared tablets showed satisfactory results for various evaluation parameters. The optimized formulation based on all the parameter A1 (Sodium starch glycolate) is selected for the immediate release layer and D3 (HPMC K4M) was selected for the controlled release layer. The drug release mechanism was found to be zero order release depends upon diffusion.
A steady two-dimensional axisymmetric flow of an incompressible viscous fluid under the influence... more A steady two-dimensional axisymmetric flow of an incompressible viscous fluid under the influence of a uniform transverse magnetic field with slip boundary condition is studied. An ordinary nonlinear differential equation is formed by transforming the Navier-Stokes equations using the transformation (,) = 2 (). Differential transform and optimal homotopy analysis methods have been used to obtain the solutions by varying pertinent flow parameters. By using residuals in each case, the validity of solutions is established. Excellent results are obtained by using the proposed schemes. The influence of different parameters on the flow is shown through graphs.
We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHA... more We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHAM). The proposed method is capable to handle a wide variety of linear and nonlinear problems effectively. The numerical results given by OHAM are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM), variational iteration (VIM), homotopy perturbation (HPM), and variational iteration decomposition method (VIDM). The results show that the proposed method is more effective and reliable.
Research in any field is a daunting task requiring perseverance and constant struggle against all... more Research in any field is a daunting task requiring perseverance and constant struggle against all odds and setbacks. I am thankful to almighty Allah for giving me courage and competence to complete my dissertation within the stipulated timeframe. I attribute my inquisitiveness to the last Messenger of Allah (SAW) who happened to be the beacon of light for all humanity. His preaching and actions are a great source of inspiration for all of us to pursue knowledge for human development.
We investigate squeezing flow between two large parallel plates by transforming the basic governi... more We investigate squeezing flow between two large parallel plates by transforming the basic governing equations of the first grade fluid to an ordinary nonlinear differential equation using the stream functionsur(r,z,t)=(1/r)(∂ψ/∂z)anduz(r,z,t)=−(1/r)(∂ψ/∂r)and a transformationψ(r,z)=r2F(z). The velocity profiles are investigated through various analytical techniques like Adomian decomposition method, new iterative method, homotopy perturbation, optimal homotopy asymptotic method, and differential transform method.
International Journal of Technology Diffusion, 2010
This work presents an improvement on the simple algorithms of matrix inversion (Farooq & Hamid, 2... more This work presents an improvement on the simple algorithms of matrix inversion (Farooq & Hamid, 2010). This generalized algorithm supports selection of pivot randomly in the matrix thus supporting partial and full pivoting. The freedom in pivot selection can be used in minimizing the numerical error and prioritizing the variable to find the solution first. The algorithm is more suitable for finding inverse and determinant of dense matrices. The algorithm requires a mechanism for selection of pivot (e.g., selection of absolute maximum value) in the available sub-matrix and the mechanism to get the inverse from the final resultant matrix by rearranging the rows and columns. A method for assigning the sign of the determinant is also given. The algorithm is explained through solved examples. The number of arithmetic calculations performed by the algorithm is of O () however. The efficiency and simplicity of coding remains the same as of the original algorithm.
In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed betw... more In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is studied with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations is reduced to a single fourth order ordinary differential equation. The resulting boundary value problems are solved by optimal homotopy asymptotic method (OHAM) and fourth order explicit Runge-Kutta method (RK4). It is observed that the results obtained from OHAM are in good agreement with numerical results by means of residuals. Furthermore, the effects of various dimensionless parameters on the velocity profiles are investigated graphically.
A general investigation has been made and analytic solutions are provided corresponding to the fl... more A general investigation has been made and analytic solutions are provided corresponding to the flows of an Oldroyd-B fluid, under the consideration of slip condition at the boundary. The fluid motion is generated by the flat plate which has a translational motion in its plane with a time-dependent velocity. The adequate integral transform approach is employed to find analytic solutions for the velocity field. Solutions for the flows corresponding to Maxwell fluid, second-grade fluid, and Newtonian fluid are also determined in both cases, namely, flows with slip on the boundary and flows with no slip on the boundary, respectively. Some of our results were compared with other results from the literature. The effects of several emerging dimensionless and pertinent parameters on the fluid velocity have been studied theoretically as well as graphically in the paper.
In the present work, in the presence of magnetic field and with slip boundary condition, squeezin... more In the present work, in the presence of magnetic field and with slip boundary condition, squeezing flow of a Newtonian fluid in a porous medium between two large parallel plates is investigated. The governing equations are transformed to a single nonlinear boundary value problem. Daftardar Jafari Method (DJM) is used to solve the problem in order to obtain the velocity profile of the fluid. By using residual of the problem, the validity of solution is established. The velocity profile is argued through graphs for various values of parameters.
The flow between two large parallel plates approaching each other symmetrically in a porous mediu... more The flow between two large parallel plates approaching each other symmetrically in a porous medium is studied. The Navier-Stokes equations have been transformed into an ordinary nonlinear differential equation using a transformation ψ r, z r 2 F z. Solution to the problem is obtained by using differential transform method DTM by varying different Newtonian fluid parameters and permeability of the porous medium. Result for the stream function is presented. Validity of the solutions is confirmed by evaluating the residual in each case, and the proposed scheme gives excellent and reliable results. The influence of different parameters on the flow has been discussed and presented through graphs.
International Journal of Technology Diffusion, 2010
In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix ... more In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix in one instance. The algorithm is straightforward in understanding and manual calculations. Computer implementation of the algorithm is extremely simple and is quite efficient in time and memory utilization. The algorithm is supported by an example. The number of multiplication/division performed by the algorithm is exactly; however, its efficiency lies in the simplicity of coding and minimal utilization of memory. Simple applicability and reduced execution time of the method is validated form the numerical experiments performed on test problems. The algorithm is applicable in the cases of pseudo inverses for non-square matrices and solution of system of linear equations with minor modification.
In this article, the velocity profile of squeezing flow of an incompressible viscous fluid in a p... more In this article, the velocity profile of squeezing flow of an incompressible viscous fluid in a porous medium under the influence of uniform transverse magnetic field is argued. Both the continuity and momentum equations, with the help of vorticity and stream functions, are simultaneously transformed to an ordinary differential equation. Boundary conditions are also transformed with the help of transformation ψ(r, z) = rT (z). Homotopy Perturbation Method (HPM) is used to solve the boundary value problem obtained. Efficiency of the proposed scheme is examined with the help of residual. Effect of different parameters on the velocity profile is discussed through graphs. It is observed that both imposed magnetic field and electro conductivity are directly proportional to the velocity of fluid.
Advances in Materials Science and Engineering, 2017
This article presents a scheme for the analysis of an unsteady axisymmetric flow of incompressibl... more This article presents a scheme for the analysis of an unsteady axisymmetric flow of incompressible Newtonian material in the form of liquid squeezed between two circular plates. The scheme combines traditional perturbation technique with homotopy using an adaptation of the Laplace Transform. The proposed method is tested against other schemes such as the Regular Perturbation Method (RPM), Homotopy Perturbation Method (HPM), Optimal Homotopy Asymptotic Method (OHAM), and the fourth-order Explicit Runge-Kutta Method (ERK4). Comparison of the solutions along with absolute residual errors confirms that the proposed scheme surpasses HPM, OHAM, RPM, and ERK4 in terms of accuracy. The article also investigates the effect of Reynolds number on the velocity profile and pressure variation graphically.
An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing th... more An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing through porous medium channel is modeled and investigated. Similarity transformations are used to convert the partial differential equations (PDEs) of non-Newtonian fluid to a highly nonlinear fourth-order ordinary differential equation (ODE). The obtained boundary value problem is solved analytically by Homotopy Perturbation Method (HPM) and numerically by explicit Runge-Kutta method of order 4. For validity purpose, we compare the analytical and numerical results which show excellent agreement. Furthermore, comprehensive graphical analysis has been made to investigate the effects of various fluid parameters on the velocity profile. Analysis shows that positive and negative squeeze numberSqhave opposite effect on the velocity profile. It is also observed that Casson parameterβshows opposite effect on the velocity profile in case of positive and negative squeeze numberSq. MHD parameterMgan...
In this work, we present the approximate solutions of higher order nonlinear boundary value probl... more In this work, we present the approximate solutions of higher order nonlinear boundary value problems by an efficient numerical algorithm. The New Iterative Method (NIM) will by used to find such solutions. The solutions thus obtained by NIM, are in the form of rapidly convergent series. The approach developed is tested through examples, which gives the stability and efficiency of the proposed algorithm.
We investigate the unsteady flow of a viscous fluid near a vertical heated plate. The momentum an... more We investigate the unsteady flow of a viscous fluid near a vertical heated plate. The momentum and energy equations are considered as fractional differential equations with respect to the time t. Solutions of the initial-boundary values problem are determined by means of the Laplace transform technique and are represented by means of the Wright functions. The fundamental solution for the temperature field is obtained. This allows obtaining the temperature field for different conditions on the wall temperature. A numerical case is analyzed in order to obtain information regarding the influence of the fractional parameters on the temperature and velocity fields. Some physical aspects of the fluid behavior are presented by graphical illustrations.
The aim of this article is to model and analyze an unsteady axisymmetric flow of nonconducting, N... more The aim of this article is to model and analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.
Objective: The objective of the present investigation is to design formulate and characterized th... more Objective: The objective of the present investigation is to design formulate and characterized the bilayer tablet containing Diclofenac sodium and Aloe Vera gel powder. In which diclofenac sodium is sustained release and Aloe Vera gel powder is immediate release. In order to produce a single dosage form containing two different classes, drug are widely prescribed by the physician to have better patient compliance. Methods: Bilayer tablet was prepared by direct compression, The immediate release layer of Aloe Vera gel powder was prepared by using different excipients such as starch, sodium starch glycolate, lactose, talc etc. sustained release layer of diclofenac sodium was prepared by using HPMC K4M, lactose, Talc Magnesium stearate, talc etc. for preparation of bilayer tablet sodium starch glycolate are use as super disintegrants in immediate release tablet and HPMC K4M are use as controlled release polymer. Various Preformulation parameter i.e. Identification, melting point, compatibility study, solubility are checked. Micromeritics properties of powder blend such as bulk density, tapped density, hausner's ratio, Carr's index, angle of repose are performed. Post-compression parameter was done such as hardness, friability, weight variation, drug content uniformity, thickness, in vitro drug release. Results: Result was found within the limit of the standard of optimized formulation. The drug release of the tablet was in the range of 82 to 92%in 8 h. Conclusion: Bilayer tablet was prepared by optimized batches of both layers. The prepared tablets showed satisfactory results for various evaluation parameters. The optimized formulation based on all the parameter A1 (Sodium starch glycolate) is selected for the immediate release layer and D3 (HPMC K4M) was selected for the controlled release layer. The drug release mechanism was found to be zero order release depends upon diffusion.
A steady two-dimensional axisymmetric flow of an incompressible viscous fluid under the influence... more A steady two-dimensional axisymmetric flow of an incompressible viscous fluid under the influence of a uniform transverse magnetic field with slip boundary condition is studied. An ordinary nonlinear differential equation is formed by transforming the Navier-Stokes equations using the transformation (,) = 2 (). Differential transform and optimal homotopy analysis methods have been used to obtain the solutions by varying pertinent flow parameters. By using residuals in each case, the validity of solutions is established. Excellent results are obtained by using the proposed schemes. The influence of different parameters on the flow is shown through graphs.
We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHA... more We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHAM). The proposed method is capable to handle a wide variety of linear and nonlinear problems effectively. The numerical results given by OHAM are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM), variational iteration (VIM), homotopy perturbation (HPM), and variational iteration decomposition method (VIDM). The results show that the proposed method is more effective and reliable.
Research in any field is a daunting task requiring perseverance and constant struggle against all... more Research in any field is a daunting task requiring perseverance and constant struggle against all odds and setbacks. I am thankful to almighty Allah for giving me courage and competence to complete my dissertation within the stipulated timeframe. I attribute my inquisitiveness to the last Messenger of Allah (SAW) who happened to be the beacon of light for all humanity. His preaching and actions are a great source of inspiration for all of us to pursue knowledge for human development.
We investigate squeezing flow between two large parallel plates by transforming the basic governi... more We investigate squeezing flow between two large parallel plates by transforming the basic governing equations of the first grade fluid to an ordinary nonlinear differential equation using the stream functionsur(r,z,t)=(1/r)(∂ψ/∂z)anduz(r,z,t)=−(1/r)(∂ψ/∂r)and a transformationψ(r,z)=r2F(z). The velocity profiles are investigated through various analytical techniques like Adomian decomposition method, new iterative method, homotopy perturbation, optimal homotopy asymptotic method, and differential transform method.
International Journal of Technology Diffusion, 2010
This work presents an improvement on the simple algorithms of matrix inversion (Farooq & Hamid, 2... more This work presents an improvement on the simple algorithms of matrix inversion (Farooq & Hamid, 2010). This generalized algorithm supports selection of pivot randomly in the matrix thus supporting partial and full pivoting. The freedom in pivot selection can be used in minimizing the numerical error and prioritizing the variable to find the solution first. The algorithm is more suitable for finding inverse and determinant of dense matrices. The algorithm requires a mechanism for selection of pivot (e.g., selection of absolute maximum value) in the available sub-matrix and the mechanism to get the inverse from the final resultant matrix by rearranging the rows and columns. A method for assigning the sign of the determinant is also given. The algorithm is explained through solved examples. The number of arithmetic calculations performed by the algorithm is of O () however. The efficiency and simplicity of coding remains the same as of the original algorithm.
In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed betw... more In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is studied with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations is reduced to a single fourth order ordinary differential equation. The resulting boundary value problems are solved by optimal homotopy asymptotic method (OHAM) and fourth order explicit Runge-Kutta method (RK4). It is observed that the results obtained from OHAM are in good agreement with numerical results by means of residuals. Furthermore, the effects of various dimensionless parameters on the velocity profiles are investigated graphically.
A general investigation has been made and analytic solutions are provided corresponding to the fl... more A general investigation has been made and analytic solutions are provided corresponding to the flows of an Oldroyd-B fluid, under the consideration of slip condition at the boundary. The fluid motion is generated by the flat plate which has a translational motion in its plane with a time-dependent velocity. The adequate integral transform approach is employed to find analytic solutions for the velocity field. Solutions for the flows corresponding to Maxwell fluid, second-grade fluid, and Newtonian fluid are also determined in both cases, namely, flows with slip on the boundary and flows with no slip on the boundary, respectively. Some of our results were compared with other results from the literature. The effects of several emerging dimensionless and pertinent parameters on the fluid velocity have been studied theoretically as well as graphically in the paper.
In the present work, in the presence of magnetic field and with slip boundary condition, squeezin... more In the present work, in the presence of magnetic field and with slip boundary condition, squeezing flow of a Newtonian fluid in a porous medium between two large parallel plates is investigated. The governing equations are transformed to a single nonlinear boundary value problem. Daftardar Jafari Method (DJM) is used to solve the problem in order to obtain the velocity profile of the fluid. By using residual of the problem, the validity of solution is established. The velocity profile is argued through graphs for various values of parameters.
The flow between two large parallel plates approaching each other symmetrically in a porous mediu... more The flow between two large parallel plates approaching each other symmetrically in a porous medium is studied. The Navier-Stokes equations have been transformed into an ordinary nonlinear differential equation using a transformation ψ r, z r 2 F z. Solution to the problem is obtained by using differential transform method DTM by varying different Newtonian fluid parameters and permeability of the porous medium. Result for the stream function is presented. Validity of the solutions is confirmed by evaluating the residual in each case, and the proposed scheme gives excellent and reliable results. The influence of different parameters on the flow has been discussed and presented through graphs.
International Journal of Technology Diffusion, 2010
In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix ... more In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix in one instance. The algorithm is straightforward in understanding and manual calculations. Computer implementation of the algorithm is extremely simple and is quite efficient in time and memory utilization. The algorithm is supported by an example. The number of multiplication/division performed by the algorithm is exactly; however, its efficiency lies in the simplicity of coding and minimal utilization of memory. Simple applicability and reduced execution time of the method is validated form the numerical experiments performed on test problems. The algorithm is applicable in the cases of pseudo inverses for non-square matrices and solution of system of linear equations with minor modification.
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