Analytical Solutions for a General Mixed Boundary Value Problem Associated with Motions of Fluids with Linear Dependence of Viscosity on the Pressure

AbstractAn unsteady flow of incompressible Newtonian fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates is analytically studied. The fluid motion is induced by the upper plate that applies an arbitrary time-dependent shear stress to the fluid. General expressions for the dimensionless velocity and shear stress fields are established using a suitable change of independent variable and the finite Hankel transform. These expressions, that satisfy all imposed initial and boundary conditions, can generate exact solutions for any motion of this type of the respective fluids. For illustration, three special cases with technical relevance are considered and some important observations and graphical representations are provided. An interesting relationship is found between the solutions corresponding to motions induced by constant or ramptype shear stresses on the boundary. Furthermore, for validation of the results, the steady-state solutions corresponding to oscillatory motions are presented in different forms whose equivalence is graphically proved.

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General solutions for the mixed boundary value problem associated to hydromagnetic flows of a viscous fluid between symmetrically heated parallel plates

Thermal Science ◽

10.2298/tsci190608384j ◽

2020 ◽

Vol 24 (2 Part B) ◽

pp. 1389-1405 ◽

Cited By ~ 2

Author(s):

Maria Javaid ◽

Muhammad Imran ◽

Constantin Fetecau ◽

Dumitru Vieru

Keyword(s):

Shear Stress ◽

Steady State ◽

Viscous Fluid ◽

Mixed Boundary ◽

Fluid Velocity ◽

Fluid Motion ◽

Shear Stresses ◽

Parallel Plates ◽

General Solutions ◽

Special Cases

Exact general solutions for hydromagnetic flows of an incompressible viscous fluid between two horizontal infinite parallel plates are established when the upper plate is fixed and the inferior one applies a time-dependent shear stress to the fluid. Porous effects are taken into consideration and the problem in discussion is completely solved for moderate values of the Hartman number. It is found that the fluid velocity and the non-trivial shear stress satisfy PDE of the same form and the motion characteristics do not depend of magnetic and porous parameters independently but only by a combination of them that is called the effective permeability. For illustration, as well as to bring to light some physical insight of results that have been obtained, three special cases are considered and the influence of Reynolds number as well as combined porous and magnetic effects on the fluid motion are graphically underlined and discussed for motions due to constant or ramped-type shear stresses on the boundary. The starting solutions corresponding to motions induced by the lower plate that applies constant or oscillatory shear stresses to the fluid are presented as sum of steady-state and transient solutions and the required time to reach the steady-state is graphically determined. This time is greater for motions due to sine as compared to cosine oscillating shear stresses on the boundary. The steady-state is rather obtained in the presence of a magnetic field or porous medium.

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Mixed initial-boundary value problems describing motions of Maxwell fluids with linear dependence of viscosity on the pressure

Zeitschrift für Naturforschung A ◽

10.1515/zna-2021-0212 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Constantin Fetecau ◽

Dumitru Vieru ◽

Abdul Rauf ◽

Tahir Mushtaq Qureshi

Keyword(s):

Shear Stress ◽

Steady State ◽

Boundary Value Problems ◽

Linear Dependence ◽

Boundary Value ◽

Fluid Motion ◽

Initial Boundary ◽

Shear Stresses ◽

Initial Boundary Value Problems ◽

Initial Boundary Value

Abstract Some mixed initial-boundary value problems are analytically studied. They correspond to unsteady motions of the incompressible upper-convected Maxwell (IUCM) fluids with linear dependence of viscosity on the pressure between infinite horizontal parallel plates. The fluid motion is generated by the upper plate that applies time-dependent shear stresses to the fluid. Exact solutions are established for the dimensionless velocity and nontrivial shear stress fields using a suitable change of the spatial variable and the Laplace transform technique. They are presented as sum of the steady-state and transient components and are used to determine the required time to reach the permanent state. Comparisons between exact and numerical solutions indicate an excellent agreement. Analytical solutions for the unsteady motion of the same fluids induced by an exponential shear stress on the boundary are obtained as limiting cases of the general solutions. Moreover, the steady-state solutions corresponding to the ordinary IUCM fluids performing the initial motions are provided by means of asymptotic approximations of standard Bessel functions. Finally, spatial variation of starting solutions and the influence of physical parameters on the fluid motion are graphically underlined and discussed.

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Permanent solutions for some oscillatory motions of fluids with power-law dependence of viscosity on the pressure and shear stress on the boundary

Zeitschrift für Naturforschung A ◽

10.1515/zna-2020-0135 ◽

2020 ◽

Vol 75 (8) ◽

pp. 757-769

Author(s):

Constantin Fetecau ◽

Abdul Rauf ◽

Tahir Mushtaq Qureshi ◽

Masood Khan

Keyword(s):

Shear Stress ◽

Power Law ◽

Fluid Motion ◽

Viscosity Coefficient ◽

Shear Stresses ◽

Parallel Plates ◽

Constant Shear Stress ◽

Starting Solutions ◽

Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure

Mathematics ◽

10.3390/math9040334 ◽

2021 ◽

Vol 9 (4) ◽

pp. 334

Author(s):

Constantin Fetecau ◽

Dumitru Vieru ◽

Tehseen Abbas ◽

Rahmat Ellahi

Keyword(s):

Fluid Motion ◽

Maxwell Fluid ◽

Newtonian Fluids ◽

Exponential Dependence ◽

Physical Parameters ◽

Shear Stresses ◽

Parallel Plates ◽

Pressure Dependent Viscosity ◽

Laplace Transform Technique ◽

Dependent Viscosity

Some unsteady motions of incompressible upper-convected Maxwell (UCM) fluids with exponential dependence of viscosity on the pressure are analytically studied. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies time-dependent shear stresses to the fluid. Exact expressions, in terms of standard Bessel functions, are established both for the dimensionless velocity fields and the corresponding non-trivial shear stresses using the Laplace transform technique and suitable changes of the unknown function and the spatial variable in the transform domain. They represent the first exact solutions for unsteady motions of non-Newtonian fluids with pressure-dependent viscosity. The similar solutions corresponding to the flow of the same fluids due to an exponential shear stress on the boundary as well as the solutions of ordinary UCM fluids performing the same motions are obtained as limiting cases of present results. Furthermore, known solutions for unsteady motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory or constant shear stresses on the boundary are also obtained as limiting cases. Finally, the influence of physical parameters on the fluid motion is graphically illustrated and discussed. It is found that fluids with pressure-dependent viscosity flow are slower when compared to ordinary fluids.

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Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2014.m588 ◽

2016 ◽

Vol 8 (5) ◽

pp. 784-794 ◽

Cited By ~ 2

Author(s):

Vatsala Mathur ◽

Kavita Khandelwal

Keyword(s):

Shear Stress ◽

Velocity Field ◽

Newtonian Fluid ◽

Outer Cylinder ◽

Fluid Motion ◽

Maxwell Fluid ◽

Circular Cylinders ◽

Generalized Solutions ◽

Special Cases ◽

Graphical Illustration

AbstractThis paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress are also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.

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Analytical Solutions for Two Mixed Initial-Boundary Value Problems Corresponding to Unsteady Motions of Maxwell Fluids through a Porous Plate Channel

Mathematical Problems in Engineering ◽

10.1155/2021/5539007 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Constantin Fetecau ◽

Dumitru Vieru ◽

Ahmed Zeeshan

Keyword(s):

Steady State ◽

Porous Plate ◽

Fluid Motion ◽

Initial Boundary ◽

Newtonian Fluids ◽

Physical Parameters ◽

Shear Stresses ◽

Parallel Plates ◽

Fourier Cosine ◽

Maxwell Fluids

Two unsteady motions of incompressible Maxwell fluids between infinite horizontal parallel plates embedded in a porous medium are analytically studied to get exact solutions using the finite Fourier cosine transform. The motion is induced by the lower plate that applies time-dependent shear stresses to the fluid. The solutions that have been obtained satisfy all imposed initial and boundary conditions. They can be easily reduced as limiting cases to known solutions for the incompressible Newtonian fluids. For a check of their correctness, the steady-state solutions are presented in different forms whose equivalence is graphically proved. The effects of physical parameters on the fluid motion are graphically emphasized and discussed. Required time to reach the steady-state is also determined. It is found that the steady-state is rather obtained for Newtonian fluids as compared with Maxwell fluids. Furthermore, the effect of the side walls on the fluid motion is more effective in the case of Newtonian fluids.

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Exact Solutions of Heat and Mass Transfer with MHD Flow in a Porous Medium under Time Dependent Shear Stress and Temperature

Abstract and Applied Analysis ◽

10.1155/2015/975201 ◽

2015 ◽

Vol 2015 ◽

pp. 1-16 ◽

Cited By ~ 1

Author(s):

Arshad Khan ◽

Ilyas Khan ◽

Farhad Ali ◽

Asma Khalid ◽

Sharidan Shafie

Keyword(s):

Mass Transfer ◽

Porous Medium ◽

Shear Stress ◽

Heat And Mass Transfer ◽

Fluid Motion ◽

Mhd Flow ◽

Convection Flow ◽

Closed Form Solutions ◽

Special Cases ◽

Laplace Transform Technique

This paper aims to study the influence of thermal radiation on unsteady magnetohyrdodynamic (MHD) natural convection flow of an optically thick fluid over a vertical plate embedded in a porous medium with arbitrary shear stress. Combined phenomenon of heat and mass transfer is considered. Closed-form solutions in general form are obtained by using the Laplace transform technique. They are expressed in terms of exponential and complementary error functions. Velocity is expressed as a sum of thermal and mechanical parts. Corresponding limiting solutions are also reduced from the general solutions. It is found that the obtained solutions satisfy all imposed initial and boundary conditions and reduce to some known solutions from the literature as special cases. Analytical results for the pertinent flow parameters are drawn graphically and discussed in detail. It is found that the velocity profiles of fluid decrease with increasing shear stress. The magnetic parameter develops shear resistance which reduces the fluid motion whereas the inverse permeability parameter increases the fluid flow.

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An Apparatus to Study the Response of Cultured Endothelium to Shear Stress

Journal of Biomechanical Engineering ◽

10.1115/1.3138624 ◽

1986 ◽

Vol 108 (4) ◽

pp. 332-337 ◽

Cited By ~ 46

Author(s):

R. F. Viggers ◽

A. R. Wechezak ◽

L. R. Sauvage

Keyword(s):

Finite Element ◽

Endothelial Cells ◽

Shear Stress ◽

Laminar Flow ◽

Fluid Shear Stress ◽

Shear Stresses ◽

Parallel Plates ◽

Dimensional Flow ◽

Wide Range ◽

Wall Shear

An apparatus which has been developed to study the response of cultured endothelial cells to a wide range of shear stress levels is described. Controlled laminar flow through a rectangular tube was used to generate fluid shear stress over a cell-lined coverslip comprising part of one wall of the tube. A finite element method was used to calculate shear stresses corresponding to cell position on the coverslip. Validity of the finite element analysis was demonstrated first by its ability to generate correctly velocity profiles and wall shear stresses for laminar flow in the entrance region between infinitely wide parallel plates (two-dimensional flow). The computer analysis also correctly predicted values for pressure difference between two points in the test region of the apparatus for the range of flow rates used in these experiments. These predictions thus supported the use of such an analysis for three-dimensional flow. This apparatus has been used in a series of experiments to confirm its utility for testing applications. In these studies, endothelial cells were exposed to shear stresses of 60 and 128 dynes/cm2. After 12 hr at 60 dynes/cm2, cells became aligned with their longitudinal axes parallel to the direction of flow. In contrast, cells exposed to 128 dynes/cm2 required 36 hr to achieve a similar reorientation. Interestingly, after 6 hr at 128 dynes/cm2, specimens passed through an intermediate phase in which cells were aligned perpendicular to flow direction. Because of its ease and use and the provided documentation of wall shear stress, this flow chamber should prove to be a valuable tool in endothelial research related to atherosclerosis.

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Radiative and Porous Effects on Free Convection Flow near a Vertical Plate that Applies Shear Stress to the Fluid

Zeitschrift für Naturforschung A ◽

10.5560/zna.2012-0083 ◽

2013 ◽

Vol 68 (1-2) ◽

pp. 130-138 ◽

Cited By ~ 13

Author(s):

Corina Fetecau ◽

Mehwish Rana ◽

Constantin Fetecau

Keyword(s):

Shear Stress ◽

Free Convection ◽

Viscous Fluid ◽

Thermal Radiation ◽

Vertical Plate ◽

Fluid Motion ◽

Convection Flow ◽

Free Convection Flow ◽

Special Cases ◽

Infinite Vertical Plate

General solutions for the unsteady free convection flow of an incompressible viscous fluid due to an infinite vertical plate that applies a shear stress f (t) to the fluid are established when thermal radiation and porous effects are taken into consideration. They satisfy all imposed initial and boundary conditions and can generate a large class of exact solutions corresponding to different motions with technical relevance. The velocity is presented as a sum of thermal and mechanical components. Finally, some special cases are brought to light, and effects of pertinent parameters on the fluid motion are graphically underlined.

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Axisymmetric motion of a spherical porous particle perpendicular to two parallel plates with slip surfaces

Canadian Journal of Physics ◽

10.1139/cjp-2014-0549 ◽

2015 ◽

Vol 93 (7) ◽

pp. 784-795 ◽

Cited By ~ 3

Author(s):

E.I. Saad

Keyword(s):

Fluid Motion ◽

Brinkman Equation ◽

Parallel Plates ◽

Porous Sphere ◽

Field Equations ◽

Clear Fluid ◽

Stress Jump ◽

Spherical Coordinate ◽

Good Convergence ◽

Special Cases

A combined analytical–numerical approach to the problem of the low Reynolds number motion of a porous sphere normal to one of two infinite parallel plates at an arbitrary position between them in a viscous fluid is investigated. The clear fluid motion governed by the Stokes equation and the Darcy–Brinkman equation is used to model the flow inside the porous material. The motion in each of the homogeneous regions is coupled with the continuity of the velocity components, the continuity of the normal stress, and the tangential stress jump condition. The fluid is allowed to slip at the surface of the walls. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The collocation solutions for the hydrodynamic interactions between the porous sphere and the walls are calculated with good convergence for various values of the slip coefficient of the walls, the separation between the porous sphere and the walls, the stress jump coefficient, and a coefficient that is proportional to the permeability. For the special cases of a solid sphere, our drag results show excellent agreement with the available solutions in the literature for all relative particle-to-wall spacing.

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