General Solutions for the Unsteady Flow of Second-Grade Fluids over an Infinite Plate that Applies Arbitrary Shear to the Fluid

2011 ◽  
Vol 66 (12) ◽  
pp. 753-759 ◽  
Author(s):  
Constantin Fetecau ◽  
Corina Fetecau ◽  
Mehwish Rana
Keyword(s):  
Shear Stress ◽  
Unsteady Flow ◽  
Infinite Plate ◽  
Unsteady Motion ◽  
Newtonian Fluids ◽  
Second Grade ◽  
Moving Plate ◽  
General Solutions ◽  
Similar Solutions ◽  
Second Grade Fluids

General solutions corresponding to the unsteady motion of second-grade fluids induced by an infinite plate that applies a shear stress ƒ (t) to the fluid are established. These solutions can immediately be reduced to the similar solutions for Newtonian fluids. They can be used to obtain known solutions from the literature or any other solution of this type by specifying the function ƒ (.). Furthermore, in view of a simple remark, general solutions for the flow due to a moving plate can be developed.

Effect of side walls on the motion of a viscous fluid induced by an infinite plate that applies an oscillating shear stress to the fluid

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Constantin Fetecau ◽  
Dumitru Vieru ◽  
Corina Fetecau
Keyword(s):  
Shear Stress ◽  
Steady State ◽  
Viscous Fluid ◽  
Fourier Transforms ◽  
Fluid Motion ◽  
Infinite Plate ◽  
Unsteady Motion ◽  
Side Walls ◽  
Oscillating Shear Stress ◽  
Similar Solutions

AbstractThe velocity field corresponding to the unsteady motion of a viscous fluid between two side walls perpendicular to a plate is determined by means of the Fourier transforms. The motion of the fluid is produced by the plate which after the time t = 0, applies an oscillating shear stress to the fluid. The solutions that have been obtained, presented as a sum of the steady-state and transient solutions satisfy the governing equation and all imposed initial and boundary conditions. In the absence of the side walls they are reduced to the similar solutions corresponding to the motion over an infinite plate. Finally, the influence of the side walls on the fluid motion, the required time to reach the steady-state, as well as the distance between the walls for which the velocity of the fluid in the middle of the channel is unaffected by their presence, are established by means of graphical illustrations.

scholarly journals First Problem of Stokes for Generalized Burgers' Fluids

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Muhammad Jamil
Keyword(s):  
Shear Stress ◽  
Steady State ◽  
Velocity Field ◽  
Integral Transforms ◽  
Newtonian Fluids ◽  
Rheological Parameters ◽  
Second Grade ◽  
Material Parameters ◽  
Transient Solutions ◽  
Similar Solutions

The velocity field and the adequate shear stress corresponding to the first problem of Stokes for generalized Burgers’ fluids are determined in simple forms by means of integral transforms. The solutions that have been obtained, presented as a sum of steady and transient solutions, satisfy all imposed initial and boundary conditions. They can be easily reduced to the similar solutions for Burgers, Oldroyd-B, Maxwell, and second-grade and Newtonian fluids. Furthermore, as a check of our calculi, for small values of the corresponding material parameters, their diagrams are almost identical to those corresponding to the known solutions for Newtonian and Oldroyd-B fluids. Finally, the influence of the rheological parameters on the fluid motions, as well as a comparison between models, is graphically illustrated. The non-Newtonian effects disappear in time, and the required time to reach steady-state is the lowest for Newtonian fluids.

Unsteady flow of a generalized Oldroyd-B fluid due to an infinite plate subject to a time-dependent shear stress

2010 ◽  
Vol 88 (9) ◽  
pp. 675-687 ◽  
Author(s):  
D. Vieru ◽  
Corina Fetecau ◽  
C. Fetecau
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Boundary Conditions ◽  
Unsteady Flow ◽  
Infinite Plate ◽  
Time Dependent ◽  
Fourier Cosine ◽  
Series Form ◽  
Maxwell Fluids ◽  
Similar Solutions

The unsteady flow of an incompressible generalized Oldroyd-B fluid induced by an infinite plate subject to a time-dependent shear-stress is studied by means of the Fourier cosine and Laplace transforms. The solutions that have been obtained, written under integral and series form in terms of the generalized Ga,b,c(·,t) functions, are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. They satisfy all imposed initial and boundary conditions, and for λ and λr → 0 reduce to the Newtonian solutions. Furthermore, the similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are also obtained as limiting cases of general solutions. Finally, to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field v(y, t) have been depicted against y for different values of t and of the material and fractional parameters.

scholarly journals Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear

2011 ◽  
Vol 16 (2) ◽  
pp. 135-151 ◽  
Author(s):  
Muhammad Athar ◽  
Corina Fetecau ◽  
Muhammad Kamran ◽  
Ahmad Sohail ◽  
Muhammad Imran
Keyword(s):  
Shear Stress ◽  
Positive Real Number ◽  
Fractional Integration ◽  
Maxwell Fluid ◽  
Unsteady Motion ◽  
Positive Real ◽  
Special Cases ◽  
Multiple Integrals ◽  
Similar Solutions ◽  
R Functions

The velocity field and the adequate shear stress corresponding to the flow of a fractional Maxwell fluid (FMF) between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is produced by the inner cylinder that at time t = 0+ applies a shear stress fta (a ≥ 0) to the fluid. The solutions that have been obtained, presented under series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as special cases of general solutions. The unsteady solutions corresponding to a = 1, 2, 3, ... can be written as simple or multiple integrals of similar solutions for a = 0 and we extend this for any positive real number a expressing in fractional integration. Furthermore, for a = 0, 1 and 2, the solutions corresponding to Maxwell fluid compared graphically with the solutions obtained in [1–3], earlier by a different technique. For a = 0 and 1 the unsteady motion of a Maxwell fluid, as well as that of a Newtonian fluid ultimately becomes steady and the required time to reach the steady-state is graphically established. Finally a comparison between the motions of FMF and Maxwell fluid is underlined by graphical illustrations.

On the Unsteady Flow of Two Incompressible Immiscible Second Grade Fluids between Two Parallel Plates

2014 ◽  
Vol 1016 ◽  
pp. 546-553
Author(s):  
Abdul M. Siddiqui ◽  
Maya K. Mitkova ◽  
Ali R. Ansari
Keyword(s):  
Unsteady Flow ◽  
Pressure Gradient ◽  
Analytical Solutions ◽  
Interface Velocity ◽  
Time Dependent ◽  
Second Grade ◽  
Parallel Plates ◽  
Governing Equations ◽  
Layer Flow ◽  
Second Grade Fluids

Unsteady, pressure driven in the gap between two parallel plates flow of two non-Newtonian incompressible second grade fluids is considered. The governing equations are established for the particular two-layer flow and analytical solutions of the equations that satisfy the imposed boundary conditions are obtained. The velocity of each fluid is expressed as function of the material constants, time dependent pressure gradient and other characteristics of the fluids. As part of the solution, an expression for the interface velocity is derived. We analyze the shift of the velocity maximum from one to another fluid as a function of variety of values of fluids’ parameters.

scholarly journals A note on “Taylor–Couette flow of a generalized second grade fluid due to a constant couple”

2010 ◽  
Vol 15 (2) ◽  
pp. 155-158 ◽  
Author(s):  
C. Fetecau ◽  
A. U. Awan ◽  
M. Athar
Keyword(s):  
Exact Solution ◽  
Steady State ◽  
Unsteady Flow ◽  
Couette Flow ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid ◽  
New Form ◽  
Second Grade Fluids

In this brief note, we show that the unsteady flow of a generalized second grade fluid due to a constant couple, as well as the similar flow of Newtonian and ordinary second grade fluids, ultimately becomes steady. For this, a new form of the exact solution for velocity is established. This solution is presented as a sum of the steady and transient components. The required time to reach the steady-state is obtained by graphical illustrations.

Some Unsteady Flows of a Jeffrey Fluid between Two Side Walls over a Plane Wall

2011 ◽  
Vol 66 (12) ◽  
pp. 745-752 ◽  
Author(s):  
Masood Khan ◽  
Faiza Iftikhar ◽  
Asia Anjum
Keyword(s):  
Infinite Plate ◽  
Unsteady Flows ◽  
Integral Transforms ◽  
Plane Wall ◽  
Second Grade ◽  
Jeffrey Fluid ◽  
Transient Solutions ◽  
Side Walls ◽  
Similar Solutions ◽  
Oscillating Plate

In this paper, some time-dependent flows of a non-Newtonian fluid between two side walls over a plane wall are investigated. The following three problems have been studied: (i) flow due to an oscillating plate, (ii) flow due to an accelerating plate, and (iii) flow due to applied constant stress. The explicit expressions for the velocity field are determined by using the integral transforms. The solutions that have been obtained, depending on the initial and boundary conditions, are written as sum of the steady state and transient solutions. The similar solutions for second-grade and Newtonian fluids can be deduced as limiting cases of our solutions. Furthermore, in absence of the side walls they reduce to the similar solutions over an infinite plate. The effects of some important parameters due to side walls on the flow field are investigated.

scholarly journals On analytical study of factional Oldroyd-B flow in annular region of two torsionally oscillating cylinders

2012 ◽  
Vol 16 (2) ◽  
pp. 411-421 ◽  
Author(s):  
A. Mahmood
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Analytical Study ◽  
Analytic Solutions ◽  
Circular Cylinders ◽  
Second Grade ◽  
General Solutions ◽  
Exact Analytic Solutions ◽  
Governing Differential Equation ◽  
R Functions

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a fractional Oldroyd-B fluid, also called generalized Oldroyd-B fluid (GOF), between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate around their common axis with different angular frequencies of their velocities. The exact analytic solutions of the velocity field and associated shear stress, that have been obtained, are presented under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of classical Oldroyd-B, generalized Maxwell, classical Maxwell, generalized second grade, classical second grade and Newtonian fluids are also obtained as limiting cases of our general solutions.

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