FLOW DOWN AN INCLINED PLANE OF A FLUID WITH PRESSURE- DEPENDENT VISCOSITY THROUGHT A POROUS MEDIUM WITH CONSTANT PERMEABILITY

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J. Modern Technology & Engineering Vol., No., 07, pp.55-66 FLOW DOWN AN INCLINED PLANE OF A FLUID WITH PRESSURE- DEPENDENT VISCOSITY THROUGHT A POROUS MEDIUM WITH CONSTANT PERMEABILITY S.M. Alzahrani, I.

Hamdan * Department of Mathematics and Statistics, University of New Brunswick, Saint John, New Brunswick, Canada (On leave from University of Umm Al-Qura, Kingdom of Saudi Arabia) Department of

1 J. Modern Technology & Engineering Vol., No., 07, pp FLOW DOWN AN INCLINED PLANE OF A FLUID WITH PRESSURE- DEPENDENT VISCOSITY THROUGHT A POROUS MEDIUM WITH CONSTANT PERMEABILITY S.M. Alzahrani, I. Gadoura, M.H. Hamdan * Department of Mathematics and Statistics, University of New Brunswick, Saint John, New Brunswick, Canada (On leave from University of Umm Al-Qura, Kingdom of Saudi Arabia) Department of Electrical and Computer Engineering, University of New Brunswick, Saint John, New Brunswick, Canada Department of Mathematics and Statistics, University of New Brunswick, Saint John, New Brunswick, Canada Abstract. Flow of a fluid with pressure-dependent viscosity through a porous channel with constant permeability, inclined at an angle, is considered to illustrate the effects of Darcy number on the flow characteristics. The momentum equations are based on Brinkman s equation with a drag coefficient expressed in terms of the permeability, and the viscosity is expressed as an exponential function of pressure in accordance with Barus relation. Keywords: pressure-dependent viscosity, Brinkman equation, constant permeability, inclined plane. Corresponding Author: M.H. Hamdan, professor, Dept. of Mathematics and Statistics, University of New Brunswick, P.O. Box 5050, Saint John, New Brunswick, Canada EL 4L5 hamdan@unb.ca Manuscript received: March 07. Introduction When the viscous stresses arising from fluid flow are proportional to the local strain (or the rate of change of velocity vector) at every point, the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. While the behaviour of some fluids such as water and air can be approximated by this Newtonian behaviour, many fluids such as paint and polymers exhibit behaviours in which a fluid becomes either stiffer and thicker, or thinner when sheared, (cf. [, 4] and the references therein). In cases where a fluid viscosity increases with pressure, the fluid is said to undergo pressure-thickening, []. This behaviour has long been known and studied, and goes back to the mid-nineteenth century s work of Stokes []. The dependence of viscosity on pressure has been reported to have many applications [9, 0] that include applications in food and polymer processing, in the pharmaceutical industry, in fluidics and in thin film lubrication, [7, 0, ]. Models of dependence of viscosity on pressure date back to Barus, [5, 6], who suggested an exponential relationship of the form: ap 0 e () 55

2 J. MODERN TECHNOLOGY & ENGINEERING, V., N., 07 where is the fluid viscosity, 0 is a reference viscosity, p is the fluid pressure and a 0 is a constant. In variations to Barus relation (), many authors have implemented either a polynomial (linear or non-linear) or an exponential form of the expression of viscosity in terms of pressure (cf. [-, -5] and the references there in), and take the following forms, where in p 0 is a reference pressure, b > 0 and n > 0: f ( p) () n n n p a ( p p0 ap, ap, ) 0( ap), 0( ap ), 0[ a( p p0)], 0( be ), 0e The above relationships between viscosity and pressure are important in the modelling and study of pressure-dependent fluid flow through porous media [8-0]. These models include Darcian, Forchheimer and Brinkman s effects in their momentum equations (cf. [, ] and the references therein). A model based on Brinkman s equation has been derived by [] and is written here as the following continuity and momentum equations, respectively: u 0 () u t u u p u ( u) T ( p) u G (4) where u is the velocity vector field, p is the pressure, is the fluid density, G is the gravitational vector field, ( p) is the variable viscosity, and ( p) is a drag function that has been given various forms as discussed by Subramanian and Rajagopal [4] and include exponential and polynomial forms in terms of pressure. The drag function can also be expressed as the ratio between viscosity of the fluid and permeability of the porous medium [6, 8], namely ( p) (5) k which has the advantage of modifying momentum equations (4) to explicitly contain the permeability of the porous medium. This in turn facilitates studies of flow through variable permeability porous media, in a manner that was carried out by Alzahrani et al. []. It is worth noting that momentum equations (4) with ( p) expressed by (5) were derived by Alharbi et al. [] using intrinsic volume averaging. Kanaan and Rajagopal [] provided elegant analysis, and obtained the solution to equations (4) in their study of flow through a porous channel inclined to the horizontal, for various forms of viscosity as a function of pressure, and for different forms of ( p). However, their analysis does not explicitly address the effects of permeability of the porous medium. In order to study the effects of constant permeability on the flow of the fluid with pressure-dependent viscosity, we consider in this work flow through the same configuration treated in the work of Kanaan and Rajagopal [] as governed by the model derived by Alharbi et al. [], namely, equations (4) with ( p) given in terms of the permeability by (5). Variations in viscosity as a function of pressure is taken to be according to Barus model, given by equation (). We point out here that this same problem was previously considered for variable 56

3 S.M. ALZAHRANI et al.: FLOW DOWN AN INCLINED PLANE permeability media with variations in viscosity taken as proportional to pressure. The current work is intended to illustrate the effects of Darcy number on the flow.. Problem formulation and solution The flow of a fluid with pressure-dependent viscosity through a Brinkman porous structure with constant permeability is governed by the continuity and momentum equations ()-(5). For the flow down the inclined plane shown in Figure (which is the same configuration given in [] and reproduced herein for illustration) the governing equations reduce to: h p x Angle Figure.. Representative sketch u u g sin u 0 k (6) p y g cos 0 (7) Boundary conditions are those of no-slip velocity on solid the boundary, y=0 and y=h, and a prescribed (atmospheric) pressure at y=h, namely: p ( h) p0; u(0) u( h) 0 (8) Assuming p p(y) and introducing the dimensionless quantities y y / h, u u / U, conditions (8) take the form: p ( ) p0; u(0) u() 0 (9) and governing equations (6) and (7) can be written, respectively, as d u d du gh h sin u 0 dy dy dy U k (0) dp ghcos dy () General solution to () takes the form p ( ghcos) y c () where c is an arbitrary constant. Using pressure condition p( ) p0, we find that c p 0 ghcos, hence () takes the form y x 57

4 J. MODERN TECHNOLOGY & ENGINEERING, V., N., 07 p p0 ( y)ghcos () In order to solve (7) for u (y), we assume that the viscosity varies with pressure exponentially according to the following, [5, 6, ], wherein A and are constants. p ( p) Ae (4) From (4) we obtain d dp P Ae (5) dy dy and equation (0) can be written as: d u dp du h u gh P sine (6) dy dy dy k UA Using () and (5), we can write (6) as: d u du h u gh ghcos sin exp[ { p0 ghcos( y)} (7) dy dy k UA General solution of (7) takes the form m y m y gk sin { p0 ghcos ( y)} u ae ae e (8) UA wherein a and a are arbitrary constants, and m and m are the characteristic roots given by and satisfy ghcos gh h m ( ) cos k ghcos gh h m ( ) cos k (9) (0) m m ghcos () The non-zero Wronskain, W, of the linearly independent solutions u e m y u e m y and to the homogeneous part of (7), is given by: ( m m ) y W( u, u) ( m m ) e () Upon using conditions (9), the following values of arbitrary constants a and a are obtained: gk sin p0 m ghcos a e e e m m UA( e e ) () gk sin p0 m ghcos a e e e m m UA( e e ) (4) Defining Darcy number as k Da h mm (5) and letting 58

5 S.M. ALZAHRANI et al.: FLOW DOWN AN INCLINED PLANE ghcos (6) gh sin 0 UA we can express (9), (0), (), (4) and (8), respectively, as: p e (7) m ( ) (8) Da m ( ) Da (9) Da m a e e ( e e ) m m (0) Da m a e e m m ( e e ) () m y m y ( y) u ae ae Da e () From (), dimensionless vorticity,, of the flow is obtained as: m y m y ( y) u [ a m e a m e Da e ] () y It should be noted that the expressions for and of equations (6) and (7) are the same as those used by Kanaan and Rajagopal, [], so that we can choose their same numerical values for computations.. Results and discussion Results have been computed for the combination of parameters = 0, and 0, and = 0, and 0. The range of Da considered is Da=, 0., 0.0, 0.00, and If 0 then either 0, in which case equation (4) implies that viscosity is a constant function of pressure, or cos 0, which implies that the inclined plane is vertical and equation () implies that p p0 for all y. An increase in can be interpreted as an increase in the viscosity as a function of pressure, or a decrease in the angle of inclination of the plane. If 0 then (7) implies sin 0, or the plane is horizontal. An increase in implies an increase in the angle of inclination or a decrease in the value of (hence an decrease in the viscosity as a function of pressure) or a decrease in the viscosity magnification factor, A, thus decreasing the viscosity. Values of the characteristic roots computed from (8) and (9), and values of the arbitrary constants, computed using (0) and (), are listed in Table for the different combinations of, and Da. Table shows that as Da becomes small, the value of a becomes extremely small. The effects of parameters, and Da on mid-channel velocity u ( y 0.5) and on wall vorticity ( y ) are illustrated in Table, which shows 59

6 J. MODERN TECHNOLOGY & ENGINEERING, V., N., 07 a decrease in both velocity and vorticity with decreasing Da, for all combinations of and. For a given Da and a given, Table shows an increase in velocity and vorticity proportional to (due to an increase in the angle of inclination resulting in faster flow). For a given Da and a given, Table shows a decrease in velocity with increasing (due to the associated viscosity increases with increasing ). Table. Characteristic Roots and Arbitrary Constants for Various, and Da. Da m m a a

7 S.M. ALZAHRANI et al.: FLOW DOWN AN INCLINED PLANE Table. Wall Vorticity and Mid-channel Velocity for Various, and Da Da at y u at y Velocity profiles across the channel for 0, and 0 are illustrated in Figure (a) and (b) for Da=, 0., 0.0, and 0.00, and in Figure (c) and (d) for Da=0.000 and Figure (a) and (b) demonstrate the increase in the velocity across the channel with increasing Da, for all values of and tested. With 0 interpreted as corresponding to the viscosity being a constant function of pressure, the profiles in Figure (a) and (b) are symmetric about the centerline of the channel, y The effect of increasing on increasing the velocity is demonstrated in Figure (a) and (b). As increases from to 0, there is a ten-fold increase in the velocity. 6

8 J. MODERN TECHNOLOGY & ENGINEERING, V., N., 07 Figure (a): Velocity for, 0 Figure (b): Vorticity profiles for 0, 0 Figure (a): Vorticity profiles for Figure (b): Vorticity profiles for, for Da=, 0., 0.0,, 0 for Da=, 0., 0.0, 0 0 and 0.00 and

9 S.M. ALZAHRANI et al.: FLOW DOWN AN INCLINED PLANE Figure 4(a): Velocity profiles for Figure 4(b): Velocity profiles for, Da=, 0., 0.0, and 0.00, 0, Da=, 0., 0.0, and 0.00 Figure 4(c): Velocity profiles for Figure 4(d): Velocity profiles for 0, Da=, 0., 0.0, and ,, Da=, 0., 0.0, and

10 J. MODERN TECHNOLOGY & ENGINEERING, V., N., 07 Figure 5(a): Vorticity profiles for Figure 5(b): Vorticity profiles for for Da=, 0., 0.0, and 0.00, 0, Da=, 0., 0.0, and 0.00 Figure 5(c): Vorticity profiles for, Figure 5(d): Vorticity profiles for 0 Da=, 0., 0.0, and ,, Da=, 0., 0.0, and 0.00 Velocity profiles across the channel for combinations of and 0, and and 0 are illustrated in Figure (a), (b), and 4(a)-4(d) and show a loss of symmetry in the profiles as takes a non-zero value, and a shift upward in the location of the maximum velocity with increasing. Increasing, for all values of, results in increasing the velocity. This holds true for the full range of Da. 64

11 S.M. ALZAHRANI et al.: FLOW DOWN AN INCLINED PLANE The corresponding vorticity profiles across the channel are illustrated in Figure (a), (d) and Figure 5(a)-5(d), and show an increase in the magnitude of vorticity with increasing Da and increasing. 4. Conclusion In this work, we obtained the solution to flow of a fluid with pressure dependent viscosity through a porous medium with constant permeability permeability inclined at an angle to the horizontal. Barus relation between viscosity and pressure was implemented and the flow characteristics were obtained to illustrate the effects of Darcy number. For a given Darcy number, velocity and vorticity qualitative behaviours are similar to those obtained in []. References. Abu Zaytoon M.S., Allan F.M., Alderson T.L., Hamdan, M.H., (06) Averaged equations of flow of fluid with pressure-dependent viscosity through porous media, Elixir Appl. Math., 96, Alharbi S.O., Alderson T.L., Hamdan M.H., (06) Flow of a fluid with pressuredependent viscosity through porous media, Advances in Theoretical and Applied Mechanics, 9(), -9.. Alzahrani S.M., Gadoura I., Hamdan M.H., (07) A note on the flow of a fluid with pressure-dependent viscosity through a porous medium with variable permeability, Journal of Modern Technology and Engineering, (), Bair S., Kottke P., (00) Pressure viscosity relationship for elastohydrodynamics, Tribology Trans., 46, Barus C.J., (89) Note on dependence of viscosity on pressure and temperature, Proceedings of the American Academy, 7, Barus C.J., (89) Isothermals, isopiestics and isometrics relative to viscosity, American Journal of Science, 45, Bridgman P.W., (9) The Physics of High Pressure, MacMillan, New York. 8. Fusi L., Farina A., Rosso F., (05) Mathematical models for fluids with pressuredependent viscosity flowing in porous media, International Journal of Engineering Science, 87, Housiadas K.D., Georgiou G.C., (06) New analytical solutions for weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity, International Journal of Engineering Science, 07, Housiadas K.D., Georgiou G.C., Tanner R.I., (05) A note on the unbounded creeping flow past a sphere for Newtonian fluids with pressure-dependent viscosity, International Journal of Engineering Science, 86, 9.. Hron J., Malek J., Rajagopal, K.R., (00) Simple flows of fluids with pressuredependent viscosities, Proceedings of the Royal Society, 457, Kannan K., Rajagopal K.R., (008) Flow through porous media due to high pressure gradients, Applied Mathematics and Computation, 99, Lanzendörfer M., (009) On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate, Nonlinear Analysis: Real World Applications, 0,

12 J. MODERN TECHNOLOGY & ENGINEERING, V., N., Málek J., Rajagopal K.R., (007) Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities, in: Handbook of Mathematical Fluid Dynamics, Elsevier. 5. Martinez-Boza F.J., Martin-Alfonso M.J., Callegos C., Fernandez M., (0) Highpressure behavior of intermediate fuel oils, Energy Fuels, 5, Nakshatrala K.B., Rajagopal K.R., (0) A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium, Int. J. Numer. Meth. Fluids, 67, Rajagopal K.R., Saccomandi G., Vergori L., (0) Flow of fluids with pressureand shear-dependent viscosity down an inclined plane, Journal of Fluid Mechanics, 706, Savatorova V.L., Rajagopal K.R., (0) Homogenization of a generalization of Brinkman s equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid, ZAMM, 9(8), Singh A.K., Sharma P.K., Singh N.P., (009) Free convection flow with variable viscosity through horizontal channel embedded in porous medium, The Open Applied Physics Journal,, Srinivasan S., Bonito A., Rajagopal K.R., (0) Flow of a fluid through a porous solid due to high pressure gradient, Journal of Porous Media, 6, Srinivasan S., Rajagopal K.R., (04) A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations, International Journal of Non-Linear Mechanics, 58, Stokes G.G., (845) On the theories of the internal friction of fluids in motion, and of the equillibrium and motion of elastic solids, Trans. Camb. Philos. Soc., 8, Szeri A.Z., (998) Fluid Film Lubrication: Theory and Design, Cambridge University Press. 4. Subramanian S.C., Rajagopal K.R., (007) A note on the flow through porous solids at high pressures, Computers and Mathematics with Applications, 5, Vergne P.H., (99) Pressure viscosity behavior of various fluids, High Press. Res., 8,

Emden-Fowler Equation and Inverse Analysis of Simple Flows through Variable Permeability Porous Layers

4 Int. J. Open Problems Compt. Math., Vol. 0, No., June 07 ISSN 998-66; Copyright ICSRS Publication, 07 www.i-csrs.org Emden-Fowler Equation and Inverse Analysis of Simple Flows through Variable Permeability