Similarity solutions of the equations of unleaded mixed boundary-layer convection over a fl"l porous vertical plate were investigated by the one-parameter method of continuous group theory. By invalidated application of this method, which reduces the number of independent variables~, first [from three to two and finally to two 10 one, the partial differential equations governing the boundary layer are transformed into a pair of nonlinear ordinary differential equations with an appropriate boundary condition~ None of the material in this paper is to be submitted in support of any other study or degree at any other university or to the institute, except for publications.
I would like to express my sincere sense of gratitude to my supervisor Dr. I am indebted to all my teachers, especially all the lecturers and lecturers; of Parliament of Mathematics, BUET, Dhaka-WOO of their kind of cooperation. Finally, it will be a detriment if I express my gratitude to my wife, brothers and sister for their sacrifice, (;()0pcTalionand Inspiration during the preparation of this dissertation.
Velocity component in the boundary layer along the plate Velocity component in the boundary layer along the normal plate External velocity.
Greek symbols
Subscripts
General Introduction
Chapter One
Free (Natural) convection
Free convection heat transfer is only important when there is no external flow. From the dimensional analysis it is expected that for large Reynolds number (ie, for high flow velocity) and small Grashof number, the influence of free convection on it. ~t transfer can be neglected. On the other hand, for large (jrashof number ~ small Reynolds number) free convection would be a dominant factor. Buoyancy causes changes in the velocity and temperature fields of the forced convection currents leading to changes in the Nusselt number and in the wall shear stress or friction coefficient, parameters that are important for most engineering problems.
That is, if the relative importance of forced and free convection is of comparable order; these phenomena can now be called mixed convection. Laminar boundary layer flow due to a mixed fluid has received considerable attention in the estimation of flow parameters for engineering purposes in both steady and unsteady conditions. In recent years, heat transfer to and from enclosed or semi-enclosed areas by means of natural convection or a combination of natural and forced convection has gained new importance in the fields of aeronautics, automatic power, electronics and chemistry. engineering and electrical engineering.
He showed that the smooth motion with the small friction can be divided into two regions (i): a very thin region close to the body over which fluid flows, also called a boundary layer, where the viscous effect dominates. ii) the area outside the boundary line, where the viscous effect is negligible.
Porous plate
Platc with suction and injection
Most of the work on the effect of subsidence and swelling of a free convective boundary layer was limited to cases where the described wnlllcmpcralurc was considered. Williams et al (1987) studied unsteady free convection flow over a vertical plate under the assumption of variations of waH temperature with time and distance Boundary layer flows of unsteady mixed convection on vertical surfaces were studied by Harris (1999) and many others.
Zakerullah (2001) has derived similarity solutions of some of the possible cases of unsteady mixed convection from Group Theory without absorption. The numerical solutions of one of the formed representative equations for different values of the controlling parameters are obtained. Harris (1999) has performed an analysis of the unsteady boundary layer flow of mixed convection from a vertical flat plate embedJeJ in porous mcdium.
Experimental measurement of air viscosity is reluted with temperature by the Southerland equation.
Thermal conductivity
The viscosity coefficient of a Newtonian fluid is directly related to the molecular embeddings and thus can be considered as a thermodynamic characteristic in thc macroscopic scene, varying with temperature and pressure i, i.e., .u'~ /t'(T', 1 ''). Normally, viscosity decreases rapidly with temperature for liquids, increases with temperature for low-pressure (dilute) gases, and always increases with pressure. Bul a fluid is isotropic, i.e., it has no directional characteristics and tllUS K is a thermodynamic property, The value of K' for a substance depends on its chemical composition, physical state, temperature, and pressure.
The variation of thermal conductivity of gases with temperature is the same as that of dynamic viscosity.
The Prandtl number
Coefficient of thennal expansion
Fundamental equations of the fluid flows
Assumptions
Before proceeding to obtain the solutions of the equations, we will first find the dimensionless group on which the solution must depend. The density will be dimensioned ~ with respect to p~, the press will be dimensionedC8' at p:,U' and I1letemperature at AT' The other transport properties of the fluid 1(',1>',cp, and gravitational components g', and g ' will be made dimensionless by!,:, K,'"c and grespectively. We shall now discuss the physical meaning of nOJl(Jimensional parameters on which equality solutions depend.
Important nondimensional parameters Prandtl number
Eckert number
Froude number
Reynolds showed that half flow in a circular pipe becomes turbulent when the Reynolds number of the flow exceeds 2300. where u is the average fluid velocity and d is the Jiameler of the pipe.
Boussincsq approximation and governing equations
The boundary layer equations
Order of magnitude analysis
CASE-l
Unsteady mixed convection with surface temperature varying inversely as a linear combination of x and I, the I'ree current velocity is constant and the suction velocity varies incly as a square root 01' the linear combination of x and I. An alternative form of Ihe Prandtl boundary layer equations are derived by representing the velocity field in terms of a scalar field", ' calibrated stream function. The existence of this function is a mere consequence of the incomprehensibility of the fluid in two-dimensional flow.
Finding the similarity solutions of the equation, (3.1.1) and (3.1.2), is equivalent to determining the invariant solutions of these equations under a particular collinear one parameter group. One of the simplest methods to search for a transformation group from the elementary set of one parameter transformation, defined by the next group (GI). Here a(•••O) is the parameter of the group and a's arc the arbitrary real numbers whose mutual relationship will be determined by the subsequent analysis.
By solving equations (3.1.8) we have the following relationship between the exponents. We shall now show that if>" ifJ, can be expressed in terms of new independent variable 7) (equality variable), F,G,l,U: and their derivatives with respect to 7). The solution of the new system will be a certain set of invariant solutions of the original system in terms of x, y, u, v, etc. The variable II must be an absolute invariant of the subgroup of the transformation of the independent variables.
Given the boundary layer concepts, it would be a good idea to assume that "I could be written in terms of powers of x and t".
Dependent variable transformation
3,2CASE.2
Unsteady mixed convection with surface temperature, frce stream velocity and suction velocity varying directly with any power of linear
Variable Transformation
Independent variable transformation
We now express all dependent variables in terms of rr, and 'h. Since there are five dependent variables, we look for five functions g (i) that are absolutely invariant (3.2.1).
Comparison with the published results
Results and Discussion
CASE-3
Unsteady mixed convection with surface temperature varying directly with linear function of x inversely with square of the linear
JII -2[']
CASE-4(pURELY UNSTEADY CASE)
Unsteady mixed convection with surface temperature and free stream velocity varying any power of linear function of I, and the suction velocity varying inversely as a square 1'00t of the linear function of t. In this case, to seek the invariant solution to the governing equation, we choose the following transformation group Gl). He,e a( ••O);s the parameter of the group and the a's are (the arbitrary real numbers whose interrelationship will be determined by the subsequent analysis.
Variable Transfonnation
Independent variable Transformation
We now express all dependent variables in terms of '10 and /), since there are five dependent variables, we look for five functions S, (i which are absolutely invariant under (3.4.3).
Dependent variable Transformation
CASE-5
Unsteady mixed convection with surface temperature and free stream velocity varying with an exponential function of function of I and suction velocity is zero. By direct substitution, we can show that 1/>,I/>, are conformally invariant under the spiral group transformation (3.5,1). m ~~)a are two invariant equality variables. Unsteady mixed convection with surface temperature, flow velocity and suction velocity varying with an exponential function of the x-function.
In this case, to find the invariant solution to the set of governmental equalities, we set up the following spiral group (Gl). By direct substitution we can show that I/l" I/l is informally invariant under the spiral group transformation (3.6.1). Applied in a similar way as before, the following absolute invariants involving dependent variables are found.
If the analysis were to stop at this slag and denote H ~constant=l, I ~constant=l and one of the similarity variables (say) 1J2 ~ 0, then we have,. It can be seen from equations (3.6.4) and (3.6.5) that they are still partial differential equations with two independent variables IJ, and lJ,. Here b(", a '"0) is the group parameter and fi are the real timbers determined in the previously stated manner.
A TABLE FOR SIMILARITY REQUIREMENTS
Chapter Four
Results and Discussions
Discussions
Chapter Five Conclusion
By substituting V;l1;OUS transformations into the original system of equations, the new system stands for mth number of independent variables reduced by one. Thus, the reduction of variables in the problem carries more and more constraints to develop various types of possible cases. Finally, we can reach 10 a po~illion to give the analytical solution of the problem under limited conditions.
Choaudhary, MA., Merkin, J.H. and Pop, t Similarity ~ojulion~ in Boundary Layer Free Convection Flow Adjacent to Permeable Surface 10 Vertical in Porous Medium". Hansen, A.G Similarity Analysis of Boundary Layer Problems in Engineering,"Prcntice-Hal1, lnc., Englewood Cliff, New Jers~y_. Hossain, M.A. and Mandai, A.C. Free hydromagnetic convection now unsteady across an accelerated infinite porous plate," A~trophysics and Space Science, Vol.
Ludlow, D.K., Clarkson, P.A, and 8assom, A.P. New equality solutions of unstable, incompressible boundary layer equations, “ Quarterly Journal of Mech. Morgan, AJ.A The reduction of one of the number of independent variables in a system of partial differential equations ”, Quarter. Muloani, L and Rahman, M Similarity analysis for natural convection of a vertical plate with distributed wall concentration, "InL Journal of Math and Math Sci".
Ostraeh, S Combined natural and forced convection laminar flows and heat transfer of nuids with and without heat sources in the ducts with linearly varying waH temperature,” NACA.
