# Similarity Requirements for Mixed Convective Boundary Layer Flow over Vertical Curvilinear Porous Surfaces with Heat Generation/Absorption

Authors: Kh. Abdul Maleque
Journal: International Journal of Aerospace Engineering
 (2020)

---

## Abstract

Similarity requirements for three dimensional combined forced and free convective laminar boundary layer flows over the porous inclined vertical curvilinear surfaces with buoyancy effects and heat absorption/generation effects are investigated theoretically. The potential flow in the mainstream and Gabriel lame coefficients outside of the boundary layer are the function ofξ,η. Hence, the external velocity components (Ue, Ve) and Gabriel lame coefficients h1,h2,h3 are independent of ζ. Here, h3ξ,η=1 has been set such that ζ represents actual distance measured normal to the surface. Similarity requirements for an incompressible fluid are sought on the basis of detailed analyses in order to reduce the governing partial differential equations into a set of ordinary differential equations. Finally, different possible cases are exhibited in a tabular form with the inclusion of ΔT variations for onward flow study that are helpful to the future researchers for the flow over the orthogonal curvilinear surfaces.


---

## Body

## 1. Introduction

When a viscous fluid flows over a heated (or cooled) vertical orthogonal curved surface in inclined position, then velocity and thermal boundary layers are formed over the surface. The similarity equations exist only for a very special type of the component of main stream flowUeξ,η,Veξ,η,0 and temperature difference ΔTξ,η between the wall and ambient fluid. Howarth [1] investigated the equations of boundary layer flow in the vicinity of a separation point on a general three-dimensional case. He showed that the equations were reducible to a pair of simultaneous ordinary differential equations. Devy and Schofield [2] studied the same case and presented numerical results in a tabular form. Hansen [3, 4] presented a table showing the nature of the variations in the mainstream components for which the governing equations were reducible to a set of ordinary differential equations. Restrictions on main flow velocity components and coordinate system which leads to similarity solutions were given in a tabular form. Wang [5] discussed general methods for calculating three-dimensional laminar boundary layers over inclined blunt bodies (not necessarily bodies of revolution). He used the traditional integral method and presented complete incompressible boundary layer results for a prolate spheroid at 30° incidence. Cheng [6] investigated the combined free and forced convection boundary layer flow along inclined surfaces embedded in porous media. It was found that similarity solutions exit when both the wall temperature distribution of the plate outside of the boundary layer vary according to some power function of distance. Langston et al. [7] studied the detailed measurements of the subsonic flow in a large scale; plane turbine cascade was made to evaluate the three-dimensional nature of the flow field. Raju et al. [8] considered the boundary layer flow over a semi-infinite vertical and horizontal flat plate, heated to a constant temperature in a uniform free stream. They discussed when the buoyancy forces either aid or oppose the development of the boundary layer. Different mixed convection parameters were introduced in the formation of the respective problems involving horizontal and vertical surfaces such that smooth transition from one convection limit to the other was possible; the governing equations for the purely forced and free convection cases were, respectively, recovered from the zero values of Grashof and Reynolds numbers. A quite good number of theoretical studies of such a flow under free convection are found in literature. Merkin and Mahmood [9] introduced the logarithmic terms in their perturbation technique, the leading order solution being of Olog−1. A general classification is made by the three-dimensional boundary layer flow for both compressible and incompressible fluids by Chong and Perry [10]. Chen and Chang [11] presented an analysis to predict the heat transfer rate between two natural convection systems at different temperature separated by a vertical plate. They opined that their predicted results matched well with the reliable experimental data. The effects of temperature difference in order to predict the frictional resistance and heat transfer rate by introducing the components of the fluid velocity generated by buoyancy effects in the direction parallel to the edges of the surface were discussed by Zakerullah and Maleque [12]. In their another paper [13], they also investigated the nature of free stream velocities and temperature difference variations between the ambient fluid and the three-dimensional rectangular surfaces theoretically and numerically. Numerical solution of MHD three-dimensional free convection and mass transfer flow with Hall current, viscous dissipation, and Joule heating was discussed by Sattar and Maleque [14]. The effects of magnetic field and heat absorption on three-dimensional transient convective boundary layer flow were studied by Maleque and Sattar [15]. They also studied the combined convective boundary layer flow due to a rotating disk with a Hall effect [16]. Steady three-dimensional forced convective compressible fluid flow with variable properties and Hall current was discussed by Maleque and Sattar [17, 18]. Effects of combined temperature- and depth-dependent viscosity and Hall current on unsteady fully three-dimensional magnetohydrodynamic boundary layer flow were studied by Maleque [19]. Receptivity of localized surface roughness and free-stream vorticity in three-dimensional boundary layer flow was considered by Schrader et al. [20]. Maleque [21] investigated the effect of thermal diffusion on MHD convective heat and mass transfer flow due to a rotating disk. Steady hydrodynamic flow of a power-law non-Newtonian fluid from a rotating disk to a saturated non-Darcian porous medium was studied by Beg et al. [22]. Nadeem et al. [23] studied three-dimensional Casson fluid flow past a porous linearly stretching sheet. In their another paper [24], they also investigated the MHD effect with the fluid and convective condition along the three-dimensional surface. The three-dimensional boundary layer flow and heat transfer of a Cu-water on a permeable shrinking flat surface were studied by Mohd Noor and Roslinda [25]. MHD three-dimensional incompressible boundary layer flow that is generated by a bidirectional stretching surface was recently investigated by Hayat et al. [26]. Maleque [27, 28] investigated the effect of a uniform electric field and Hall current on unsteady MHD three-dimensional non-Newtonian Casson fluid flow due to a porous rotating disk. Recently, Hayat et al. [29] studied the effect of chemical reaction on three-dimensional combined convective boundary layer flow of Williamson nanofluid. More recently, Rashid et al. [30] considered three-dimensional nanomaterial flow of Maxwell material over a stretchable moving rotating frame that was generated by a linear stretched sheet.The present paper concerned with the systematic analyses in reducing the governing partial differential equations for a combined convective laminar incompressible flow over the inclined orthogonal curvilinear surfaces into a set of ordinary differential equations. Finally, a tabular form of similarity requirements is made forΔT,h1,h2,Ue, and Ve variations. Two parameters UF2/Ue2 and VF2/Ve2 are introduced to show the dominancy of buoyancy effect over the forced flow effect, where UFξ,η,VFξ,η,0 are the local fluid velocity components generated by buoyancy effect only and Ueξ,η,Veξ,η,0 are the components of the main stream velocity.


## 2. Geometry of the Problem and the Governing Equations

An orthogonal curvilinear heated surface inclined in its own position is held vertically in the free-stream velocityUeξ,η,Veξ,η,0 perpendicular to ζ axis shown in Figure 1.Figure 1
The flow configuration and the coordinate system.g¯−gξξ,η,−gηξ,η,0 is the acceleration vector of the fluid particle due to buoyancy effect of the fluid along ξ and η increasing owing to heated surface. With the introduction of Prandtl boundary layer concept, the equations governing the flow field in general orthogonal curvilinear coordinates are
(1)∂∂ξh2u+∂∂ηh1v+∂∂ζh1h2w=0,(2)ρuh1∂u∂ξ+vh2∂u∂η+w∂u∂ζ+uvh1h2∂h1∂η−v2h1h2∂h2∂ξ=−1h1∂p∂ξ+ρgξ+μ∂2u∂ζ2,(3)ρuh1∂v∂ξ+vh2∂v∂η+w∂v∂ζ+uvh1h2∂h2∂ξ−u2h1h2∂h1∂η=−1h2∂p∂η+ρgη+μ∂2v∂ζ2,(4)∂p∂ζ=0,(5)ρuh1∂T∂ξ+vh2∂T∂η+w∂T∂ζ=μPr∂2T∂ζ2+Q0CpT−Te.Here,h3ξ,η=1 has been set such that ζ represents actual distance measured normal to the surface. The body force terms are retained in equations (2) and (3) in addition to the equations used by Davy and Schofield [2]. The potential flow in the mainstream outside of the boundary layer is the function of ξ,η. Hence, the external velocity components Ue and Ve independent of ζ, u,v,w are the components of fluid velocity, and T is the temperature in the boundary layer. μ and ρ are the viscosity and the density of the fluid, respectively, Pr is the Prandtl number, Cp is the specific heat with constant pressure, and Q0 is the heat generation >0 or absorption <0 coefficient.The appropriate boundary conditions for the velocities and temperature fields are(6)u=0,v=0,w=w0ξ,η,T=Twξ,η,atζ=0,u⟶Ueξ,η,v⟶Veξ,η,T⟶Teasζ⟶∞.w0 is considered here to be the surface suction w0>0 or injection w0<0 velocity for the developable surface. Tw and Te are the wall temperature and wall temperature, respectively.To eliminate pressure terms in equations (2) and (3), the conditions outside the boundary layer are imposed. External conditions u⟶Ue,v⟶Ve, ρ⟶ρe, T⟶Te (=constant), ∂/∂ζ⟶0, and the negligible pressure variation yield the following equations.
(7)ρeUeh1∂Ue∂ξ+Veh2∂Ue∂η+UeVeh1h2∂h1∂η−Ve2h1h2∂h2∂ξ−ρegξ=−1h1∂p∂ξ,(8)ρeUeh1∂Ve∂ξ+Veh2∂Ve∂η+UeVeh1h2∂h2∂ξ−Ue2h1h2∂h1∂η−ρegη=−1h2∂p∂η,(9)ρeUeh1∂Te∂ξ+Veh2∂Te∂η=0.Te=T0 (=constant) is also a solution of equation (9).In view of equations (7) to (9), T−T0=Tw−T0θξ,η,ζ and Tw−T0=ΔT, the boundary layer and energy equations with the elimination of pressure terms are
(10)∂∂ξh2u+∂∂ηh1v+∂∂ζh1h2w=0,(11)uh1∂u∂ξ+vh2∂u∂η+w∂u∂ζ+uvh1h2∂h1∂η−v2h1h2∂h2∂ξ=−gξβTΔTθ+υ∂2u∂ζ2+Ueh1∂Ue∂ξ+Veh2∂Ue∂η+UeVeh1h2∂h1∂η−Ve2h1h2∂h2∂ξ,(12)uh1∂v∂ξ+vh2∂v∂η+w∂v∂ζ+uvh1h2∂h2∂ξ−u2h1h2∂h1∂η=−gηβTΔTθ+υ∂2v∂ζ2+Ueh1∂Ve∂ξ+Veh2∂Ve∂η+UeVeh1h2∂h2∂ξ−Ue2h1h2∂h1∂η,(13)uh1∂θ∂ξ+vh2∂θ∂η+w∂θ∂ζ+θuh1lnΔTξ+vh2lnΔTη=υPr∂2θ∂ζ2+Q0ρCpθ.Consideringρ=ρP,T. For Boussinesq fluid approximations, ρ−ρe=−ρβTΔTθ is substituted in the above equations where βT=−1/ρ∂ρ/∂Tp is the coefficient of thermal expansion.


## 3. Similarity Transformations

Equations (10)–(13) are nonlinear, simultaneous partial differential equations, and the solutions of these equations are extremely difficult to obtain. Hence, our aim is to reduce equations (11)–(13) into ordinary differential equations with the help of equation (10). Let the set of variables (ξ,η,ζ) be changed to a new set of variables X,Y,ϕ¯ where relations between two sets of variables are
(14)X=ξ,Y=η,ζγX,Y,γX,Y is considered here to be proportional to the square root of the local boundary layer thickness.Let two stream functionsψξ,η,ζ and χξ,η,ζ be defined as the mass flow components within the boundary layer for the case of incompressible flow. The following substitutions may be chosen
(15)ψζ=h2u,χζ=h1v,−ψξ+χη=h1h2w,to satisfy the equation of continuity (10).Guided by the idea of similarity, it can be allowed to write(16)∫0ϕ¯uUX,Ydϕ¯=FX,Y,ϕ¯,∫0ϕ¯vVX,Ydϕ¯=SX,Y,ϕ¯.In attempting separation of variables ofFX,Y,ϕ¯, SX,Y,ϕ¯, and θX,Y,ϕ¯, let it be assumed that
(17)FX,Y,ϕ¯=LX,YF¯ϕ¯,SX,Y,ϕ¯=MX,YS¯ϕ¯,θX,Y,ϕ¯=NX,Yθ¯ϕ¯,where F¯, S¯, and θ¯ are the functions of the single variable ϕ¯. Then, it can be found that
(18)u=ULF¯ϕ¯,v=VMS¯ϕ¯,w=−1h1h2h2γULXF¯−ϕ¯γXh2ULF¯ϕ¯+h1γVMYS¯−ϕ¯γYh1VMS¯ϕ¯+h1h2w0X,Y,0,ψX,Y,ϕ¯=h2γULF¯ϕ¯+ψX,Y,0,χX,Y,ϕ¯=h1γULS¯ϕ¯+χX,Y,0,where w0X,Y,0=−1/h1h2ψXX,Y,0+χYX,Y,0.When the surface is porous,w0X,Y,0 represents the suction w0>0 or injection w0<0 velocity normal to the surface, since Ue,Ve are independent of ζ, then Ueϕ¯=Veϕ¯=0. Equations (10), (11), (12), and (13) take the following forms:u− momentum equation:
(19)υF¯ϕ¯ϕ¯ϕ¯+γh1h2h2γULXF¯F¯ϕ¯ϕ¯+γh1h2h1γVMYS¯F¯ϕ¯ϕ¯+γw0F¯ϕ¯ϕ¯−γ2h1ULXF¯ϕ¯2−γ2VMh2ULYUL+h1Yh1F¯ϕ¯S¯ϕ¯+γ2VM2h1h2ULh2XS¯ϕ¯2+γ2h1ULUeUeX+γ2h2ULVeUeY+γ2UeVeh1h2ULh1Y−γ2Ve2h1h2ULh2X−gXβTγ2ULΔTNθ¯=0.v− momentum equation:
(20)υS¯ϕ¯ϕ¯ϕ¯+γh1h2γh1VMYS¯S¯ϕ¯ϕ¯+γh1h2γh2ULXF¯S¯ϕ¯ϕ¯+γw0S¯ϕ¯ϕ¯−γ2h2VMYS¯ϕ¯2−γ2ULh1VMXVM+h2Xh2F¯ϕ¯S¯ϕ¯+γ2UL2h1h2VMh1YF¯ϕ¯2+γ2h2VMUeVeX+γ2h1VMVeVeY+γ2UeVeh1h2VMh2X−γ2Ue2h1h2VMh1Y−gYβTγ2VMΔTNθ¯=0.Energy equation:(21)υPrθ¯ϕ¯ϕ¯+γh1h2γh2ULXF¯θ¯ϕ¯+γh1h2γh1VMYS¯θ¯ϕ¯+γw0θ¯ϕ¯+γ2Q0ΔTρCpθ¯−γ2ULh1lnNX+lnΔTXF¯ϕ¯θ¯−γ2VMh2lnNY+lnΔTYS¯ϕ¯θ¯=0.In order to derive the generalized boundary conditions, we are to use the no slip conditions, exterior conditions for the velocity components, and finally the condition of the temperature function. Without loss of generality, we may put(22)UL=Ue,VM=Ve,N=1.In view of (22), the equations (19)–(21) take the following formu− momentum equation:
(23)υF¯ϕ¯ϕ¯ϕ¯+12a0+a1−a2F¯F¯ϕ¯ϕ¯+12a3+a4−a5S¯F¯ϕ¯ϕ¯−a6F¯ϕ¯ϕ¯+a71−F¯ϕ¯2+a8+a91−F¯ϕ¯S¯ϕ¯−a101−S¯ϕ¯2+a11θ¯=0.v− momentum equation:
(24)υS¯ϕ¯ϕ¯ϕ¯+12a0+a1−a2F¯S¯ϕ¯ϕ¯+12a3+a4−a5S¯S¯ϕ¯ϕ¯−a6S¯ϕ¯ϕ¯+a121−S¯ϕ¯2+a13+a141−F¯ϕ¯S¯ϕ¯−a151−F¯ϕ¯2+a16θ¯=0.Energy equation:(25)υPrθ¯ϕ¯ϕ¯+12a0+a1−a2F¯θ¯ϕ¯+12a3+a4−a5S¯θ¯ϕ¯−a6θ¯ϕ¯−a17F¯ϕ¯+a18S¯ϕ¯θ¯+a19θ¯=0,where a0,a1,a2.⋯a19 are the contractions in X and Y given by
(26)ia0=γ2Ueh1X,iia1=γ2h2UeXh1h2,iiia3=γ2h2Ue1h1h2X,iva3=γ2Veh2Y,va4=γ2h1VeYh1h2,via5=γ2h1Ve1h1h2Y,viia6=−γw0,viiia7=γ2UeXh1,ixa8=γ2VeUeYh2Ue,xa9=γ2Veh1Yh1h2,xia10=γ2Ve2h2Xh1h2Ue,xiia11=−γ2UegXβTΔT,xiiia12=γ2VeYh2,xiva13=γ2UeVeXh1Ve,xva14=γ2Ueh2Xh1h2,xvia15=γ2Ue2h1Yh1h2Ve,xviia16=−γ2VegYβTΔT,xviiia17=γ2Ueh1lnΔTX,xixa18=γ2Veh2lnΔTY,xxa19=γ2Q0ΔTρCp.Similarity solutions exist only when all thea’s in equation (26) are finite and independent of X and Y; that is to say that all a’s must be constants.Thus, the boundary layer momentum equations and energy equation will become nonlinear ordinary differential equations ifUeX,Y, VeX,Y, h1X,Y, h2X,Y, γX,Y, and ΔTX,Y satisfy equation (26). From equations (26)(i) and (26)(iv) one may have
(27)γ2Ueh1=a0X+AY,γ2Veh2=a3Y+BX,where AY is the function of Y alone and BX is the function of X alone. Using the equations in (26), AY and BX are determined by the following pair of equations
(28)dAdY=h2Ueh1Vea3−a4−a5+a8−a9,(29)dBdX=h1Veh2Uea0−a1−a2+a13−a14.Equations (28) and (29) furnish us the possibilities of similarity cases for which we can find the nature of Ue, Ve, h1, h2, and finally ΔT in terms of X and Y.Case 1.
LetdA/dY=constant and dB/dX=constant.This assumption yields(30)VeUe=k1=constant,h2h1=k2=constant.Let(31)dAdY=k2k1l1,dBdX=k1k2l2,where l1=a3−a4−a5+a8−a9 and l2=a0−a1−a2+a13−a14.From equation (31), we have
(32)AY=k2k1l1Y+A0,BX=k1k2l2X+B0.Then, equation (27) becomes
(33)γ2Ueh1=a0X+k2k1l1Y+A0,(34)γ2Veh2=a3Y+k1k2l2X+B0,orγ2Ueh1=l2X+k2k1Y+B0′.Comparing equation (33) and equation (34), we have
(35)l2=a0,l1=a3implya4+a5=a8−a9anda1+a2=a13−a14,therefore
(36)γ2Ueh1=a0X+k2k1a3Y+A0.In view of requirements (x) and (xi) of equation (26) and equation (36), one may derive
(37)h1x,y=ax+byn,where
(38)a10a0k12=n,a0=a,a3=k1k2b,A0=aX0+bY0,X+X0=x,Y+Y0=y.Using equation (36) and similarity requirement (viii) of equation (26), one may obtain
(39)Uex,y=U0ax+bym,where m=a7/a0. From other requirements, γ2x,y and ΔTx,y are found to be in this situation
(40)γ2x,y=ax+byn+1−m,ΔTx,y=ΔT0ax+by2m−n−1,here, U0 and ΔT0 are integrating constants that represent velocity and temperature difference, respectively.Substituting equations (37), (38), (39), and (40) in the similarity requirements, the constants in terms of a and b associated with m,n,k1 are
(41)a0=a,a1=m+na,a2=−2na,a3=k1k2a,a4=m+nk1k2b,a=5−2na,a7=ma,a8=mk1k2b,a9=nk1k2b,a10=k12na,a12=mk1k2b,a13=ma,a14=na,a15=nak12,a17=2m−n−1a,a18=2m−n−1bk1k2.Hence, transform equations (23)–(25) become
(42)υF¯ϕ¯ϕ¯ϕ¯+123n+m+1aF¯F¯ϕ¯ϕ¯+123n+m+1k1k2bS¯F¯ϕ¯ϕ¯+ma1−F¯ϕ¯2+m+nk1k2b1−F¯ϕ¯S¯ϕ¯−a6F¯ϕ¯ϕ¯−k12na1−S¯ϕ¯2+a11θ¯=0,(43)υS¯ϕ¯ϕ¯ϕ¯+a23n+m+1F¯S¯ϕ¯ϕ¯+bk12k23n+m+1S¯S¯ϕ¯ϕ¯+bmk12k21−S¯ϕ¯2+am+n1−F¯ϕ¯S¯ϕ¯−a6S¯ϕ¯ϕ¯−mbk1k21−F¯ϕ¯2+a16θ¯=0,(44)υPr−1θ¯ϕ¯ϕ¯+123n+m+1aF¯+k1k2bS¯θ¯ϕ¯−a6θ¯ϕ¯−2m−n−1aF¯ϕ¯+k1k2bS¯ϕ¯θ¯+a19θ¯=0.In order to simplify the above type of equations, we substituteF¯=αf, S¯=αg, θ¯=ϑ, and ϕ¯=αϕ. Choosing a3n+m+1α2/2υ=1, k1b/k2a=c, 2a6/α3n+m+1a=−Pw, and β=2a19/3n+m+1a, the final form of the similarity solution stands as
(45)fϕϕϕ+f+cgfϕϕ+2m3n+m+11−fϕ2+2cm+n3n+m+11−fϕgϕ+Pwfϕϕ−2nk123n+m+11−gϕ2+UF2Ue2ϑ=0,(46)gϕϕϕ+f+cggϕϕ+2mc3n+m+11−gϕ2+2m+n3n+m+11−fϕgϕ+Pwgϕϕ−2ck12n3n+m+11−fϕ2+VF2Ve2ϑ=0,(47)Pr−1ϑϕϕ+f+cgϑϕ+Pwϑϕ−22m−n−13n+m+1fϕ+cgϕϑ+βϑ=0.α is introduced here in order to reduce the types of equations (42), (43), and (44) in a simplified form as equations (45), (46), and (47).
(48)UF2=−gxβTΔTh1⋅characteristiclength,VF2=−gyβTΔTh2⋅characteristiclength,characteristiclength=ax+bya3n+m+1.The boundary conditions are(49)f0+cg0=Pw,fφ0=0,fφ∞=1,g0=g0,gϕ0=0,gϕ∞=1,ϑ0=1,ϑ∞=0.In this case, velocity componentw perpendicular to the surface and the similarity variable ϕ are found to be
(50)w=υa21/23n+m+121/2ax+bym−n−1/2f+cs−ϕn+1−m3n+m+1fϕ+csϕ,ϕ=za3n+m+12υ1/2ax+bym−n−1/2.For a real situation, the condition of irrotationality in the main streamh2Vex−h1Uey=0, yields m=−n or c=k12. If m≠−n, equations (45), (46), and (47) with boundary condition (49) are
(51)fϕϕϕ+f+cgfϕϕ+2m3n+m+11−fϕ2+2cm+n3n+m+11−fϕgϕ+Pwfϕϕ−2nc3n+m+11−gϕ2+UF2Ue2ϑ=0,(52)gϕϕϕ+f+cggϕϕ+2mc3n+m+11−gϕ2+2m+n3n+m+11−fϕgϕ+Pwgϕϕ−2n3n+m+11−fϕ2+VF2Ve2ϑ=0,(53)Pr−1ϑϕϕ+f+cgϑϕ+Pwϑϕ−22m−n−13n+m+1fϕ+cgϕϑ+βϑ=0.Equations (37), (38), (39), and (40) are the requirement for the existence of one of the possible similarity solutions. Equations (51), (52), and (53) with the boundary conditions (49) are the similarity solutions in which the mainstream has the velocity components Ue,Ve,0 satisfying the condition of irrotationality. In similar way, other possible cases in curvilinear coordinates like equations (37), (38), (39), and (40) are given a tabular form.


## 4. Conclusions

General discussion for similarity requirements, detail similarity analyses have been done presented in Table1 to reduce the governing partial differential equations into the set of ordinary differential equations for an incompressible fluid flow over the curvilinear surfaces. Detail discussion of one case has been presented in this paper, and different possible cases are exhibited in a tabular form with the inclusion of ΔT, h1, h2, Ue, and Ve variations for onward flow study that are helpful to the future researchers for the flow over the orthogonal curvilinear surfaces. If we put c=0, n=0, and Pw=0, the above equations (51)–(52) become well-known Falkner-Skan equation in the absence of UF=0 and β=0, i.e., buoyancy and heat absorption effects. If c=0, n=0, k1=0, UF=0, β=0, and Pw=0, the equations coincide with those dealt with by Sparrow et al. [31].Table 1
Similarity requirements in curvilinear coordinates.
h1
h2
Ue
Ve
ΔT
ax+byn
k2h1
ax+bym
k1Ue
ax+by2m−n−1
ax+byn
k2h1
expnax+by
k1Ue
expax+byax+byn
expax+by
k2h1
expax+by
k1Ue
expax+by
1
xn
xm
k1Ue
x2m−1
1
expnx
expmx
k1Ue
exp2mx
yn
1
ym
k1Ue
y2m−1
expny
1
expmy
k1Ue
exp2my



---

*Source: 7486971-2020-03-03.xml*