Impact of Velocity Slip and Temperature Jump of Nanofluid in the Flow over a Stretching Sheet with Variable Thickness

1 Introduction

Since proposed by Prandtl in 1904, the boundary theory has a critical influence on the study of flow behaviour of multifarious fluids. Among them, the flow with heat transfer over the stretching sheet is the common one, and it originates from some certain physical and industrial backgrounds such as metal cooling, glass fibre, and plastic extrusion processes. An extension to the flow due to a sheet issuing with constant speed from a slit into a quiescent fluid was studied by Crane [1], in which the velocity of a stretching sheet became linear. Following it, a number of researchers investigated the flow and heat transfer over a stretching sheet with different surface velocity by using various methods, and the collection of related papers is given in [2–10].

However, it seems that the flow and heat transfer on a stretching sheet with variable thickness were seldom mentioned and explored. Historically, Fang et al. investigated the flow over the stretching sheet at power law velocity with variable thickness [11]. Then based on the study of Fang et al., Elbashbeshy et al. researched the flow and heat transfer over a continuous variable thickness surface saturated in three different nanofluids [12]. Khader et al. presented a numerical solution for the flow of Newtonian fluid over an impermeable stretching sheet with power law surface velocity and variable thickness [13]. Abdel-wahed et al. studied the effects of variable thickness, hydromagnetic flow, Brownian motion, and heat generation on heat transfer characteristics and mechanical properties of a moving surface embedded into cooling medium consists of nanofluid [14].

It was well known that because of the non-adherence of the fluid to a solid boundary, the velocity slip and the temperature jump between fluid and moving sheet may happen in industrial process, so taking velocity slip and temperature jump into account is indispensable. A simple theory based on replacing the effect of the boundary layer with a slip velocity property to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results by Beavers and Joseph [15]. Andersson considered the slip flow of a Newtonian fluid past a linearly stretching sheet [16]. Hong et al. gave out some physical explanation on thermal boundary condition of slip flow [17]. Wang investigated stagnation slip flow and heat transfer on a moving plate [18]. An exact solution of slip MHD viscous flow over a stretching sheet was obtained by Fang et al. [19]. Besides, Aman et al. researched the mixed convection boundary layer flow near stagnation point on vertical surface with the partial slip at the boundary [20].

The above-mentioned term “nanofluid” was proposed in 1995 by Choi at the ASME Winter Annual Meeting [21]. Subsequently, the studies of flow and heat transfer of nanofluid were carried out greatly because of the good properties of nanofluid in the thermal conductivity and heat transfer. In the recent years, there are some interesting achievements obtained by many scientific workers. Rohni et al. studied flow and heat transfer of nanofluids over an unsteady shrinking sheet with suction [22]. Malvandi and Ganji made a theoretical investigation on the effects of nanoparticle migration and asymmetrical heating on the forced convective heat transfer of nanofluids in a uniform magnetic field [23–25]. The effects of nanoparticle migration on mixed convection of nanofluids inside a vertical microchannel were investigated by Hedayati and Domairry [26]. By the lattice Boltzmann method, Karimipour et al. studied laminar-forced convection heat transfer of water–Cu nanofluids in a microchannel [27]. Malvandi et al. investigated the mutual thermal effects of solid body and nanofluid flow over a flat plate [28]. Besides, Malvant et al. also investigated slip effects on the flow of nanofluids in some other circumstances [29, 30].

The homotopy analysis method (HAM) introduced by Liao [31–35] is one of the effective mathematic methods for solving nonlinear equations, and the effectiveness of HAM has been approved by himself and many researchers. In this work, we utilise the HAM to investigate the effects of the generalised velocity slip and the generalised temperature jump on the flow and heat transfer resulted from a stretching sheet with variable thickness in nanofluid. The involved parameters such as velocity slip parameter, temperature jump parameter, Prandtl number, magnetic field parameter, permeable parameter, Lewis number, thermophoresis parameter, and Brownian motion parameter are also investigated and analysed.

2 Formulation of the Problem

We consider here an incompressible and steady flow and heat transfer of nanofluid over a permeable stretching sheet in the presence of Brownian motion and thermophoresis. The physical model and coordinate system are shown in Figure 1. The governing equations of the flow and heat transfer in the Descartes coordinate are as follows:

with the boundary conditions

where u and v, respectively, are the velocity components in the x and y directions, ν is the kinematic viscosity, ρ is the density of nanofluid, σ is the electrical conductivity, α is the thermal diffusion, D_B is the Brownian diffusion coefficient, D_T is the thermophoretic diffusion coefficient, τ is the ratio between the effective heat capacity of nanoparticle and heat capacity of nanofluid, B(x) is the strength of magnetic field, and K(x) is the permeability of sheet surface. The special forms of magnetic field B(x) = B0(x + b)n − 12 and permeability K(x) = k0(x + b)2n − 1 are chosen to obtain similarity solution of the problem. N1 = γ1(x + b)1 − n2 changes with the value of x, so is called the generalised velocity slip factor, in which γ1 = 2 − σνσνλ0 is the traditional velocity slip factor.D1 = γ2(x + b)1 − n2 changes with the value of x, so is called the generalised temperature jump factor, in which γ2 = 2 − σTσT(2rr + 1)λ0Pr is the traditional temperature jump factor. σ_v is the tangential momentum accommodation coefficient, σ_T is the thermal accommodation coefficient, λ₀ is the molecular mean free path, r is the specific heat ratio, T is the temperature of nanofluid, T_w is the temperature of sheet surface, T_∞ is the temperature of ambient nanofluid, C is the concentration of nanofluid, C_w is the concentration of nanofluid at sheet surface, and C_∞ is the concentration of ambient nanofluid. In addition, a, b, and δ are constants, and together with n decide the sheet surface velocity, the shape of sheet surface, and the sheet thickness.

Figure 1:

The physical model of a stretching sheet with variable thickness.

3 Nonlinear Boundary Value Problem

Making use of the following similar transformation,

where η is the similarity variable, ψ is the stream function and defined as u = ∂ψ∂y, v = − ∂ψ∂x, which automatically satisfy (1), and substituting (7) into (2)–(4) and boundary conditions (5) and (6), we obtain the following nonlinear ordinary different equations:

with the corresponding boundary conditions,

and

where primes mean differentiation with respect to η, and with these definitions, the velocities are expressed as u=a(x+b)ⁿφ′(η) and v = − aν(n + 1)2(x + b)n − 12[φ(η) + n − 1n + 1ηφ′(η)]; meanwhile, the temperature and the concentration are written as T=T_∞+(T_w–T_∞)θ(η) and C=C_∞+(C_w–C_∞)ϕ(η), respectively. β = δn + 12 ⋅ aν refers to the sheet thickness parameter, and η = β = δn + 12aν denotes the sheet surface, M is the magnetic field parameter, S is the permeable parameter, Pr is the Prandtl number, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter, Le is the Lewis number, l₁ is the generalised velocity slip (called the velocity slip for short in the following) parameter, l₂ is the generalised temperature jump (called the temperature jump for short in the following) parameter, and they are defined as the following formulas:

Furthermore, we make deformations as φ(η)=f(η–β)=f(ξ), θ(η)=g(η–β)=g(ξ), and ϕ(η)=w(η–β)=w(ξ), so the similar equations (8)–(10) and the boundary conditions (11) and (12) become

with the boundary conditions,

and

where primes mean differentiation with respect to ξ, and the velocities are expressed as u=a(x+b)ⁿf′(ξ) and v = − aν(n + 1)2(x + b)n − 12[f(ξ) + n − 1n + 1(ξ + β)f′(ξ)]; meanwhile, the temperature and the concentration are written as T=T_∞+(T_w–T_∞)g(ξ) and C=C_∞+(C_w–C_∞)w(ξ), respectively, again.

Under the similarity deformation, some engineering parameters of the flow and heat transfer such as the local skin friction coefficient Cf_x, the local Nusselt number Nu_x, and the local Sherwood number Sh_x, which indicate physically to surface shear stress, rate of heat transfer, and rate of mass transfer, respectively, are defined as the following forms, respectively [14]:

Cfx = 2τwρUw2, Nux = (x + b)qwk(Tw − T∞), Shx = (x + b)qmk(Cw − C∞).

Furthermore, the reduced skin friction coefficient Cfr, the reduced Nusselt number Nur, and the reduced Sherwood number Shr may be written as follows:

where the local Reynolds number Re_x is defined as Rex = Uw(x + b)ν, and the velocity of sheet surface is U_w=a(x+b)ⁿ.

4 HAM Solution

Before employing the technique [31–35], we make comparison with results in Fang et al. [11] and Abdel-wahed et al. [14], which is shown in Table 1, and testifying the validity of HAM. Then we utilise the HAM to solve the above nonlinear problem (14)–(18). The set of base functions is chosen as follows:

According to the rule of solution expression and the boundary conditions, the initial approximations are properly selected as follows:

Moreover, the auxiliary linear operators are employed as follows:

and satisfy the following forms:

where C_i, i=1–7, refer to arbitrary constants. The nonlinear operators are set by the following form:

where p∈[0,1] is regarded as the embedding parameter. The zero-order deformation equations are constructed as the following forms:

with the boundary conditions,

where h_f≠0, h_g≠0, and h_w≠0 are the auxiliary parameters, and H_f(ξ), H_g(ξ), and H_w(ξ) are the auxiliary functions. By Taylor’s theorem, expanding F(ξ, p), G(ξ, p), and W(ξ, p) into power series of p=0 as follows,

where if h_f, h_g, and h_w are exactly provided, series (33)–(35) are convergent at p=1, and then we obtain the series solutions as follows:

Furthermore, we derive the high-order deformation equations in the following forms:

where f_k(ξ), g_k(ξ), and w_k(ξ) satisfy the following boundary conditions:

and

Under consideration of the rules of solution expression and solution existence described in [31–35], we choose the auxiliary functions as

5 Discussion of the Convergence and the Accuracy of Approximation Solution

The nonlinear ordinary differential equations (14)–(16) with the boundary conditions (17) and (18) are solved by the HAM [31–35] with the software Mathematica. In order to facilitate computation, we set h_f=h_g=h_w=h and figure out the tenth-order approximation solution. However, according to the introduction of Liao [31–35], the convergence of approximation solution depends on the appropriate value of h, and the convergence interval for h corresponds to the line segment nearly parallel to the horizontal line. Then taking the case of the velocity slip parameter l₁=0.1 (when n=2, β=0.1, l₂=0.1, Pr=7, M=0.5, S=0.5, Nb=0.1, Nt=0.1, and Le=3) as an example, it is seen that the convergence interval for h_f is –1.5≤h≤–0.1; the convergence interval for h_g is –1.5≤h≤0.1; the convergence interval for h_w is –1.3≤h≤–0.4 from Figure 2a. Furthermore, we determine the appropriate value of h via the averaged residual error curves [34], which is shown in Figure 2b. The value of h almost simultaneously minimising the averaged residual errors of f(ξ), g(ξ), and w(ξ) is considered to be appropriate. Hence, h=–1.2 is defined as the appropriate value of h; meanwhile, the accuracy tends to 0.0005. By this way, the appropriate value of h is obtained at the tenth-order approximation and listed in others cases, keeping that the accuracy is <0.005, as the following: (i) h=–1.2 when l₁=0.1, 0.3, 0.5, 0.7, (ii) h=–1.2 when l₂=0.1, 0.3, 0.5, 0.7, (iii) h=–1.3 when Pr=7, 9, 11, 13, (vi) h=–1 when S=0.5, 1, 1.5, 2, (v) h=–1 when M=0.5, 1, 1.5, 2, (vi) h=–1.2 when Le=3, 5, 7, 9, (vii) h=–1.2 when Nt=0.05, 0.1, 0.15, 0.2, 0.3, 0.4, and (viii) h=–1.2 when Nb=0.05, 0.1, 0.15, 0.2, 03, 0.4.

Figure 2:

(a) The h curves of f′(0), g′(0), and w′(0) obtained by tenth-order approximation solution of the HAM when n=2, β=0.1, l₁=0.1, l₂=0.1, Pr=7, M=0.5, S=0.5, Nb=0.1, Nt=0.1, and Le=3. (b) The averaged residual error curves of f(ξ), g(ξ), and w(ξ) obtained by tenth-order approximation solution of the HAM when n=2, β=0.1, l₁=0.1, l₂=0.1, Pr=7, M=0.5, S=0.5, Nb=0.1, Nt=0.1, and Le=3.

6 Results and Discussion

6.1 Effects of the Velocity Slip Parameter

Figures 3–6 present the effects of the velocity slip on the velocity u, the velocity v, the temperature, and the concentration profiles. Figure 3 indicates that increasing of the velocity slip decreases the velocity u at the surface and thickness of the velocity u boundary layer. Figure 4 indicates that increasing of the velocity slip decreases the velocity v. Figure 5 indicates that increasing of the velocity slip increases the temperature and thickness of the temperature boundary layer. Figure 6 indicates that increasing of the velocity slip increases the concentration and thickness of the concentration boundary layer.

Figure 3:

Effects of l₁ on the velocity u profile.

Figure 4:

Effects of l₁ on the velocity v profile.

Figure 5:

Effects of l₁ on the temperature profile.

Figure 6:

Effects of l₁ on the concentration profile.

Moreover, the effects of the velocity slip on the skin friction, Nusselt number, and Sherwood number are revealed in Table 2 when n=2, indicating that thickness of the sheet becomes smaller, and in Table 3 when n=0.5, indicating that thickness of the sheet gets bigger. In both cases, we uniformly observe Cfr increases, while that Nur and Shr decrease with increase in the velocity slip l₁. Consequently, the surface shear stress, rates of heat transfer, and mass transfer decrease with an increase in the velocity slip.

Table 2:

The effects of the velocity slip and the temperature jump on velocity gradient, temperature gradient, and concentration gradient at the sheet surface and the corresponding values of skin friction, Nusselt number, and Sherwood number when n=2, β=0.1, Pr=7, M=0.5, S=0.5, Nt=0.1, Nb=0.1, Le=3, and h=–1.2.

l1	l2	f″(0)	g′(0)	w′(0)	Cfr	Nur	Shr
0.5	0.5	–0.740728	–0.60748	–0.402868	–1.8144	0.744008	0.49341
	1.0	–	–0.471874	–0.485941	–	0.577925	0.595154
	1.5	–	–0.384169	–0.540451	–	0.47051	0.661915
1.0	0.5	–0.525146	–0.55235	–0.33276	–1.28634	0.676488	0.407546
	1.0	–	–0.439006	–0.400671	–	0.537671	0.49072
	1.5	–	–0.362564	–0.447183	–	0.444049	0.547685
1.5	0.5	–0.409244	–0.517695	–0.29146	–1.00244	0.634044	0.356964
	1.0	–	–0.417868	–0.349801	–	0.511782	0.428416
	1.5	–	–0.34847	–0.391065	–	0.426787	0.478955

Table 3:

The effects of the velocity slip and the generalised temperature jump on velocity gradient, temperature gradient, and concentration gradient at the sheet surface and the corresponding values of skin friction coefficient, Nusselt number, and Sherwood number when n=0.5, β=0.1, Pr=7, M=0.5, S=0.5, Nt=0.1, Nb=0.1, Le=3, and h=–1.

l1	l2	f″(0)	g′(0)	w′(0)	Cfr	Nur	Shr
0.5	0.5	–0.806126	–0.636949	–0.371836	–1.36925	0.551614	0.32202
	1.0	–	–0.488513	–0.470896	–	0.423064	0.407808
	1.5	–	–0.394737	–0.534137	–	0.341852	0.462576
1.0	0.5	–0.565046	–0.57096	–0.289176	–0.978689	0.494466	0.250434
	1.0	–	–0.449889	–0.367468	–	0.389615	0.318236
	1.5	–	–0.369606	–0.419846	–	0.320088	0.363597
1.5	0.5	–0.437215	–0.531677	–0.245036	–0.757278	0.460446	0.212207
	1.0	–	–0.426294	–0.311085	–	0.369181	0.269408
	1.5	–	–0.354008	–0.356807	–	0.30658	0.309004

6.2 Effects of the Temperature Jump Parameter

Figures 7 and 8 show the effects of the temperature jump on the temperature and the concentration profiles. Figure 7 indicates that increasing of the temperature jump decreases the temperature at the surface and thickness of the temperature boundary layer. Figure 8 indicates that increasing of the temperature jump decreases the concentration and thickness of the concentration boundary layer.

Figure 7:

Effects of l₂ on the temperature profile.

Figure 8:

Effects of l₂ on the concentration profile.

Moreover, the effects of the generalised temperature jump on Nusselt number and Sherwood number are revealed in Table 2 when n=2, indicating that thickness of the sheet becomes smaller, and in Table 3 when n=0.5, indicating that thickness of the sheet gets bigger. In both cases, we uniformly observe Nur decreases, while that Shr increases with increase in the temperature jump l₂. Consequently, the rate of heat transfer decreases, whereas the rate of mass transfer increases with increase in the temperature jump.

6.3 Effects of Prandtl Number

Figures 9 and 10 reveal the influence of Prandtl number on the temperature and the concentration profiles. Figure 9 indicates that increase in Prandtl number makes a negative effect on thickness of the temperature boundary layer. Figure 10 indicates that increase in Prandtl number results in weak enhancing in the beginning, while slight reduction of the concentration in the concentration boundary layer when it develops to some extent, and finally leads to decrease in thickness of the concentration boundary layer. It should be pointed out that Prandtl number has no effect on the velocity profiles.

Figure 9:

Effects of Pr on the temperature profile.

Figure 10:

Effects of Pr on the concentration profile.

Moreover, Table 4 presents the effects of Prandtl number on Nusselt number and Sherwood number. We observe that Nur increases, whereas Shr decreases with increase in Prandtl number. Consequently, the rate of heat transfer increases, whereas the rate of mass transfer decreases with increase in Prandtl number.

Table 4:

The effects of Prandtl number on temperature gradient and concentration gradient at the sheet surface and the corresponding values of skin friction coefficient, Nusselt number, and Sherwood number when n=2, β=0.1, l₁=0.1, l₂=0.1, M=0.5, S=0.5, Nt=0.1, Nb=0.1, Le=3, and h=–1.2.

Pr	g′(0)	w′(0)	Nur	Shr
7.0	–0.919446	–0.379318	1.12609	0.464568
7.5	–0.939259	–0.363348	1.15035	0.445009
8.0	–0.957572	–0.348556	1.17278	0.426892
8.5	–0.97454	–0.334828	1.19356	0.410078
9.0	–0.9903	–0.32206	1.21286	0.394442

6.4 Effects of the Permeable Parameter

Figures 11–14 show the effects of the permeable parameter on the velocities, the temperature, and the concentration profiles. Figure 11 indicates that increase in the permeable parameter decreases the velocity u and thickness of the velocity u boundary layer. Figure 12 indicates that increase in the permeable parameter decreases the velocity v. Figure 13 indicates that increase in the permeable parameter increases the temperature and thickness of the temperature boundary layer. Figure 14 indicates that increase in the permeable parameter increases the concentration and thickness of the concentration boundary layer.

Figure 11:

Effects of S on the velocity u profile.

Figure 12:

Effects of S on the velocity v profile.

Figure 13:

Effects of S on the temperature profile.

Figure 14:

Effects of S on the concentration profile.

6.5 Effects of the Magnetic Field Parameter

Figures 15–18 show the effects of the magnetic field parameter on the velocities, the temperature, and the concentration profiles. We observe that the results are the same as the effects of the permeable parameter.

Figure 15:

Effects of M on the velocity u profile.

Figure 16:

Effects of M on the velocity v profile.

Figure 17:

Effects of M on the temperature profile.

Figure 18:

Effects of M on the concentration profile.

6.6 Effects of Lewis Number

Figures 19 and 20 show the effects of Lewis number on the temperature and the concentration profiles. We find that the effect of Lewis number on the concentration boundary layer is stronger than that on the temperature boundary layer. Figure 19 indicates that the temperature and thickness of the temperature boundary layer increase with increase in Lewis number. But Figure 20 indicates that the concentration and thickness of the concentration boundary layer decrease with increase in Lewis number.

Figure 19:

Effects of Le on the temperature profile.

Figure 20:

Effects of Le on the concentration profile.

Moreover, Table 5 presents the effects of Lewis number on Nusselt number and Sherwood number. We observe that Nur decreases and Shr increases with increase in Lewis number. Consequently, the rate of heat transfer decreases, whereas the rate of mass transfer increases with increase in Lewis number.

Table 5:

The effects of Lewis number on temperature gradient and concentration gradient at the sheet surface and the corresponding values of skin friction coefficient, Nusselt number, and Sherwood number when n=2, β=0.1, l₁=0.1, l₂=0.1, Pr=7, M=0.5, S=0.5, Nt=0.1, Nb=0.1, and h=–1.2.

Le	g′(0)	w′(0)	Nur	Shr
3.0	–0.919446	–0.379318	1.12609	0.464568
3.5	–0.902092	–0.514923	1.10483	0.63065
4.0	–0.886304	–0.640727	1.0855	0.784727
4.5	–0.872089	–0.756483	1.06809	0.926498
5.0	–0.859394	–0.862758	1.05254	1.05666

6.7 Effect of the Thermophoresis Parameter

The thermophoresis is a physical phenomenon that the nanoparticles migrate from warmer to colder field because of the temperature difference. Figures 21 and 22 show the effects of the thermophoresis parameter Nt on the temperature and the concentration profiles. It is observed that the increase in Nt promotes the development of the temperature and the concentration boundary layer.

Figure 21:

Effects of Nt on the temperature profile.

Figure 22:

Effects of Nt on the concentration profile.

On the other hand, the effects of the thermophoresis parameter Nt on Nusselt number and Sherwood number are presented in Table 6. It is found that the increase in Nt leads to the decline in Nur, meaning the decline of heat transfer from the surface to the boundary layer. Although the increase in Nt results in the decrease in Shr, it indicates that the mass transfer from the surface to the boundary layer declines when Nt is smaller, and the mass transfer is strengthened when Nt is larger.

Table 6:

The effects of thermophoresis on temperature gradient and concentration gradient at the sheet surface and the corresponding values of skin friction coefficient, Nusselt number, and Sherwood number when n=2, β=0.1, l₁=0.1, l₂=0.1, Pr=7, M=0.5, S=0.5, Nb=0.1, Le=3, and h=–1.2.

Nt	g′(0)	w′(0)	Nur	Shr
0.05	–0.974543	–0.643196	1.19357	0.787752
0.1	–0.919446	–0.379318	1.12609	0.464568
0.15	–0.866778	–0.158867	1.06158	0.194571
0.2	–0.816497	0.0214188	1.0	–0.0262325
0.3	–0.722914	0.274005	0.885386	–0.335587
0.4	–0.63829	0.402321	0.781743	–0.492741

6.8 Effect of the Brownian Motion Parameter

The Brownian diffusion refers to the random drifting of suspended nanoparticle in the base fluid, which is caused from continuous collision between nanoparticles and liquid molecules. Figures 23 and 24 present the effects of the Brownian motion parameter Nb on the temperature and the concentration profiles. It is observed that the increase in Nb promotes the development of the temperature boundary layer, while restrains the concentration boundary.

Figure 23:

Effects of Nb on the temperature profile.

Figure 24:

Effects of Nb on the concentration profile.

On the other hand, as Table 7 shows, the increase in Nb leads to the decline in Nur and the increase in Shr. Consequently, the increase in Nb controls the heat transfer and promotes the mass transfer from the surface to the boundary layer. But the mass transfers in the opposite direction if Nb is too small.

Table 7:

The effects of Brownian motion on temperature gradient and concentration gradient at the sheet surface and the corresponding values of skin friction coefficient, Nusselt number, and Sherwood number when n=2, β=0.1, l₁=0.1, l₂=0.1, Pr=7, M=0.5, S=0.5, Nt=0.1, Le=3, and h=–1.2.

Nb	g′(0)	w′(0)	Nur	Shr
0.05	–1.01844	0.366701	1.24783	–0.449115
0.1	–0.919446	–0.379318	1.12609	0.464568
0.15	–0.82482	–0.625336	1.01019	0.765878
0.2	–0.734605	–0.746344	0.899704	0.914081
0.3	–0.567556	–0.863335	0.695111	1.05736
0.4	–0.418478	–0.917811	0.512529	1.12409

7 Conclusions

In this article, we study the flow, heat transfer, and mass transfer of nanofluid over a stretching sheet with variable thickness, considering the generalised velocity slip and the generalised temperature jump at the surface of sheet. The governing equations are transformed into ordinary differential equations by proper transformations and then solved by applying the HAM. The effects of involved parameters such as the generalised velocity slip and the temperature jump, Prandtl number, magnetic field parameter, permeable parameter, Lewis number, thermophoresis parameter, and Brownian motion parameter are investigated and analysed graphically.