Two-Dimensional Mathematical Models

Section 10.4 Two-Dimensional Mathematical Models

We now consider problems involving steady state heat flow, electrostatics, and ideal fluid flow that can be solved with conformal mapping techniques. Conformal mapping transforms a region in which the problem is posed to one in which the solution is easy to obtain. As our solutions involve only two independent variables, \(x\) and \(y\text{,}\) we first mention a basic assumption needed for the validity of the model.

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The physical problems we just mentioned are real-world applications and involve solutions in three-dimensional Cartesian space. Such problems generally would involve the Laplacian in three variables and the divergence and curl of three-dimensional vector functions. Since complex analysis involves only \(x\) and \(y\text{,}\) we consider the special case in which the solution does not vary with the coordinate along the axis perpendicular to the \(xy\) plane. For steady state heat flow and electrostatics, this assumption means that the temperature, \(T\text{,}\) or the potential, \(V\text{,}\) varies only with \(x\) and \(y\text{.}\) Thus for the flow of ideal fluids, the fluid motion is the same in any plane that is parallel to the \(z\) plane. Curves drawn in the \(z\) plane are to be interpreted as cross sections that correspond to infinite cylinders perpendicular to the \(z\) plane. An infinite cylinder is the limiting case of a “long” physical cylinder, so the mathematical model that we present is valid provided the three-dimensional problem involves a physical cylinder long enough that the effects at the ends can be reasonably neglected.

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In Sections 10.1 and Section 10.2, we showed how to obtain solutions \(\phi(x,y)\) for harmonic functions. For applications, we need to consider the family of level curves

\begin{equation} \{\phi(x,y) =K_1:K_1 \text{ is a real constant } \}\tag{10.4.1} \end{equation}

and the conjugate harmonic function \(\psi(x,y)\) and its family of level curves

\begin{equation} \{\psi(x,y) =K_2:K_2 \text{ is a real constant } \}\text{.}\tag{10.4.2} \end{equation}

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For convenience, we introduce the term complex potential for the analytic function

\begin{equation*} F(z) = \phi(x,y) + i\psi(x,y) \end{equation*}

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We use Theorem 10.4.1, regarding the orthogonality of the families of level curves (Equations (10.4.1) and (10.4.2)), to develop ideas concerning the physical applications that we will consider.

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Theorem 10.4.1. Orthogonal families of level curves.

Suppose \(\phi(x,y)\) is harmonic in a domain \(D\text{,}\) \(\psi(x,y)\) is its harmonic conjugate, and \(F(z)=\phi(x,y)+i\psi(x,y)\) is the complex potential. Then the two families of level curves given in Equations (10.4.1) and (10.4.2), respectively, are orthogonal in the sense that if \((a,b)\) is a point common to the two curves \(\phi(x,y)=K_1\) and \(\psi(x,y)=K_2\text{,}\) and if \(F,'(a+ib) \ne 0\text{,}\) then these two curves intersect orthogonally.

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Proof.

Since \(\phi(x,y)=K_1\) is an implicit equation of a plane curve, the gradient vector grad \(\phi\text{,}\) evaluated at \((a,b)\text{,}\) is perpendicular to the curve at \((a,b)\text{.}\) This vector is given by

\begin{equation*} \mathbf{N}_1=\phi_x(a,b) +i\phi_y(a,b) \end{equation*}

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Similarly, the vector N\(_2\) defined by

\begin{equation*} \mathbf{N}_2=\psi_x(a,b) +i\psi_y(a,b) \end{equation*}

is orthogonal to the curve \(\psi(x,y) =K_2\) at \((a,b)\text{.}\) Using the Cauchy-Riemann equations, \(\phi_x=\psi_y\) and \(\phi_y=-\psi_x\text{,}\) we have

\begin{align} \mathbf{N}_1 \cdot \mathbf{N}_2 \amp = \phi_x(a,b)\psi_x(a,b)+\phi_y(a,b)\psi_y(a,b)\tag{10.4.3}\\ \amp = -\phi_x (a,b) \phi_y(a,b) + \phi_y(a,b) \phi_x(a,b)\notag\\ \amp = 0\text{.}\notag \end{align}

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In addition, \(F,'(a+ib) \ne 0\text{,}\) so we have

\begin{equation*} \phi_x(a,b) +i\psi_x(a,b) \ne 0\text{.} \end{equation*}

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The Cauchy-Riemann equations and the facts \(\phi_x(a,b) \ne 0\) and \(\psi_x(a,b) \ne 0\) imply that both \(\mathbf{N}_1\) and \(\mathbf{N}_2\) are nonzero. Therefore Equation (10.4.3) implies that \(\mathbf{N}_1\) is perpendicular to \(\mathbf{N}_2\text{,}\) and hence the curves are orthogonal.

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The complex potential \(F(z)=\phi(x,y)+i\psi(x,y)\) has many physical interpretations. Suppose, for example, that we have solved a problem in steady state temperatures. Then we can obtain the solution to a similar problem with the same boundary conditions in electrostatics by interpreting the isothermals as equipotential curves and the heat flow lines as flux lines. This implies that heat flow and electrostatics correspond directly.

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Or suppose that we have solved a fluid flow problem. Then we can obtain a solution to an analogous problem in heat flow by interpreting the equipotentials as isothermals and streamlines as heat flow lines. Various interpretations of the families of level curves given in Equations (10.4.1) and (10.4.2) and correspondences between families are summarized in Table 10.4.2

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Table 10.4.2. Interpretations for Level Curves

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Physical Phenomenon	\(\phi(x,y)=\) constant	\(\psi(x,y) =\) constant

Heat flow	Isothermals	Heat flow lines
Electrostatics	Equipotential curves	Flux lines
Fluid flow	Equipotentials	Streamlines
Gravitational field	Gravitational potential	Lines of force
Magnetism	Potential	Lines of force
Diffusion	Concentration	Lines of flow
Elasticity	Strain function	Stress lines
Current flow	Potential	Lines of flow

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