A theory about the essential spectrum of operator pencils is presented with application to the Orr-Sommerfeld eigenvalue problem for instability of incompressible viscous flow with the Blasius flow profile. Let \(A\), \(B\) be linear operators in Banach spaces \(X,Y,A:D(A)\to Y\) \((D(A)\subset X)\) is a closed operator, and \(B:X\to Y\) bounded. Consider the eigenvalue problem \[ Au=\lambda Bu,\quad \lambda\in\mathbb{C},\quad u\in D(A),\quad u\neq 0.\tag{i} \] The resolvent set for (i) is \[ \rho(A,B)= \{\lambda\in\mathbb{C}\mid A-\lambda B:D(A)\to Y\text{ is bijective}\} \] and the spectral set is \(\sigma(A,B)= \mathbb{C}\setminus \rho(A, B)\). The author identifies five subsets \(\sigma_{ek}(A, B)\), \(k= 1,2,\dots, 5\) of \(\sigma(A, B)\) as essential spectral sets and show that \(\sigma_{ek}(A, B)\subset\sigma_{ek+ 1}(A, B)\), \(k= 1,\dots, 4\). Eigenvalues of finite multiplicity are contained in \(\sigma(A,B)\setminus \sigma_{e5}(A, B)\). The Orr-Sommerfeld equation with Blasius flow profile is formulated abstractly in the form (i) with \(A= A_0+ K:D(A_0)\subset D(K)\to Y\), \(B: X\to Y\), \(B\) is bounded bijective, \(K\) is \(A_0\)-compact, \(A\), \(B\) are respectively fourth-order and second-order ordinary differential operators and \(A_0\) is a differential operator with constant coefficients. The author determines \(\sigma_{e4}(A, B)= M\) explicitly as a ray in \(\mathbb{C}\) and proves that \(\sigma_{ek}(A, B)= M\), \(k= 1,2,\dots, 5\).